domain listlengths 0 3 | difficulty float64 1 9.5 | problem stringlengths 18 9.01k | solution stringlengths 2 11.1k | answer stringlengths 0 3.77k | source stringclasses 67
values |
|---|---|---|---|---|---|
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 6 | Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations
\[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50, \\ y^3 \equal{} 12y\plus{}3z\minus{}2, \\ z^3 \equal{} 27z \plus{} 27x. \end{cases}\]
[i]Razvan Gelca.[/i] |
We are given the system of equations:
\[
\begin{cases}
x^3 = 3x - 12y + 50, \\
y^3 = 12y + 3z - 2, \\
z^3 = 27z + 27x.
\end{cases}
\]
To find all triples \((x, y, z)\) of real numbers that satisfy these equations, we analyze the behavior of the functions involved.
1. **Case \(x > 2\):**
- For \(x > 2\), the funct... | (2, 4, 6) | usa_team_selection_test |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 6.5 | There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn. |
Given 7 arbitrary points in the plane, we need to determine the maximum number of circles that can be drawn through every 4 possible concyclic points.
To solve this, we consider the combinatorial aspect of selecting 4 points out of 7. The number of ways to choose 4 points from 7 is given by the binomial coefficient:
... | 7 | china_team_selection_test |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Inequalities"
] | 9 | Number $a$ is such that $\forall a_1, a_2, a_3, a_4 \in \mathbb{R}$, there are integers $k_1, k_2, k_3, k_4$ such that $\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a$. Find the minimum of $a$. |
Let \( a \) be such that for all \( a_1, a_2, a_3, a_4 \in \mathbb{R} \), there exist integers \( k_1, k_2, k_3, k_4 \) such that
\[
\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a.
\]
We aim to find the minimum value of \( a \).
Consider the numbers \( a_i = \frac{i}{4} \) for \( i = 1, 2, 3, 4 \). L... | 1.25 | china_team_selection_test |
[
"Mathematics -> Number Theory -> Congruences"
] | 7 | For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that:
\[ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc \]
then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number. |
For a positive integer \( M \), we need to determine if it is a GOOD or BAD number based on the existence of integers \( a, b, c, \) and \( d \) such that:
\[ M \leq a < b \leq c < d \leq M + 49, \qquad ad = bc. \]
We aim to find the greatest GOOD number and the smallest BAD number.
### Greatest GOOD Number
**Lemma... | 576 | china_team_selection_test |
[
"Mathematics -> Algebra -> Abstract Algebra -> Other",
"Mathematics -> Number Theory -> Congruences"
] | 7 | Let $\lfloor \bullet \rfloor$ denote the floor function. For nonnegative integers $a$ and $b$, their [i]bitwise xor[/i], denoted $a \oplus b$, is the unique nonnegative integer such that $$ \left \lfloor \frac{a}{2^k} \right \rfloor+ \left\lfloor\frac{b}{2^k} \right\rfloor - \left\lfloor \frac{a\oplus b}{2^k}\right\rf... |
Let \(\lfloor \bullet \rfloor\) denote the floor function. For nonnegative integers \(a\) and \(b\), their bitwise xor, denoted \(a \oplus b\), is the unique nonnegative integer such that
\[ \left \lfloor \frac{a}{2^k} \right \rfloor + \left \lfloor \frac{b}{2^k} \right \rfloor - \left \lfloor \frac{a \oplus b}{2^k} ... | \text{All odd positive integers} | usa_team_selection_test_for_imo |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Discrete Mathematics -> Algorithms"
] | 7.5 | Let $n$ be a positive integer. Initially, a $2n \times 2n$ grid has $k$ black cells and the rest white cells. The following two operations are allowed :
(1) If a $2\times 2$ square has exactly three black cells, the fourth is changed to a black cell;
(2) If there are exactly two black cells in a $2 \times 2$ square, t... |
Let \( n \) be a positive integer. Initially, a \( 2n \times 2n \) grid has \( k \) black cells and the rest white cells. The following two operations are allowed:
1. If a \( 2 \times 2 \) square has exactly three black cells, the fourth is changed to a black cell.
2. If there are exactly two black cells in a \( 2 \ti... | n^2 + n + 1 | china_team_selection_test |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Other",
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 8 | Given two integers $m,n$ which are greater than $1$. $r,s$ are two given positive real numbers such that $r<s$. For all $a_{ij}\ge 0$ which are not all zeroes,find the maximal value of the expression
\[f=\frac{(\sum_{j=1}^{n}(\sum_{i=1}^{m}a_{ij}^s)^{\frac{r}{s}})^{\frac{1}{r}}}{(\sum_{i=1}^{m})\sum_{j=1}^{n}a_{ij}^r)^... |
Given two integers \( m, n \) which are greater than 1, and two positive real numbers \( r, s \) such that \( r < s \), we aim to find the maximal value of the expression
\[
f = \frac{\left( \sum_{j=1}^{n} \left( \sum_{i=1}^{m} a_{ij}^s \right)^{\frac{r}{s}} \right)^{\frac{1}{r}}}{\left( \sum_{i=1}^{m} \sum_{j=1}^{n} ... | \min(m, n)^{\frac{1}{r} - \frac{1}{s}} | china_team_selection_test |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 9 | Let $a, b, c, p, q, r$ be positive integers with $p, q, r \ge 2$. Denote
\[Q=\{(x, y, z)\in \mathbb{Z}^3 : 0 \le x \le a, 0 \le y \le b , 0 \le z \le c \}. \]
Initially, some pieces are put on the each point in $Q$, with a total of $M$ pieces. Then, one can perform the following three types of operations repeatedly:
(1... |
Let \(a, b, c, p, q, r\) be positive integers with \(p, q, r \ge 2\). Denote
\[ Q = \{(x, y, z) \in \mathbb{Z}^3 : 0 \le x \le a, 0 \le y \le b, 0 \le z \le c\}. \]
Initially, some pieces are placed on each point in \(Q\), with a total of \(M\) pieces. The following three types of operations can be performed repeatedl... | p^a q^b r^c | china_team_selection_test |
[
"Mathematics -> Number Theory -> Prime Numbers"
] | 7 | Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers. |
We need to find the smallest prime number \( p \) that cannot be represented in the form \( |3^a - 2^b| \), where \( a \) and \( b \) are non-negative integers.
First, we verify that all primes less than 41 can be expressed in the form \( |3^a - 2^b| \):
- For \( p = 2 \): \( 2 = |3^0 - 2^1| \)
- For \( p = 3 \): \(... | 41 | china_team_selection_test |
[
"Mathematics -> Algebra -> Abstract Algebra -> Field Theory"
] | 9 | Find all functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, such that
1) $f(0,x)$ is non-decreasing ;
2) for any $x,y \in \mathbb{R}$, $f(x,y)=f(y,x)$ ;
3) for any $x,y,z \in \mathbb{R}$, $(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0$ ;
4) for any $x,y,a \in \mathbb{R}$, $f(x+a,y+a)=f(x,y)+a$ . |
Let \( f: \mathbb{R}^2 \rightarrow \mathbb{R} \) be a function satisfying the following conditions:
1. \( f(0,x) \) is non-decreasing.
2. For any \( x, y \in \mathbb{R} \), \( f(x,y) = f(y,x) \).
3. For any \( x, y, z \in \mathbb{R} \), \( (f(x,y) - f(y,z))(f(y,z) - f(z,x))(f(z,x) - f(x,y)) = 0 \).
4. For any \( x, y,... | f(x,y) = a + \min(x,y) \quad \text{or} \quad f(x,y) = a + \max(x,y) \quad \text{for any } a \in \mathbb{R}. | china_team_selection_test |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Algebra -> Abstract Algebra -> Group Theory"
] | 8 | Find all functions $f,g$:$R \to R$ such that $f(x+yg(x))=g(x)+xf(y)$ for $x,y \in R$. |
We are given the functional equation \( f(x + y g(x)) = g(x) + x f(y) \) for all \( x, y \in \mathbb{R} \). We aim to find all functions \( f \) and \( g \) from \(\mathbb{R} \to \mathbb{R}\) that satisfy this equation.
First, assume \( g(0) = 0 \).
### Fact 1: \( f(x) = g(x) \) for every \( x \).
**Proof:**
Set \(... | f(x) = g(x) = 0 \text{ for all } x \in \mathbb{R} \text{ or } f(x) = g(x) \text{ with } f(0) = 0 | china_team_selection_test |
[
"Mathematics -> Number Theory -> Congruences"
] | 7 | A positive integer $n$ is known as an [i]interesting[/i] number if $n$ satisfies
\[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \]
for all $k=1,2,\ldots 9$.
Find the number of interesting numbers. |
A positive integer \( n \) is known as an interesting number if \( n \) satisfies
\[
\left\{ \frac{n}{10^k} \right\} > \frac{n}{10^{10}}
\]
for all \( k = 1, 2, \ldots, 9 \), where \( \{ x \} \) denotes the fractional part of \( x \).
To determine the number of interesting numbers, we can use a computational approach... | 999989991 | china_team_selection_test |
[
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions",
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 8 | Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that
$$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$
for all $x,y \in \mathbb{Q}$. |
Let \( f: \mathbb{Q} \to \mathbb{Q} \) be a function such that
\[
f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}
\]
for all \( x, y \in \mathbb{Q} \).
First, we denote the given functional equation as \( P(x, y) \):
\[
P(x, y): f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}.
\]
By considering \( P(... | f(x) = x^2 + \frac{1}{2} | china_team_selection_test |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Number Theory -> Congruences"
] | 9 | Does there exist $ 2002$ distinct positive integers $ k_1, k_2, \cdots k_{2002}$ such that for any positive integer $ n \geq 2001$, one of $ k_12^n \plus{} 1, k_22^n \plus{} 1, \cdots, k_{2002}2^n \plus{} 1$ is prime? |
We need to determine whether there exist \( 2002 \) distinct positive integers \( k_1, k_2, \ldots, k_{2002} \) such that for any positive integer \( n \geq 2001 \), at least one of \( k_1 2^n + 1, k_2 2^n + 1, \ldots, k_{2002} 2^n + 1 \) is prime.
To address this, we generalize the problem for \( F > 2002 \). Consid... | \text{No} | china_team_selection_test |
[
"Mathematics -> Number Theory -> Factorization"
] | 9 | Let $a,b$ be two integers such that their gcd has at least two prime factors. Let $S = \{ x \mid x \in \mathbb{N}, x \equiv a \pmod b \} $ and call $ y \in S$ irreducible if it cannot be expressed as product of two or more elements of $S$ (not necessarily distinct). Show there exists $t$ such that any element of $S$ c... |
Let \( a \) and \( b \) be two integers such that their greatest common divisor (gcd) has at least two prime factors. Define the set \( S = \{ x \mid x \in \mathbb{N}, x \equiv a \pmod{b} \} \) and consider an element \( y \in S \) to be irreducible if it cannot be expressed as the product of two or more elements of \... | t = \max \{ 2q, q - 1 + 2M \} | china_team_selection_test |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 4.5 | Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$. |
Given a square \(ABCD\) with side length \(1\), points \(P\) and \(Q\) are on sides \(AB\) and \(AD\) respectively. We are to find the angle \( \angle PCQ \) given that the perimeter of \( \triangle APQ \) is \(2\).
Let \( AP = x \) and \( AQ = y \). Then, \( PB = 1 - x \) and \( QD = 1 - y \). We need to find \( \ta... | 45^\circ | china_team_selection_test |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 7.5 | A(x,y), B(x,y), and C(x,y) are three homogeneous real-coefficient polynomials of x and y with degree 2, 3, and 4 respectively. we know that there is a real-coefficient polinimial R(x,y) such that $B(x,y)^2-4A(x,y)C(x,y)=-R(x,y)^2$. Proof that there exist 2 polynomials F(x,y,z) and G(x,y,z) such that $F(x,y,z)^2+G(x,y,z... |
Given that \( A(x,y), B(x,y), \) and \( C(x,y) \) are homogeneous real-coefficient polynomials of \( x \) and \( y \) with degrees 2, 3, and 4 respectively, and knowing that there exists a real-coefficient polynomial \( R(x,y) \) such that \( B(x,y)^2 - 4A(x,y)C(x,y) = -R(x,y)^2 \), we need to prove that there exist t... | F(x,y,z)^2 + G(x,y,z)^2 = A(x,y)z^2 + B(x,y)z + C(x,y) | china_team_selection_test |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 6 | Let $ P$ be a convex $ n$ polygon each of which sides and diagnoals is colored with one of $ n$ distinct colors. For which $ n$ does: there exists a coloring method such that for any three of $ n$ colors, we can always find one triangle whose vertices is of $ P$' and whose sides is colored by the three colors respectiv... |
Let \( P \) be a convex \( n \)-polygon where each side and diagonal is colored with one of \( n \) distinct colors. We need to determine for which \( n \) there exists a coloring method such that for any three of the \( n \) colors, we can always find one triangle whose vertices are vertices of \( P \) and whose side... | n \text{ must be odd} | china_national_olympiad |
[
"Mathematics -> Number Theory -> Congruences"
] | 8 | Does there exist a finite set $A$ of positive integers of at least two elements and an infinite set $B$ of positive integers, such that any two distinct elements in $A+B$ are coprime, and for any coprime positive integers $m,n$, there exists an element $x$ in $A+B$ satisfying $x\equiv n \pmod m$ ?
Here $A+B=\{a+b|a\in... |
To determine whether there exists a finite set \( A \) of positive integers of at least two elements and an infinite set \( B \) of positive integers such that any two distinct elements in \( A+B \) are coprime, and for any coprime positive integers \( m, n \), there exists an element \( x \) in \( A+B \) satisfying ... | \text{No} | china_team_selection_test |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 9 | Find the smallest positive number $\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$ ,if $z_1+z_2+z_3=0$, then $$\left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda .$$ |
We aim to find the smallest positive number \(\lambda\) such that for any complex numbers \(z_1, z_2, z_3 \in \{z \in \mathbb{C} \mid |z| < 1\}\) with \(z_1 + z_2 + z_3 = 0\), the following inequality holds:
\[
\left|z_1z_2 + z_2z_3 + z_3z_1\right|^2 + \left|z_1z_2z_3\right|^2 < \lambda.
\]
First, we show that \(\lam... | 1 | china_team_selection_test |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions"
] | 9 | Define the sequences $(a_n),(b_n)$ by
\begin{align*}
& a_n, b_n > 0, \forall n\in\mathbb{N_+} \\
& a_{n+1} = a_n - \frac{1}{1+\sum_{i=1}^n\frac{1}{a_i}} \\
& b_{n+1} = b_n + \frac{1}{1+\sum_{i=1}^n\frac{1}{b_i}}
\end{align*}
1) If $a_{100}b_{100} = a_{101}b_{101}$, find the value of $a_1-b_1$;
2) If $a_{100} = b_{99}... |
Define the sequences \( (a_n) \) and \( (b_n) \) by
\[
\begin{align*}
& a_n, b_n > 0, \forall n \in \mathbb{N_+}, \\
& a_{n+1} = a_n - \frac{1}{1 + \sum_{i=1}^n \frac{1}{a_i}}, \\
& b_{n+1} = b_n + \frac{1}{1 + \sum_{i=1}^n \frac{1}{b_i}}.
\end{align*}
\]
1. If \( a_{100} b_{100} = a_{101} b_{101} \), find the value... | 199 | china_national_olympiad |
[
"Mathematics -> Algebra -> Linear Algebra -> Linear Transformations",
"Mathematics -> Algebra -> Abstract Algebra -> Other"
] | 9 | Let $\alpha$ be given positive real number, find all the functions $f: N^{+} \rightarrow R$ such that $f(k + m) = f(k) + f(m)$ holds for any positive integers $k$, $m$ satisfying $\alpha m \leq k \leq (\alpha + 1)m$. |
Let \(\alpha\) be a given positive real number. We aim to find all functions \( f: \mathbb{N}^{+} \rightarrow \mathbb{R} \) such that \( f(k + m) = f(k) + f(m) \) holds for any positive integers \( k \) and \( m \) satisfying \( \alpha m \leq k \leq (\alpha + 1)m \).
To solve this, we first note that the given functi... | f(n) = cn | china_team_selection_test |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 6.5 | There is a frog in every vertex of a regular 2n-gon with circumcircle($n \geq 2$). At certain time, all frogs jump to the neighborhood vertices simultaneously (There can be more than one frog in one vertex). We call it as $\textsl{a way of jump}$. It turns out that there is $\textsl{a way of jump}$ with respect to 2n-g... |
Let \( n \) be a positive integer such that \( n \geq 2 \). We aim to find all possible values of \( n \) for which there exists a way of jump in a regular \( 2n \)-gon such that the line connecting any two distinct vertices having frogs on it after the jump does not pass through the circumcenter of the \( 2n \)-gon.
... | 2^k \cdot m \text{ where } k = 1 \text{ and } m \text{ is an odd integer} | china_team_selection_test |
[
"Mathematics -> Number Theory -> Congruences",
"Mathematics -> Algebra -> Abstract Algebra -> Group Theory"
] | 8 | Determine whether or not there exists a positive integer $k$ such that $p = 6k+1$ is a prime and
\[\binom{3k}{k} \equiv 1 \pmod{p}.\] |
To determine whether there exists a positive integer \( k \) such that \( p = 6k + 1 \) is a prime and
\[
\binom{3k}{k} \equiv 1 \pmod{p},
\]
we proceed as follows:
Let \( g \) be a primitive root modulo \( p \). By definition, \( g^{6k} \equiv 1 \pmod{p} \). For any integer \( a \) such that \( p \nmid a \), by Fer... | \text{No, there does not exist such a prime } p. | usa_team_selection_test |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Number Theory -> Congruences"
] | 8 | Let $a=2001$. Consider the set $A$ of all pairs of integers $(m,n)$ with $n\neq0$ such that
(i) $m<2a$;
(ii) $2n|(2am-m^2+n^2)$;
(iii) $n^2-m^2+2mn\leq2a(n-m)$.
For $(m, n)\in A$, let \[f(m,n)=\frac{2am-m^2-mn}{n}.\]
Determine the maximum and minimum values of $f$. |
Let \( a = 2001 \). Consider the set \( A \) of all pairs of integers \((m, n)\) with \( n \neq 0 \) such that:
1. \( m < 2a \),
2. \( 2n \mid (2am - m^2 + n^2) \),
3. \( n^2 - m^2 + 2mn \leq 2a(n - m) \).
For \((m, n) \in A\), let
\[ f(m, n) = \frac{2am - m^2 - mn}{n}. \]
We need to determine the maximum and minimum... | 2 \text{ and } 3750 | china_national_olympiad |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 8 | Given a positive integer $n$, find all $n$-tuples of real number $(x_1,x_2,\ldots,x_n)$ such that
\[ f(x_1,x_2,\cdots,x_n)=\sum_{k_1=0}^{2} \sum_{k_2=0}^{2} \cdots \sum_{k_n=0}^{2} \big| k_1x_1+k_2x_2+\cdots+k_nx_n-1 \big| \]
attains its minimum. |
Given a positive integer \( n \), we aim to find all \( n \)-tuples of real numbers \( (x_1, x_2, \ldots, x_n) \) such that
\[
f(x_1, x_2, \cdots, x_n) = \sum_{k_1=0}^{2} \sum_{k_2=0}^{2} \cdots \sum_{k_n=0}^{2} \left| k_1 x_1 + k_2 x_2 + \cdots + k_n x_n - 1 \right|
\]
attains its minimum.
To solve this, we first cl... | \left( \frac{1}{n+1}, \frac{1}{n+1}, \ldots, \frac{1}{n+1} \right) | china_team_selection_test |
[
"Mathematics -> Number Theory -> Factorization",
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 7 | Determine all integers $s \ge 4$ for which there exist positive integers $a$, $b$, $c$, $d$ such that $s = a+b+c+d$ and $s$ divides $abc+abd+acd+bcd$. |
We need to determine all integers \( s \geq 4 \) for which there exist positive integers \( a, b, c, d \) such that \( s = a + b + c + d \) and \( s \) divides \( abc + abd + acd + bcd \).
### Claim 1: If \( a + b + c + d = s \) divides \( abc + abd + acd + bcd \), then \( s \) must be composite.
**Proof:**
Assume, ... | \text{All composite integers } s \geq 4 | usa_team_selection_test_for_imo |
[
"Mathematics -> Number Theory -> Congruences",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 8.5 | For a given integer $n\ge 2$, let $a_0,a_1,\ldots ,a_n$ be integers satisfying $0=a_0<a_1<\ldots <a_n=2n-1$. Find the smallest possible number of elements in the set $\{ a_i+a_j \mid 0\le i \le j \le n \}$. |
For a given integer \( n \ge 2 \), let \( a_0, a_1, \ldots, a_n \) be integers satisfying \( 0 = a_0 < a_1 < \ldots < a_n = 2n-1 \). We aim to find the smallest possible number of elements in the set \( \{ a_i + a_j \mid 0 \le i \le j \le n \} \).
First, we prove that the set \( \{ a_i + a_j \mid 1 \le i \le j \le n-... | 3n | china_team_selection_test |
[
"Mathematics -> Geometry -> Plane Geometry -> Area",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | Consider a $2n \times 2n$ board. From the $i$ th line we remove the central $2(i-1)$ unit squares. What is the maximal number of rectangles $2 \times 1$ and $1 \times 2$ that can be placed on the obtained figure without overlapping or getting outside the board? | Problem assumes that we remove $2(i-1)$ squares if $i\leq n$ , and $2(2n-i)$ squares if $i>n$ .
Divide the entire board into 4 quadrants each containing $n^2$ unit squares.
First we note that the $2$ squares on the center on each of the $4$ bordering lines of the board can always be completely covered by a single tile,... | \[
\begin{cases}
n^2 + 4 & \text{if } n \text{ is even} \\
n^2 + 3 & \text{if } n \text{ is odd}
\end{cases}
\] | jbmo |
[
"Mathematics -> Algebra -> Abstract Algebra -> Other",
"Mathematics -> Discrete Mathematics -> Logic"
] | 8 | Find all functions $f: \mathbb R \to \mathbb R$ such that for any $x,y \in \mathbb R$, the multiset $\{(f(xf(y)+1),f(yf(x)-1)\}$ is identical to the multiset $\{xf(f(y))+1,yf(f(x))-1\}$.
[i]Note:[/i] The multiset $\{a,b\}$ is identical to the multiset $\{c,d\}$ if and only if $a=c,b=d$ or $a=d,b=c$. |
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that for any \( x, y \in \mathbb{R} \), the multiset \( \{ f(xf(y) + 1), f(yf(x) - 1) \} \) is identical to the multiset \( \{ xf(f(y)) + 1, yf(f(x)) - 1 \} \).
We aim to find all such functions \( f \).
Let \( P(x, y) \) denote the assertion that \( \{ f(xf(... | f(x) \equiv x \text{ or } f(x) \equiv -x | china_team_selection_test |
[
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 8 | Given a circle with radius 1 and 2 points C, D given on it. Given a constant l with $0<l\le 2$. Moving chord of the circle AB=l and ABCD is a non-degenerated convex quadrilateral. AC and BD intersects at P. Find the loci of the circumcenters of triangles ABP and BCP. |
Given a circle with radius 1 and two points \( C \) and \( D \) on it, and a constant \( l \) with \( 0 < l \leq 2 \). A moving chord \( AB \) of the circle has length \( l \), and \( ABCD \) forms a non-degenerate convex quadrilateral. Let \( AC \) and \( BD \) intersect at \( P \). We aim to find the loci of the cir... | \text{circles passing through fixed points} | china_team_selection_test |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions",
"Mathematics -> Number Theory -> Other"
] | 7 | What is the smallest integer $n$ , greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be \[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\] | Let's first obtain an algebraic expression for the root mean square of the first $n$ integers, which we denote $I_n$ . By repeatedly using the identity $(x+1)^3 = x^3 + 3x^2 + 3x + 1$ , we can write \[1^3 + 3\cdot 1^2 + 3 \cdot 1 + 1 = 2^3,\] \[1^3 + 3 \cdot(1^2 + 2^2) + 3 \cdot (1 + 2) + 1 + 1 = 3^3,\] and \[1^3 + 3... | \(\boxed{337}\) | usamo |
[
"Mathematics -> Number Theory -> Factorization",
"Mathematics -> Number Theory -> Prime Numbers"
] | 7 | For any positive integer $d$, prove there are infinitely many positive integers $n$ such that $d(n!)-1$ is a composite number. |
For any positive integer \( d \), we aim to prove that there are infinitely many positive integers \( n \) such that \( d(n!) - 1 \) is a composite number.
### Case 1: \( d = 1 \)
Assume for the sake of contradiction that for all sufficiently large \( n \in \mathbb{N} \), \( n! - 1 \) is prime. Define \( p_n = n! - ... | \text{There are infinitely many positive integers } n \text{ such that } d(n!) - 1 \text{ is a composite number.} | china_team_selection_test |
[
"Mathematics -> Algebra -> Abstract Algebra -> Other",
"Mathematics -> Number Theory -> Congruences",
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 9 | For a rational point (x,y), if xy is an integer that divided by 2 but not 3, color (x,y) red, if xy is an integer that divided by 3 but not 2, color (x,y) blue. Determine whether there is a line segment in the plane such that it contains exactly 2017 blue points and 58 red points. |
Consider the line \( y = ax + b \) where \( b = 2 \) and \( a = p_1 p_2 \cdots p_m \) for primes \( p_1, p_2, \ldots, p_m \) that will be chosen appropriately. We need to ensure that for a rational point \( (x, y) \), \( xy = z \in \mathbb{Z} \) such that \( 1 + az \) is a perfect square.
We construct the primes \( p... | \text{Yes} | china_team_selection_test |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 6.5 | [color=blue][b]Generalization.[/b] Given two integers $ p$ and $ q$ and a natural number $ n \geq 3$ such that $ p$ is prime and $ q$ is squarefree, and such that $ p\nmid q$.
Find all $ a \in \mathbb{Z}$ such that the polynomial $ f(x) \equal{} x^n \plus{} ax^{n \minus{} 1} \plus{} pq$ can be factored into 2 integral ... |
Given two integers \( p \) and \( q \) and a natural number \( n \geq 3 \) such that \( p \) is prime and \( q \) is squarefree, and such that \( p \nmid q \), we need to find all \( a \in \mathbb{Z} \) such that the polynomial \( f(x) = x^n + ax^{n-1} + pq \) can be factored into two integral polynomials of degree a... | a = -1 - pq \text{ or } a = 1 + (-1)^n pq | china_team_selection_test |
[
"Mathematics -> Number Theory -> Congruences",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 8 | For a positive integer $n$, and a non empty subset $A$ of $\{1,2,...,2n\}$, call $A$ good if the set $\{u\pm v|u,v\in A\}$ does not contain the set $\{1,2,...,n\}$. Find the smallest real number $c$, such that for any positive integer $n$, and any good subset $A$ of $\{1,2,...,2n\}$, $|A|\leq cn$. |
For a positive integer \( n \), and a non-empty subset \( A \) of \(\{1, 2, \ldots, 2n\}\), we call \( A \) good if the set \(\{u \pm v \mid u, v \in A\}\) does not contain the set \(\{1, 2, \ldots, n\}\). We aim to find the smallest real number \( c \) such that for any positive integer \( n \), and any good subset \... | \frac{6}{5} | china_team_selection_test |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Number Theory -> Prime Numbers"
] | 4.5 | Find all primes $p$ such that there exist positive integers $x,y$ that satisfy $x(y^2-p)+y(x^2-p)=5p$ . | Rearrange the original equation to get \begin{align*} 5p &= xy^2 + x^2 y - xp - yp \\ &= xy(x+y) -p(x+y) \\ &= (xy-p)(x+y) \end{align*} Since $5p$ , $xy-p$ , and $x+y$ are all integers, $xy-p$ and $x+y$ must be a factor of $5p$ . Now there are four cases to consider.
Case 1: and
Since $x$ and $y$ are positive integers... | \[
\boxed{2, 3, 7}
\] | jbmo |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Other",
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 6 | Let $R$ denote a non-negative rational number. Determine a fixed set of integers $a,b,c,d,e,f$ , such that for every choice of $R$ ,
$\left|\frac{aR^2+bR+c}{dR^2+eR+f}-\sqrt[3]{2}\right|<|R-\sqrt[3]{2}|$ | Note that when $R$ approaches $\sqrt[3]{2}$ , $\frac{aR^2+bR+c}{dR^2+eR+f}$ must also approach $\sqrt[3]{2}$ for the given inequality to hold. Therefore
\[\lim_{R\rightarrow \sqrt[3]{2}} \frac{aR^2+bR+c}{dR^2+eR+f}=\sqrt[3]{2}\]
which happens if and only if
\[\frac{a\sqrt[3]{4}+b\sqrt[3]{2}+c}{d\sqrt[3]{4}+e\sqrt[3]{2... | \[
a = 0, \quad b = 2, \quad c = 2, \quad d = 1, \quad e = 0, \quad f = 2
\] | usamo |
[
"Mathematics -> Number Theory -> Congruences",
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 9 | Given a positive integer $n \ge 2$. Find all $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$, such that $1<a_1 \le a_2 \le a_3 \le \cdots \le a_n$, $a_1$ is odd, and
(1) $M=\frac{1}{2^n}(a_1-1)a_2 a_3 \cdots a_n$ is a positive integer;
(2) One can pick $n$-tuples of integers $(k_{i,1},k_{i,2},\ldots,k_{i,n})$ fo... |
Given a positive integer \( n \ge 2 \), we aim to find all \( n \)-tuples of positive integers \((a_1, a_2, \ldots, a_n)\) such that \( 1 < a_1 \le a_2 \le a_3 \le \cdots \le a_n \), \( a_1 \) is odd, and the following conditions hold:
1. \( M = \frac{1}{2^n}(a_1-1)a_2 a_3 \cdots a_n \) is a positive integer.
2. One c... | (a_1, a_2, \ldots, a_n) \text{ where } a_1 = k \cdot 2^n + 1 \text{ and } a_2, \ldots, a_n \text{ are odd integers such that } 1 < a_1 \le a_2 \le \cdots \le a_n | china_team_selection_test |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 9 | Let $n \geq 2$ be a natural. Define
$$X = \{ (a_1,a_2,\cdots,a_n) | a_k \in \{0,1,2,\cdots,k\}, k = 1,2,\cdots,n \}$$.
For any two elements $s = (s_1,s_2,\cdots,s_n) \in X, t = (t_1,t_2,\cdots,t_n) \in X$, define
$$s \vee t = (\max \{s_1,t_1\},\max \{s_2,t_2\}, \cdots , \max \{s_n,t_n\} )$$
$$s \wedge t = (\min \{s_1... |
Let \( n \geq 2 \) be a natural number. Define
\[
X = \{ (a_1, a_2, \cdots, a_n) \mid a_k \in \{0, 1, 2, \cdots, k\}, k = 1, 2, \cdots, n \}.
\]
For any two elements \( s = (s_1, s_2, \cdots, s_n) \in X \) and \( t = (t_1, t_2, \cdots, t_n) \in X \), define
\[
s \vee t = (\max \{s_1, t_1\}, \max \{s_2, t_2\}, \cdots... | (n + 1)! - (n - 1)! | china_team_selection_test |
[
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 8 | Let the intersections of $\odot O_1$ and $\odot O_2$ be $A$ and $B$. Point $R$ is on arc $AB$ of $\odot O_1$ and $T$ is on arc $AB$ on $\odot O_2$. $AR$ and $BR$ meet $\odot O_2$ at $C$ and $D$; $AT$ and $BT$ meet $\odot O_1$ at $Q$ and $P$. If $PR$ and $TD$ meet at $E$ and $QR$ and $TC$ meet at $F$, then prove: $AE \c... |
Let the intersections of \(\odot O_1\) and \(\odot O_2\) be \(A\) and \(B\). Point \(R\) is on arc \(AB\) of \(\odot O_1\) and \(T\) is on arc \(AB\) on \(\odot O_2\). \(AR\) and \(BR\) meet \(\odot O_2\) at \(C\) and \(D\); \(AT\) and \(BT\) meet \(\odot O_1\) at \(Q\) and \(P\). If \(PR\) and \(TD\) meet at \(E\) an... | AE \cdot BT \cdot BR = BF \cdot AT \cdot AR | china_team_selection_test |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 7 | For each integer $n\ge 2$ , determine, with proof, which of the two positive real numbers $a$ and $b$ satisfying \[a^n=a+1,\qquad b^{2n}=b+3a\] is larger. | Solution 1
Square and rearrange the first equation and also rearrange the second. \begin{align} a^{2n}-a&=a^2+a+1\\ b^{2n}-b&=3a \end{align} It is trivial that \begin{align*} (a-1)^2 > 0 \tag{3} \end{align*} since $a-1$ clearly cannot equal $0$ (Otherwise $a^n=1\neq 1+1$ ). Thus \begin{align*} a^2+a+1&>3a \tag{4}\\ a^... | \[ a > b \] | usamo |
[
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 7.5 | Let the circumcenter of triangle $ABC$ be $O$. $H_A$ is the projection of $A$ onto $BC$. The extension of $AO$ intersects the circumcircle of $BOC$ at $A'$. The projections of $A'$ onto $AB, AC$ are $D,E$, and $O_A$ is the circumcentre of triangle $DH_AE$. Define $H_B, O_B, H_C, O_C$ similarly.
Prove: $H_AO_A, H_BO_B,... |
Let the circumcenter of triangle \(ABC\) be \(O\). \(H_A\) is the projection of \(A\) onto \(BC\). The extension of \(AO\) intersects the circumcircle of \(\triangle BOC\) at \(A'\). The projections of \(A'\) onto \(AB\) and \(AC\) are \(D\) and \(E\), respectively. \(O_A\) is the circumcenter of triangle \(DH_AE\). D... | \text{The lines } H_AO_A, H_BO_B, \text{ and } H_CO_C \text{ are concurrent at the orthocenter of } \triangle H_AH_BH_C. | china_team_selection_test |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Algebra -> Abstract Algebra -> Group Theory"
] | 7 | Assume $n$ is a positive integer. Considers sequences $a_0, a_1, \ldots, a_n$ for which $a_i \in \{1, 2, \ldots , n\}$ for all $i$ and $a_n = a_0$.
(a) Suppose $n$ is odd. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i \pmod{n}$ for all $i = 1, 2, \ldots, n$.
(b) Suppose $n$ is an odd prime. F... |
Let \( n \) be a positive integer. Consider sequences \( a_0, a_1, \ldots, a_n \) for which \( a_i \in \{1, 2, \ldots , n\} \) for all \( i \) and \( a_n = a_0 \).
### Part (a)
Suppose \( n \) is odd. We need to find the number of such sequences if \( a_i - a_{i-1} \not\equiv i \pmod{n} \) for all \( i = 1, 2, \ldots... | (n-1)(n-2)^{n-1} - \frac{2^{n-1} - 1}{n} - 1 | usa_team_selection_test |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons",
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 8 | In convex quadrilateral $ ABCD$, $ AB\equal{}a$, $ BC\equal{}b$, $ CD\equal{}c$, $ DA\equal{}d$, $ AC\equal{}e$, $ BD\equal{}f$. If $ \max \{a,b,c,d,e,f \}\equal{}1$, then find the maximum value of $ abcd$. |
Given a convex quadrilateral \(ABCD\) with side lengths \(AB = a\), \(BC = b\), \(CD = c\), \(DA = d\), and diagonals \(AC = e\), \(BD = f\), where \(\max \{a, b, c, d, e, f\} = 1\), we aim to find the maximum value of \(abcd\).
We claim that the maximum value of \(abcd\) is \(2 - \sqrt{3}\).
To show that this value... | 2 - \sqrt{3} | china_team_selection_test |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5.5 | 16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems. |
Let 16 students take part in a competition where each problem is multiple choice with four choices. We are to find the maximum number of problems such that any two students have at most one answer in common.
Let \( T \) denote the number of triples \((S_i, S_j, Q_k)\) such that students \( S_i \) and \( S_j \) answer... | 5 | china_team_selection_test |
[
"Mathematics -> Number Theory -> Factorization"
] | 9 | For positive integer $k>1$, let $f(k)$ be the number of ways of factoring $k$ into product of positive integers greater than $1$ (The order of factors are not countered, for example $f(12)=4$, as $12$ can be factored in these $4$ ways: $12,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3$.
Prove: If $n$ is a positive integer greater... |
For a positive integer \( k > 1 \), let \( f(k) \) represent the number of ways to factor \( k \) into a product of positive integers greater than 1. For example, \( f(12) = 4 \) because 12 can be factored in these 4 ways: \( 12 \), \( 2 \cdot 6 \), \( 3 \cdot 4 \), and \( 2 \cdot 2 \cdot 3 \).
We aim to prove that i... | \frac{n}{p} | china_team_selection_test |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Number Theory -> Factorization"
] | 9 | A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting. |
A number \( n \) is considered interesting if 2018 divides \( d(n) \), the number of positive divisors of \( n \). We aim to determine all positive integers \( k \) such that there exists an infinite arithmetic progression with common difference \( k \) whose terms are all interesting.
To solve this, we need to ident... | \text{All } k \text{ such that } v_p(k) \geq 2018 \text{ for some prime } p \text{ or } v_q(k) \geq 1009 \text{ and } v_r(k) \geq 2 \text{ for some distinct primes } q \text{ and } r. | china_team_selection_test |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations",
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 5 | Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points? | There are $\binom{2n+1}{3}$ ways how to pick the three vertices. We will now count the ways where the interior does NOT contain the center. These are obviously exactly the ways where all three picked vertices lie among some $n+1$ consecutive vertices of the polygon.
We will count these as follows: We will go clockwise ... | \[
\boxed{\frac{n+1}{4n-2}}
\] | usamo |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Number Theory -> Factorization"
] | 8 | Given a fixed positive integer $a\geq 9$. Prove: There exist finitely many positive integers $n$, satisfying:
(1)$\tau (n)=a$
(2)$n|\phi (n)+\sigma (n)$
Note: For positive integer $n$, $\tau (n)$ is the number of positive divisors of $n$, $\phi (n)$ is the number of positive integers $\leq n$ and relatively prime with ... |
Given a fixed positive integer \( a \geq 9 \), we need to prove that there exist finitely many positive integers \( n \) satisfying the following conditions:
1. \( \tau(n) = a \)
2. \( n \mid \phi(n) + \sigma(n) \)
Here, \( \tau(n) \) is the number of positive divisors of \( n \), \( \phi(n) \) is the Euler's totient... | \text{There exist finitely many positive integers } n. | china_team_selection_test |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations",
"Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Single-variable"
] | 8 | Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate
\[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\] |
Given positive integers \( k, m, n \) such that \( 1 \leq k \leq m \leq n \), we aim to evaluate the sum
\[
\sum_{i=0}^n \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.
\]
To solve this, we employ a calculus-based approach. We start by expressing the sum in terms of an integral:
\[
\sum_{i=0}^n \frac{(-1)... | 0 | china_team_selection_test |
[
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 8 | Given positive integer $ n \ge 5 $ and a convex polygon $P$, namely $ A_1A_2...A_n $. No diagonals of $P$ are concurrent. Proof that it is possible to choose a point inside every quadrilateral $ A_iA_jA_kA_l (1\le i<j<k<l\le n) $ not on diagonals of $P$, such that the $ \tbinom{n}{4} $ points chosen are distinct, and a... |
Given a positive integer \( n \geq 5 \) and a convex polygon \( P \) with vertices \( A_1, A_2, \ldots, A_n \), we need to prove that it is possible to choose a point inside every quadrilateral \( A_iA_jA_kA_l \) (where \( 1 \leq i < j < k < l \leq n \)) such that the chosen points are distinct and any segment connect... | \text{Proven} | china_team_selection_test |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Number Theory -> Prime Numbers"
] | 7 | Determine whether or not there are any positive integral solutions of the simultaneous equations \begin{align*} x_1^2 +x_2^2 +\cdots +x_{1985}^2 & = y^3,\\ x_1^3 +x_2^3 +\cdots +x_{1985}^3 & = z^2 \end{align*} with distinct integers $x_1,x_2,\cdots,x_{1985}$ . | Lemma: For a positive integer $n$ , $1^3+2^3+\cdots +n^3 = (1+2+\cdots +n)^2$ (Also known as Nicomachus's theorem)
Proof by induction: The identity holds for $1$ . Suppose the identity holds for a number $n$ . It is well known that the sum of first $n$ positive integers is $\frac{n(n+1)}{2} = \frac{n^2+n}{2}$ . Thus it... | A positive integral solution exists. | usamo |
[
"Mathematics -> Number Theory -> Factorization",
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 7 | Determine all integers $k$ such that there exists infinitely many positive integers $n$ [b]not[/b] satisfying
\[n+k |\binom{2n}{n}\] |
Determine all integers \( k \) such that there exist infinitely many positive integers \( n \) not satisfying
\[
n + k \mid \binom{2n}{n}.
\]
We claim that all integers \( k \neq 1 \) satisfy the desired property.
First, recall that \(\frac{1}{n + 1} \binom{2n}{n}\) is the \( n \)-th Catalan number. Since the Catal... | k \neq 1 | china_national_olympiad |
[
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 6.5 | If $A$ and $B$ are fixed points on a given circle and $XY$ is a variable diameter of the same circle, determine the locus of the point of intersection of lines $AX$ and $BY$ . You may assume that $AB$ is not a diameter.
[asy] size(300); defaultpen(fontsize(8)); real r=10; picture pica, picb; pair A=r*expi(5*pi/6), B=r*... | WLOG, assume that the circle is the unit circle centered at the origin. Then the points $A$ and $B$ have coordinates $(-a,b)$ and $(a,b)$ respectively and $X$ and $Y$ have coordinates $(r,s)$ and $(-r,-s)$ . Note that these coordinates satisfy $a^2 + b^2 = 1$ and $r^2 + s^2 = 1$ since these points are on a unit circle.... | The locus of the point of intersection of lines \(AX\) and \(BY\) is a circle with the equation:
\[
x^2 + \left(y - \frac{1}{b}\right)^2 = \frac{a^2}{b^2}.
\] | usamo |
[
"Mathematics -> Number Theory -> Factorization",
"Mathematics -> Number Theory -> Prime Numbers"
] | 9 | Proof that
$$ \sum_{m=1}^n5^{\omega (m)} \le \sum_{k=1}^n\lfloor \frac{n}{k} \rfloor \tau (k)^2 \le \sum_{m=1}^n5^{\Omega (m)} .$$ |
To prove the inequality
\[
\sum_{m=1}^n 5^{\omega(m)} \le \sum_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \tau(k)^2 \le \sum_{m=1}^n 5^{\Omega(m)},
\]
we define the following functions:
\[
\chi(n) = 3^{\omega(n)}, \quad \phi(n) = \sum_{d \mid n} \tau(d), \quad \psi(n) = 3^{\Omega(n)}.
\]
We claim that:
\[
\chi(n)... | \sum_{m=1}^n 5^{\omega(m)} \le \sum_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \tau(k)^2 \le \sum_{m=1}^n 5^{\Omega(m)} | china_team_selection_test |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 8 | Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Carte... |
To find all positive integers \( n \) such that there exists a convex \( n \)-lattice polygon with all side lengths being odd numbers and unequal to each other, we need to analyze the conditions given.
First, note that a lattice polygon is defined as a polygon whose vertices have integer coordinates in the Cartesian ... | \{ n \in \mathbb{Z}^+ \mid n \geq 4 \text{ and } n \text{ is even} \} | china_team_selection_test |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 7 | Given two integers $ m,n$ satisfying $ 4 < m < n.$ Let $ A_{1}A_{2}\cdots A_{2n \plus{} 1}$ be a regular $ 2n\plus{}1$ polygon. Denote by $ P$ the set of its vertices. Find the number of convex $ m$ polygon whose vertices belongs to $ P$ and exactly has two acute angles. |
Given two integers \( m \) and \( n \) satisfying \( 4 < m < n \), let \( A_1A_2\cdots A_{2n+1} \) be a regular \( 2n+1 \) polygon. Denote by \( P \) the set of its vertices. We aim to find the number of convex \( m \)-gons whose vertices belong to \( P \) and have exactly two acute angles.
Notice that if a regular \... | (2n + 1) \left[ \binom{n}{m - 1} + \binom{n + 1}{m - 1} \right] | china_national_olympiad |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Number Theory -> Other"
] | 5 | Find all positive integers $x,y$ satisfying the equation \[9(x^2+y^2+1) + 2(3xy+2) = 2005 .\] | Solution 1
We can re-write the equation as:
$(3x)^2 + y^2 + 2(3x)(y) + 8y^2 + 9 + 4 = 2005$
or $(3x + y)^2 = 4(498 - 2y^2)$
The above equation tells us that $(498 - 2y^2)$ is a perfect square.
Since $498 - 2y^2 \ge 0$ . this implies that $y \le 15$
Also, taking $mod 3$ on both sides we see that $y$ cannot be a multi... | \[
\boxed{(7, 11), (11, 7)}
\] | jbmo |
[
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 5 | Determine the triangle with sides $a,b,c$ and circumradius $R$ for which $R(b+c) = a\sqrt{bc}$ . | Solution 1
Solving for $R$ yields $R = \tfrac{a\sqrt{bc}}{b+c}$ . We can substitute $R$ into the area formula $A = \tfrac{abc}{4R}$ to get \begin{align*} A &= \frac{abc}{4 \cdot \tfrac{a\sqrt{bc}}{b+c} } \\ &= \frac{abc}{4a\sqrt{bc}} \cdot (b+c) \\ &= \frac{(b+c)\sqrt{bc}}{4}. \end{align*} We also know that $A = \tfra... | \[
(a, b, c) \rightarrow \boxed{(n\sqrt{2}, n, n)}
\] | jbmo |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 7 | Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ wi... |
Let \( n \geq 3 \) be an odd number and suppose that each square in an \( n \times n \) chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex. Two squares \( a \) and \( b \) are considered connected if there exists a sequence of square... | \left(\frac{n+1}{2}\right)^2 + 1 | china_national_olympiad |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | A $5 \times 5$ table is called regular if each of its cells contains one of four pairwise distinct real numbers, such that each of them occurs exactly once in every $2 \times 2$ subtable.The sum of all numbers of a regular table is called the total sum of the table. With any four numbers, one constructs all possible re... | Solution 1 (Official solution)
We will prove that the maximum number of total sums is $60$ .
The proof is based on the following claim:
In a regular table either each row contains exactly two of the numbers or each column contains exactly two of the numbers.
Proof of the Claim:
Let R be a row containing at least three... | \boxed{60} | jbmo |
[
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions"
] | 7 | Let $S_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$ ,
$(*)$ $\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}$
for $(m,n)=(2,3),(3,2),(2,5)$ , or $(5,2)$ . Determine all other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real numbers $x,y,z$ such that $x+y+z=0$ . | Claim Both $m,n$ can not be even.
Proof $x+y+z=0$ , $\implies x=-(y+z)$ .
Since $\frac{S_{m+n}}{m+n} = \frac{S_m S_n}{mn}$ ,
by equating cofficient of $y^{m+n}$ on LHS and RHS ,get
$\frac{2}{m+n}=\frac{4}{mn}$ .
$\implies \frac{m}{2} + \frac {n}{2} = \frac{m\cdot n}{2\cdot2}$ .
So we have, $\frac{m}{2} \biggm{|} \frac... | \((m, n) = (5, 2), (2, 5), (3, 2), (2, 3)\) | usamo |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 7 | A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minim... | Answer: $98$ .
There are $4\cdot97$ adjacent pairs of squares in the border and each pair has one black and one white square. Each move can fix at most $4$ pairs, so we need at least $97$ moves. However, we start with two corners one color and two another, so at least one rectangle must include a corner square. But suc... | \[ 98 \] | usamo |
[
"Mathematics -> Algebra -> Algebra -> Sequences and Series",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 6 | Find the maximum possible number of three term arithmetic progressions in a monotone sequence of $n$ distinct reals. | Consider the first few cases for $n$ with the entire $n$ numbers forming an arithmetic sequence \[(1, 2, 3, \ldots, n)\] If $n = 3$ , there will be one ascending triplet (123). Let's only consider the ascending order for now.
If $n = 4$ , the first 3 numbers give 1 triplet, the addition of the 4 gives one more, for 2 i... | \[
f(n) = \left\lfloor \frac{(n-1)^2}{2} \right\rfloor
\] | usamo |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons",
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 4.5 | Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$ , $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$ . Compute the area of the pentagon. | Solution 1
Let $BC = a, ED = 1 - a$
Let $\angle DAC = X$
Applying cosine rule to $\triangle DAC$ we get:
$\cos X = \frac{AC ^ {2} + AD ^ {2} - DC ^ {2}}{ 2 \cdot AC \cdot AD }$
Substituting $AC^{2} = 1^{2} + a^{2}, AD ^ {2} = 1^{2} + (1-a)^{2}, DC = 1$ we get:
$\cos^{2} X = \frac{(1 - a - a ^ {2}) ^ {2}}{(1 + a^{2})... | \[ 1 \] | jbmo |
[
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 7 | Let $A,B,C,D$ denote four points in space such that at most one of the distances $AB,AC,AD,BC,BD,CD$ is greater than $1$ . Determine the maximum value of the sum of the six distances. | Suppose that $AB$ is the length that is more than $1$ . Let spheres with radius $1$ around $A$ and $B$ be $S_A$ and $S_B$ . $C$ and $D$ must be in the intersection of these spheres, and they must be on the circle created by the intersection to maximize the distance. We have $AC + BC + AD + BD = 4$ .
In fact, $CD$ must ... | \[ 5 + \sqrt{3} \] | usamo |
[
"Mathematics -> Number Theory -> Congruences"
] | 7.5 | Consider the assertion that for each positive integer $n \ge 2$ , the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of 4. Either prove the assertion or find (with proof) a counter-example. | We will show that $n = 25$ is a counter-example.
Since $\textstyle 2^n \equiv 1 \pmod{2^n - 1}$ , we see that for any integer $k$ , $\textstyle 2^{2^n} \equiv 2^{(2^n - kn)} \pmod{2^n-1}$ . Let $0 \le m < n$ be the residue of $2^n \pmod n$ . Note that since $\textstyle m < n$ and $\textstyle n \ge 2$ , necessarily $\te... | The assertion is false; \( n = 25 \) is a counter-example. | usamo |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 7 | On a given circle, six points $A$ , $B$ , $C$ , $D$ , $E$ , and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points. | First we give the circle an orientation (e.g., letting the circle be the unit circle in polar coordinates). Then, for any set of six points chosen on the circle, there are exactly $6!$ ways to label them one through six. Also, this does not affect the probability we wish to calculate. This will, however, make calculat... | \[
\frac{3}{10}
\] | usamo |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations",
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 8 | Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1. |
Given distinct prime numbers \( p \) and \( q \) and a natural number \( n \geq 3 \), we aim to find all \( a \in \mathbb{Z} \) such that the polynomial \( f(x) = x^n + ax^{n-1} + pq \) can be factored into two integral polynomials of degree at least 1.
To solve this, we use the following reasoning:
1. **Lemma (Eise... | -1 - pq \text{ and } 1 + pq | china_team_selection_test |
[
"Mathematics -> Geometry -> Plane Geometry -> Triangulations",
"Mathematics -> Geometry -> Plane Geometry -> Angles",
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 5 | Let $ABC$ be an isosceles triangle with $AC=BC$ , let $M$ be the midpoint of its side $AC$ , and let $Z$ be the line through $C$ perpendicular to $AB$ . The circle through the points $B$ , $C$ , and $M$ intersects the line $Z$ at the points $C$ and $Q$ . Find the radius of the circumcircle of the triangle $ABC$ in term... | Let length of side $CB = x$ and length of $QM = a$ . We shall first prove that $QM = QB$ .
Let $O$ be the circumcenter of $\triangle ACB$ which must lie on line $Z$ as $Z$ is a perpendicular bisector of isosceles $\triangle ACB$ .
So, we have $\angle ACO = \angle BCO = \angle C/2$ .
Now $MQBC$ is a cyclic quadrilateral... | \[ R = \frac{2}{3}m \] | jbmo |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 8 | Find all functions $f:(0,\infty) \to (0,\infty)$ such that
\[f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{z}\right) + f\left(z+\frac{1}{x}\right) = 1\] for all $x,y,z >0$ with $xyz =1.$ | Obviously, the output of $f$ lies in the interval $(0,1)$ . Define $g:(0,1)\to(0,1)$ as $g(x)=f\left(\frac1x-1\right)$ . Then for any $a,b,c\in(0,1)$ such that $a+b+c=1$ , we have $g(a)=f\left(\frac1a-1\right)=f\left(\frac{1-a}a\right)=f\left(\frac{b+c}a\right)$ . We can transform $g(b)$ and $g(c)$ similarly:
\[g(a)+g(... | \[ f(x) = \frac{k}{1+x} + \frac{1-k}{3} \quad \left(-\frac{1}{2} \le k \le 1\right) \] | usamo |
[
"Mathematics -> Number Theory -> Congruences",
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 5 | Find all triplets of integers $(a,b,c)$ such that the number
\[N = \frac{(a-b)(b-c)(c-a)}{2} + 2\]
is a power of $2016$ .
(A power of $2016$ is an integer of form $2016^n$ ,where $n$ is a non-negative integer.) | It is given that $a,b,c \in \mathbb{Z}$
Let $(a-b) = -x$ and $(b-c)=-y$ then $(c-a) = x+y$ and $x,y \in \mathbb{Z}$
$(a-b)(b-c)(c-a) = xy(x+y)$
We can then distinguish between two cases:
Case 1: If $n=0$
$2016^n = 2016^0 = 1 \equiv 1 (\mod 2016)$
$xy(x+y) = -2$
$-2 = (-1)(-1)(+2) = (-1)(+1)(+2)=(+1)(+1)(-2)$
$(x... | \[
(a, b, c) = (k, k+1, k+2) \quad \text{and all cyclic permutations, with } k \in \mathbb{Z}
\] | jbmo |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 6 | In the polynomial $x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$ , the product of $2$ of its roots is $- 32$ . Find $k$ . | Using Vieta's formulas, we have:
\begin{align*}a+b+c+d &= 18,\\ ab+ac+ad+bc+bd+cd &= k,\\ abc+abd+acd+bcd &=-200,\\ abcd &=-1984.\\ \end{align*}
From the last of these equations, we see that $cd = \frac{abcd}{ab} = \frac{-1984}{-32} = 62$ . Thus, the second equation becomes $-32+ac+ad+bc+bd+62=k$ , and so $ac+ad+bc+b... | \[ k = 86 \] | usamo |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 7 | Let $n \geq 5$ be an integer. Find the largest integer $k$ (as a function of $n$ ) such that there exists a convex $n$ -gon $A_{1}A_{2}\dots A_{n}$ for which exactly $k$ of the quadrilaterals $A_{i}A_{i+1}A_{i+2}A_{i+3}$ have an inscribed circle. (Here $A_{n+j} = A_{j}$ .) | Lemma: If quadrilaterals $A_iA_{i+1}A_{i+2}A_{i+3}$ and $A_{i+2}A_{i+3}A_{i+4}A_{i+5}$ in an equiangular $n$ -gon are tangential, and $A_iA_{i+3}$ is the longest side quadrilateral $A_iA_{i+1}A_{i+2}A_{i+3}$ for all $i$ , then quadrilateral $A_{i+1}A_{i+2}A_{i+3}A_{i+4}$ is not tangential.
Proof:
[asy] import geometry;... | \[
k = \left\lfloor \frac{n}{2} \right\rfloor
\] | usamo |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 7 | Let $p_1,p_2,p_3,...$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between $0$ and $1$ . For positive integer $k$ , define
$x_{k}=\begin{cases}0&\text{ if }x_{k-1}=0\\ \left\{\frac{p_{k}}{x_{k-1}}\right\}&\text{ if }x_{k-1}\ne0\end{cases}$
where $\{x\}$ denotes the fractional part of... | All rational numbers between 0 and 1 inclusive will eventually yield some $x_k = 0$ . To begin, note that by definition, all rational numbers can be written as a quotient of coprime integers. Let $x_0 = \frac{m}{n}$ , where $m,n$ are coprime positive integers. Since $0<x_0<1$ , $0<m<n$ . Now \[x_1 = \left\{\frac{p_1}{\... | All rational numbers between 0 and 1 inclusive will eventually yield some \( x_k = 0 \). | usamo |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 5 | Determine all the roots , real or complex , of the system of simultaneous equations
$x+y+z=3$ , , $x^3+y^3+z^3=3$ . | Let $x$ , $y$ , and $z$ be the roots of the cubic polynomial $t^3+at^2+bt+c$ . Let $S_1=x+y+z=3$ , $S_2=x^2+y^2+z^2=3$ , and $S_3=x^3+y^3+z^3=3$ . From this, $S_1+a=0$ , $S_2+aS_1+2b=0$ , and $S_3+aS_2+bS_1+3c=0$ . Solving each of these, $a=-3$ , $b=3$ , and $c=-1$ . Thus $x$ , $y$ , and $z$ are the roots of the polyn... | The roots of the system of simultaneous equations are \(x = 1\), \(y = 1\), and \(z = 1\). | usamo |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 7 | ( Gregory Galparin ) Let $\mathcal{P}$ be a convex polygon with $n$ sides, $n\ge3$ . Any set of $n - 3$ diagonals of $\mathcal{P}$ that do not intersect in the interior of the polygon determine a triangulation of $\mathcal{P}$ into $n - 2$ triangles. If $\mathcal{P}$ is regular and there is a triangulation of $\mathcal... | We label the vertices of $\mathcal{P}$ as $P_0, P_1, P_2, \ldots, P_n$ . Consider a diagonal $d = \overline{P_a\,P_{a+k}},\,k \le n/2$ in the triangulation. We show that $k$ must have the form $2^{m}$ for some nonnegative integer $m$ .
This diagonal partitions $\mathcal{P}$ into two regions $\mathcal{Q},\, \mathcal{R}... | \[ n = 2^{a+1} + 2^b, \quad a, b \ge 0 \]
Alternatively, this condition can be expressed as either \( n = 2^k, \, k \ge 2 \) or \( n \) is the sum of two distinct powers of 2. | usamo |
[
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 4.5 | Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$ , and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$ , evaluate the angle $NMB$ . | [asy] size(15cm,0); draw((0,0)--(0,2)--(4,2)--(4,-3)--(0,0)); draw((-1,2)--(9,2)); draw((0,0)--(2,2)); draw((2,2)--(1,1)); draw((0,0)--(4,2)); draw((0,2)--(1,1)); draw(circle((0,1),1)); draw(circle((4,-3),5)); dot((0,0)); dot((0,2)); dot((2,2)); dot((4,2)); dot((4,-3)); dot((1,1)); dot((0,1)); label("A",(0,0),NW); labe... | \[
\boxed{\frac{\pi}{4}}
\] | jbmo |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Other",
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 7 | Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$ , define $b_m=\min\{n: a_n \ge m\}$ , that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$ . If $a_{19}=85$ , determine the maximum value of $a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}$ . | We create an array of dots like so: the array shall go out infinitely to the right and downwards, and at the top of the $i$ th column we fill the first $a_i$ cells with one dot each. Then the $19$ th row shall have 85 dots. Now consider the first 19 columns of this array, and consider the first 85 rows. In row $j$ , we... | \boxed{1700} | usamo |
[
"Mathematics -> Number Theory -> Congruences"
] | 8 | Two positive integers $p,q \in \mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$. Determine the smallest $n$ so that such operation can go on infinitely. |
Given two positive integers \( p \) and \( q \), we are to determine the smallest number \( n \) such that the operation of choosing two identical numbers \( a, a \) on the blackboard and replacing them with \( a+p \) and \( a+q \) can go on infinitely.
To solve this, we first note that we can assume \(\gcd(p, q) = 1... | \frac{p+q}{\gcd(p,q)} | china_team_selection_test |
[
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 7 | Let $\angle XOY = \frac{\pi}{2}$; $P$ is a point inside $\angle XOY$ and we have $OP = 1; \angle XOP = \frac{\pi}{6}.$ A line passes $P$ intersects the Rays $OX$ and $OY$ at $M$ and $N$. Find the maximum value of $OM + ON - MN.$ |
Given that \(\angle XOY = \frac{\pi}{2}\), \(P\) is a point inside \(\angle XOY\) with \(OP = 1\) and \(\angle XOP = \frac{\pi}{6}\). We need to find the maximum value of \(OM + ON - MN\) where a line passing through \(P\) intersects the rays \(OX\) and \(OY\) at \(M\) and \(N\), respectively.
To solve this problem, ... | 2 | china_team_selection_test |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Number Theory -> Congruences"
] | 7 | Find all ordered triples of primes $(p, q, r)$ such that \[ p \mid q^r + 1, \quad q \mid r^p + 1, \quad r \mid p^q + 1. \] [i]Reid Barton[/i] |
We are tasked with finding all ordered triples of primes \((p, q, r)\) such that
\[ p \mid q^r + 1, \quad q \mid r^p + 1, \quad r \mid p^q + 1. \]
Assume \( p = \min(p, q, r) \) and \( p \neq 2 \). Note the following conditions:
\[
\begin{align*}
\text{ord}_p(q) &\mid 2r \implies \text{ord}_p(q) = 2 \text{ or } 2r, \... | (2, 3, 5), (2, 5, 3), (3, 2, 5), (3, 5, 2), (5, 2, 3), (5, 3, 2) | usa_team_selection_test |
[
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 6 | Prove \[\frac{1}{\cos 0^\circ \cos 1^\circ} + \frac{1}{\cos 1^\circ \cos 2^\circ} + \cdots + \frac{1}{\cos 88^\circ \cos 89^\circ} = \frac{\cos 1^\circ}{\sin^2 1^\circ}.\] | Solution 1
Consider the points $M_k = (1, \tan k^\circ)$ in the coordinate plane with origin $O=(0,0)$ , for integers $0 \le k \le 89$ .
Evidently, the angle between segments $OM_a$ and $OM_b$ is $(b-a)^\circ$ , and the length of segment $OM_a$ is $1/\cos a^\circ$ . It then follows that the area of triangle $M_aOM_... | \[
\frac{\cos 1^\circ}{\sin^2 1^\circ}
\] | usamo |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 6.5 | Problem
Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a recta... | Let the number of stones in row $i$ be $r_i$ and let the number of stones in column $i$ be $c_i$ . Since there are $m$ stones, we must have $\sum_{i=1}^n r_i=\sum_{i=1}^n c_i=m$
Lemma 1: If any $2$ pilings are equivalent, then $r_i$ and $c_i$ are the same in both pilings $\forall i$ .
Proof: We suppose the contrary. N... | \[
\binom{n+m-1}{m}^{2}
\] | usamo |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations",
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Number Theory -> Congruences (due to use of properties of roots of unity) -> Other"
] | 7.5 | Let $p(x)$ be the polynomial $(1-x)^a(1-x^2)^b(1-x^3)^c\cdots(1-x^{32})^k$ , where $a, b, \cdots, k$ are integers. When expanded in powers of $x$ , the coefficient of $x^1$ is $-2$ and the coefficients of $x^2$ , $x^3$ , ..., $x^{32}$ are all zero. Find $k$ . | Solution 1
First, note that if we reverse the order of the coefficients of each factor, then we will obtain a polynomial whose coefficients are exactly the coefficients of $p(x)$ in reverse order. Therefore, if \[p(x)=(1-x)^{a_1}(1-x^2)^{a_2}(1-x^3)^{a_3}\cdots(1-x^{32})^{a_{32}},\] we define the polynomial $q(x)$ to b... | \[ k = 2^{27} - 2^{11} \] | usamo |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 9 | Let $x_n=\binom{2n}{n}$ for all $n\in\mathbb{Z}^+$. Prove there exist infinitely many finite sets $A,B$ of positive integers, satisfying $A \cap B = \emptyset $, and \[\frac{{\prod\limits_{i \in A} {{x_i}} }}{{\prod\limits_{j\in B}{{x_j}} }}=2012.\] |
Let \( x_n = \binom{2n}{n} \) for all \( n \in \mathbb{Z}^+ \). We aim to prove that there exist infinitely many finite sets \( A \) and \( B \) of positive integers, satisfying \( A \cap B = \emptyset \), and
\[
\frac{\prod\limits_{i \in A} x_i}{\prod\limits_{j \in B} x_j} = 2012.
\]
### Claim:
For every positive in... | \text{There exist infinitely many such sets } A \text{ and } B. | china_team_selection_test |
[
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions",
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 8 | Given positive integers $n, k$ such that $n\ge 4k$, find the minimal value $\lambda=\lambda(n,k)$ such that for any positive reals $a_1,a_2,\ldots,a_n$, we have
\[ \sum\limits_{i=1}^{n} {\frac{{a}_{i}}{\sqrt{{a}_{i}^{2}+{a}_{{i}+{1}}^{2}+{\cdots}{{+}}{a}_{{i}{+}{k}}^{2}}}}
\le \lambda\]
Where $a_{n+i}=a_i,i=1,2,\ldots,... |
Given positive integers \( n \) and \( k \) such that \( n \geq 4k \), we aim to find the minimal value \( \lambda = \lambda(n, k) \) such that for any positive reals \( a_1, a_2, \ldots, a_n \), the following inequality holds:
\[
\sum_{i=1}^{n} \frac{a_i}{\sqrt{a_i^2 + a_{i+1}^2 + \cdots + a_{i+k}^2}} \leq \lambda,
\... | n - k | china_team_selection_test |
[
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions",
"Mathematics -> Number Theory -> Congruences"
] | 7 | For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1,a_2,\cdots$ and an infinite geometric sequence of integers $g_1,g_2,\cdots$ satisfying the following properties?
$\bullet$ $a_n-g_n$ is divisible by $m$ for all integers $n>1$ ;
$\bullet$ $a_2-a_1$ is not divisible by $m... | Let the arithmetic sequence be $\{ a, a+d, a+2d, \dots \}$ and the geometric sequence to be $\{ g, gr, gr^2, \dots \}$ . Rewriting the problem based on our new terminology, we want to find all positive integers $m$ such that there exist integers $a,d,r$ with $m \nmid d$ and $m|a+(n-1)d-gr^{n-1}$ for all integers $n>1$ ... | The only positive integers \( m \) that will work are numbers in the form of \( xy^2 \), other than \( 1 \), for integers \( x \) and \( y \) (where \( x \) and \( y \) can be equal), i.e., \( 4, 8, 9, 12, 16, 18, 20, 24, 25, \dots \). | usajmo |
[
"Mathematics -> Geometry -> Plane Geometry -> Triangulations",
"Mathematics -> Algebra -> Intermediate Algebra -> Inequalities"
] | 6 | Attempt of a halfways nice solution.
[color=blue][b]Problem.[/b] Let ABC be a triangle with $C\geq 60^{\circ}$. Prove the inequality
$\left(a+b\right)\cdot\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 4+\frac{1}{\sin\frac{C}{2}}$.[/color]
[i]Solution.[/i] First, we equivalently transform the inequality in... |
Let \( \triangle ABC \) be a triangle with \( \angle C \geq 60^\circ \). We aim to prove the inequality:
\[
(a + b) \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \geq 4 + \frac{1}{\sin \frac{C}{2}}.
\]
First, we transform the given inequality:
\[
(a + b) \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)... | (a + b) \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \geq 4 + \frac{1}{\sin \frac{C}{2}} | china_team_selection_test |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Number Theory -> Other"
] | 6 | For a nonempty set $S$ of integers, let $\sigma(S)$ be the sum of the elements of $S$ . Suppose that $A = \{a_1, a_2, \ldots, a_{11}\}$ is a set of positive integers with $a_1 < a_2 < \cdots < a_{11}$ and that, for each positive integer $n \le 1500$ , there is a subset $S$ of $A$ for which $\sigma(S) = n$ . What is the... | Let's a $n$ - $p$ set be a set $Z$ such that $Z=\{a_1,a_2,\cdots,a_n\}$ , where $\forall i<n$ , $i\in \mathbb{Z}^+$ , $a_i<a_{i+1}$ , and for each $x\le p$ , $x\in \mathbb{Z}^+$ , $\exists Y\subseteq Z$ , $\sigma(Y)=x$ , $\nexists \sigma(Y)=p+1$ .
(For Example $\{1,2\}$ is a $2$ - $3$ set and $\{1,2,4,10\}$ is a $4$ - ... | The smallest possible value of \(a_{10}\) is \(248\). | usamo |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 6.5 | Let $u$ and $v$ be real numbers such that \[(u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8.\] Determine, with proof, which of the two numbers, $u$ or $v$ , is larger. | The answer is $v$ .
We define real functions $U$ and $V$ as follows: \begin{align*} U(x) &= (x+x^2 + \dotsb + x^8) + 10x^9 = \frac{x^{10}-x}{x-1} + 9x^9 \\ V(x) &= (x+x^2 + \dotsb + x^{10}) + 10x^{11} = \frac{x^{12}-x}{x-1} + 9x^{11} . \end{align*} We wish to show that if $U(u)=V(v)=8$ , then $u <v$ .
We first note tha... | \[ v \] | usamo |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 6 | Find all triples of positive integers $(x,y,z)$ that satisfy the equation
\begin{align*} 2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023 \end{align*} | We claim that the only solutions are $(2,3,3)$ and its permutations.
Factoring the above squares and canceling the terms gives you:
$8(xyz)^2 + 2(x^2 +y^2 + z^2) = 4((xy)^2 + (yz)^2 + (zx)^2) + 2024$
Jumping on the coefficients in front of the $x^2$ , $y^2$ , $z^2$ terms, we factor into:
$(2x^2 - 1)(2y^2 - 1)(2z^2 - 1... | The only solutions are \((2, 3, 3)\) and its permutations. | usajmo |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Number Theory -> Congruences",
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 7 | Two rational numbers $\frac{m}{n}$ and $\frac{n}{m}$ are written on a blackboard, where $m$ and $n$ are relatively prime positive integers. At any point, Evan may pick two of the numbers $x$ and $y$ written on the board and write either their arithmetic mean $\frac{x+y}{2}$ or their harmonic mean $\frac{2xy}{x+y}$ on t... | We claim that all odd $m, n$ work if $m+n$ is a positive power of 2.
Proof:
We first prove that $m+n=2^k$ works. By weighted averages we have that $\frac{n(\frac{m}{n})+(2^k-n)\frac{n}{m}}{2^k}=\frac{m+n}{2^k}=1$ can be written, so the solution set does indeed work. We will now prove these are the only solutions.
Assum... | All pairs \((m, n)\) such that \(m\) and \(n\) are odd and \(m+n\) is a positive power of 2. | usamo |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 7 | Let $n$ be a positive integer. There are $\tfrac{n(n+1)}{2}$ marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing $n$ marks. Initially, each mark has the black side up. An operation is to choose a line parallel to the sides of the triangle, and flippi... | This problem needs a solution. If you have a solution for it, please help us out by adding it .
The problems on this page are copyrighted by the Mathematical Association of America 's American Mathematics Competitions .
| The problem does not have a provided solution, so the final answer cannot be extracted. | usamo |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Number Theory -> Prime Numbers"
] | 5.5 | Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$ , where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$ , $2+2$ , $2+1+1$ , $1+2+1$ , $1+1+2$ , and $1+1+1+1$ . Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd. | First of all, note that $f(n)$ = $\sum_{i=0}^{k} f(n-2^{i})$ where $k$ is the largest integer such that $2^k \le n$ . We let $f(0) = 1$ for convenience.
From here, we proceed by induction, with our claim being that the only $n$ such that $f(n)$ is odd are $n$ representable of the form $2^{a} - 1, a \in \mathbb{Z}$
We... | \[ 2047 \] | usajmo |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 8 | Suppose $A_1,A_2,\cdots ,A_n \subseteq \left \{ 1,2,\cdots ,2018 \right \}$ and $\left | A_i \right |=2, i=1,2,\cdots ,n$, satisfying that $$A_i + A_j, \; 1 \le i \le j \le n ,$$ are distinct from each other. $A + B = \left \{ a+b|a\in A,\,b\in B \right \}$. Determine the maximal value of $n$. |
Suppose \( A_1, A_2, \ldots, A_n \subseteq \{1, 2, \ldots, 2018\} \) and \( |A_i| = 2 \) for \( i = 1, 2, \ldots, n \), satisfying that \( A_i + A_j \), \( 1 \leq i \leq j \leq n \), are distinct from each other. Here, \( A + B = \{a + b \mid a \in A, b \in B\} \). We aim to determine the maximal value of \( n \).
To... | 4033 | china_team_selection_test |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 5 | Given that $a,b,c,d,e$ are real numbers such that
$a+b+c+d+e=8$ ,
$a^2+b^2+c^2+d^2+e^2=16$ .
Determine the maximum value of $e$ . | By Cauchy Schwarz, we can see that $(1+1+1+1)(a^2+b^2+c^2+d^2)\geq (a+b+c+d)^2$ thus $4(16-e^2)\geq (8-e)^2$ Finally, $e(5e-16) \geq 0$ which means $\frac{16}{5} \geq e \geq 0$ so the maximum value of $e$ is $\frac{16}{5}$ .
from: Image from Gon Mathcenter.net | \[
\frac{16}{5}
\] | usamo |
[
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions"
] | 6.5 | Let $a,b,c,d,e\geq -1$ and $a+b+c+d+e=5.$ Find the maximum and minimum value of $S=(a+b)(b+c)(c+d)(d+e)(e+a).$ |
Given \( a, b, c, d, e \geq -1 \) and \( a + b + c + d + e = 5 \), we aim to find the maximum and minimum values of \( S = (a+b)(b+c)(c+d)(d+e)(e+a) \).
First, we consider the maximum value. We can use the method of Lagrange multipliers or symmetry arguments to determine that the maximum value occurs when the variab... | -512 \leq (a+b)(b+c)(c+d)(d+e)(e+a) \leq 288 | china_national_olympiad |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 8 | Find all integer $n$ such that the following property holds: for any positive real numbers $a,b,c,x,y,z$, with $max(a,b,c,x,y,z)=a$ , $a+b+c=x+y+z$ and $abc=xyz$, the inequality $$a^n+b^n+c^n \ge x^n+y^n+z^n$$ holds. |
We are given the conditions \( \max(a, b, c, x, y, z) = a \), \( a + b + c = x + y + z \), and \( abc = xyz \). We need to find all integer \( n \) such that the inequality
\[
a^n + b^n + c^n \ge x^n + y^n + z^n
\]
holds for any positive real numbers \( a, b, c, x, y, z \).
We claim that the answer is all \( n \ge 0 ... | n \ge 0 | china_team_selection_test |
Subsets and Splits
High-Difficulty Math Problems Under 20
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SQL Console for KbsdJames/Omni-MATH
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