source stringclasses 1
value | problem stringlengths 119 887 | solution stringlengths 977 6.14k | messages listlengths 2 2 |
|---|---|---|---|
aops_forum | a) Prove that for all real numbers $k,l,m$ holds : $$ (k+l+m)^2 \ge 3 (kl+lm+mk) $$ When does equality holds?
b) If $x,y,z$ are positive real numbers and $a,b$ real numbers such that $$ a(x+y+z)=b(xy+yz+zx)=xyz, $$ prove that $a \ge 3b^2$ . When does equality holds?
| ### Part (a)
1. We start with the given inequality:
\[
(k + l + m)^2 \geq 3(kl + lm + mk)
\]
2. Expand the left-hand side:
\[
(k + l + m)^2 = k^2 + l^2 + m^2 + 2kl + 2lm + 2mk
\]
3. Rewrite the inequality:
\[
k^2 + l^2 + m^2 + 2kl + 2lm + 2mk \geq 3(kl + lm + mk)
\]
4. Simplify the inequal... | [
{
"content": "a) Prove that for all real numbers $k,l,m$ holds : $$ (k+l+m)^2 \\ge 3 (kl+lm+mk) $$ When does equality holds?\n\nb) If $x,y,z$ are positive real numbers and $a,b$ real numbers such that $$ a(x+y+z)=b(xy+yz+zx)=xyz, $$ prove that $a \\ge 3b^2$ . When does equality holds?\n",
"role"... |
aops_forum | a) Find all integers that can divide both the numerator and denominator of the ratio $\frac{5m + 6}{8m + 7}$ for an integer $m$ .
b) Let $a, b, c, d, m$ be integers. Prove that if the numerator and denominator of the ratio $\frac{am + b}{cm+ d}$ are both divisible by $k$ , then so is $ad - bc$ . | ### Part (a)
1. We need to find all integers \( x \) that can divide both the numerator \( 5m + 6 \) and the denominator \( 8m + 7 \) for any integer \( m \).
2. If \( x \) divides both \( 5m + 6 \) and \( 8m + 7 \), then \( x \) must also divide any linear combination of these two expressions. Consider the following:
... | [
{
"content": "a) Find all integers that can divide both the numerator and denominator of the ratio $\\frac{5m + 6}{8m + 7}$ for an integer $m$ .\nb) Let $a, b, c, d, m$ be integers. Prove that if the numerator and denominator of the ratio $\\frac{am + b}{cm+ d}$ are both divisible by $k$ , then so is $... |
aops_forum | a) Prove that $x^2 + y^2 + z^2 = 2xyz$ for integer $x, y, z$ only if $x = y = z = 0$ .
b) Find integers $x, y, z, u$ such that $x^2 + y^2 + z^2 + u^2 = 2xyzu$ . | ### Part (a)
We need to prove that the equation \( x^2 + y^2 + z^2 = 2xyz \) holds for integer values of \( x, y, z \) only if \( x = y = z = 0 \).
1. **Assume \( x, y, z \) are integers and \( x^2 + y^2 + z^2 = 2xyz \)**:
\[
x^2 + y^2 + z^2 = 2xyz
\]
2. **Rearrange the equation**:
\[
x^2 + y^2 + z^2 -... | [
{
"content": "a) Prove that $x^2 + y^2 + z^2 = 2xyz$ for integer $x, y, z$ only if $x = y = z = 0$ .\nb) Find integers $x, y, z, u$ such that $x^2 + y^2 + z^2 + u^2 = 2xyzu$ .",
"role": "user"
},
{
"content": "### Part (a)\nWe need to prove that the equation \\( x^2 + y^2 + z^2 = 2xyz \\) ho... |
aops_forum | a) Prove that for all positive reals $u,v,x,y$ the following inequality takes place:
\[ \frac ux + \frac vy \geq \frac {4(uy+vx)}{(x+y)^2} . \]
b) Let $a,b,c,d>0$ . Prove that
\[ \frac a{b+2c+d} + \frac b{c+2d+a} + \frac c{d+2a+b} + \frac d{a+2b+c} \geq 1.\]
*Traian Tămâian* | ### Part (a)
We need to prove that for all positive reals \( u, v, x, y \), the following inequality holds:
\[
\frac{u}{x} + \frac{v}{y} \geq \frac{4(uy + vx)}{(x + y)^2}.
\]
1. **Rewrite the inequality:**
\[
\frac{u}{x} + \frac{v}{y} \geq \frac{4(uy + vx)}{(x + y)^2}.
\]
2. **Multiply both sides by \( (x + ... | [
{
"content": "a) Prove that for all positive reals $u,v,x,y$ the following inequality takes place:\r\n\\[ \\frac ux + \\frac vy \\geq \\frac {4(uy+vx)}{(x+y)^2} . \\]\r\nb) Let $a,b,c,d>0$ . Prove that \r\n\\[ \\frac a{b+2c+d} + \\frac b{c+2d+a} + \\frac c{d+2a+b} + \\frac d{a+2b+c} \\geq 1.\\]\r\n\r\n*Traia... |
aops_forum | a) Find two sets $X,Y$ such that $X\cap Y =\emptyset$ , $X\cup Y = \mathbb Q^{\star}_{+}$ and $Y = \{a\cdot b \mid a,b \in X \}$ .
b) Find two sets $U,V$ such that $U\cap V =\emptyset$ , $U\cup V = \mathbb R$ and $V = \{x+y \mid x,y \in U \}$ . | ### Part (a)
1. **Define the sets \(X\) and \(Y\):**
- Let \(p\) be an odd prime.
- Define \(X\) as the set of all positive rational numbers \(\frac{a}{b}\) (in simplest form) where one of \(a\) or \(b\) is a quadratic residue modulo \(p\) and the other is not, or one of \(a\) or \(b\) is divisible by \(p\) and ... | [
{
"content": "a) Find two sets $X,Y$ such that $X\\cap Y =\\emptyset$ , $X\\cup Y = \\mathbb Q^{\\star}_{+}$ and $Y = \\{a\\cdot b \\mid a,b \\in X \\}$ . \r\n\r\nb) Find two sets $U,V$ such that $U\\cap V =\\emptyset$ , $U\\cup V = \\mathbb R$ and $V = \\{x+y \\mid x,y \\in U \\}$ .",
"role": "... |
aops_forum | a) For integer $n \ge 3$ , suppose that $0 < a_1 < a_2 < ...< a_n$ is a arithmetic sequence and $0 < b_1 < b_2 < ... < b_n$ is a geometric sequence with $a_1 = b_1, a_n = b_n$ . Prove that a_k > b_k for all $k = 2,3,..., n -1$ .
b) Prove that for every positive integer $n \ge 3$ , there exist an integer arithm... | ### Part (a)
1. **Define the sequences:**
Let \( a_k = a_1 + (k-1)d \) for \( d \) be the common difference of the arithmetic sequence, and \( b_k = b_1 \cdot q^{k-1} \) for \( q \) be the common ratio of the geometric sequence.
2. **Given conditions:**
Since \( a_1 = b_1 \) and \( a_n = b_n \), we have:
\[
... | [
{
"content": "a) For integer $n \\ge 3$ , suppose that $0 < a_1 < a_2 < ...< a_n$ is a arithmetic sequence and $0 < b_1 < b_2 < ... < b_n$ is a geometric sequence with $a_1 = b_1, a_n = b_n$ . Prove that a_k > b_k for all $k = 2,3,..., n -1$ .\nb) Prove that for every positive integer $n \\ge 3$ , ther... |
aops_forum | a) Find the value of $\sum_{n=1}^{\infty}\frac{\phi(n)}{2^n-1}$ ;
b) Show that $\sum_k {m\choose k}{{n+k}\choose m}=\sum_k {m\choose k} {n\choose k} 2^k$ for $m,n\geq 0$ ;
c) Using the identity $(1-x)^{-\frac 12}(1-x)^{-\frac 12}=(1-x)^{-1}$ derive a combinatorial identity!
d) Express the value of $\sum... | 1. **Restate the problem**: We need to show that
\[
\sum_{k=0}^{m} \binom{m}{k} \binom{n+k}{m} = \sum_{k=0}^{m} \binom{m}{k} \binom{n}{k} 2^k \quad \text{for} \quad m, n \geq 0.
\]
2. **Double Counting Argument**:
- Consider two sets \( M \) and \( N \) with \( |M| = m \) and \( |N| = n \), respectively.... | [
{
"content": "a) Find the value of $\\sum_{n=1}^{\\infty}\\frac{\\phi(n)}{2^n-1}$ ;\r\n\r\nb) Show that $\\sum_k {m\\choose k}{{n+k}\\choose m}=\\sum_k {m\\choose k} {n\\choose k} 2^k$ for $m,n\\geq 0$ ;\r\n\r\nc) Using the identity $(1-x)^{-\\frac 12}(1-x)^{-\\frac 12}=(1-x)^{-1}$ derive a combinatorial ... |
aops_forum | a) Prove that it is possible to choose one number out of any 39 consecutive positive integers, having the sum of its digits divisible by 11;
b) Find the first 38 consecutive positive integers none of which have the sum of its digits divisible by 11. | a) We need to prove that among any 39 consecutive positive integers, there is at least one integer whose sum of digits is divisible by 11.
1. Let \( n_1, n_2, \ldots, n_{39} \) be 39 consecutive positive integers.
2. Consider the sum of the digits function \( s(x) \), which gives the sum of the digits of \( x \).
3. W... | [
{
"content": "a) Prove that it is possible to choose one number out of any 39 consecutive positive integers, having the sum of its digits divisible by 11;\r\n\r\nb) Find the first 38 consecutive positive integers none of which have the sum of its digits divisible by 11.",
"role": "user"
},
{
"conten... |
aops_forum | a) Two positive integers are chosen. The sum is revealed to logician $A$ , and the sum of squares is revealed to logician $B$ . Both $A$ and $B$ are given this information and the information contained in this sentence. The conversation between $A$ and $B$ goes as follows: $B$ starts
B: ` I can't tell ... | Let's break down the problem step by step.
### Part (a)
1. **Define the Problem:**
- Two positive integers \( u \) and \( v \) are chosen.
- Logician \( A \) knows the sum \( S = u + v \).
- Logician \( B \) knows the sum of squares \( Q = u^2 + v^2 \).
2. **Initial Information:**
- \( B \) says, "I can'... | [
{
"content": "a) Two positive integers are chosen. The sum is revealed to logician $A$ , and the sum of squares is revealed to logician $B$ . Both $A$ and $B$ are given this information and the information contained in this sentence. The conversation between $A$ and $B$ goes as follows: $B$ starts ... |
aops_forum | a) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+1) - f(n)$ for all $n \in N$ ?
b) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+2) - f(n)$ for all $n \in N$ ?
I. Voronovich | 1. **Assume the function \( f: \mathbb{N} \to \mathbb{N} \) exists such that \( f(f(n)) = f(n+1) - f(n) \) for all \( n \in \mathbb{N} \).**
2. **Consider the properties of \( f \):**
- Since \( f: \mathbb{N} \to \mathbb{N} \), \( f(n) \) maps natural numbers to natural numbers.
- Suppose \( f(n+1) > f(n) \). Th... | [
{
"content": "a) Does there exist a function $f: N \\to N$ such that $f(f(n))=f(n+1) - f(n)$ for all $n \\in N$ ?\nb) Does there exist a function $f: N \\to N$ such that $f(f(n))=f(n+2) - f(n)$ for all $n \\in N$ ?\n\nI. Voronovich",
"role": "user"
},
{
"content": "1. **Assume the function... |
aops_forum | a) Prove that
\[ \dfrac{1}{2} \plus{} \dfrac{1}{3} \plus{} ... \plus{} \dfrac{1}{2^{2n}} > n,
\]
for all positive integers $ n$ .
b) Prove that for every positive integer $ n$ we have $ \min\left\{ k \in \mathbb{Z}, k\geq 2 \mid \dfrac{1}{2} \plus{} \dfrac{1}{3} \plus{} \cdots \plus{} \dfrac{1}{k}>n \right\}... | ### Part (a)
We need to prove that:
\[ \dfrac{1}{2} + \dfrac{1}{3} + \cdots + \dfrac{1}{2^{2n}} > n \]
for all positive integers \( n \).
1. Consider the series:
\[ S = \dfrac{1}{2} + \dfrac{1}{3} + \cdots + \dfrac{1}{2^{2n}} \]
2. We can compare each term of the series with a simpler term:
\[ \dfrac{1}{2^k} \leq \df... | [
{
"content": "a) Prove that\r\n\\[ \\dfrac{1}{2} \\plus{} \\dfrac{1}{3} \\plus{} ... \\plus{} \\dfrac{1}{2^{2n}} > n,\r\n\\]\r\nfor all positive integers $ n$ . \r\n\r\nb) Prove that for every positive integer $ n$ we have $ \\min\\left\\{ k \\in \\mathbb{Z}, k\\geq 2 \\mid \\dfrac{1}{2} \\plus{} \\dfrac{1}... |
aops_forum | a) For each $n \ge 2$ , find the maximum constant $c_{n}$ such that
$\frac 1{a_{1}+1}+\frac 1{a_{2}+1}+\ldots+\frac 1{a_{n}+1}\ge c_{n}$
for all positive reals $a_{1},a_{2},\ldots,a_{n}$ such that $a_{1}a_{2}\cdots a_{n}= 1$ .
b) For each $n \ge 2$ , find the maximum constant $d_{n}$ such that
$\frac 1{... | ### Part (a)
We need to find the maximum constant \( c_n \) such that
\[
\frac{1}{a_1 + 1} + \frac{1}{a_2 + 1} + \ldots + \frac{1}{a_n + 1} \ge c_n
\]
for all positive reals \( a_1, a_2, \ldots, a_n \) such that \( a_1 a_2 \cdots a_n = 1 \).
1. **Assume \( a_1 \ge a_2 \ge \ldots \ge a_n \):**
Without loss of gener... | [
{
"content": "a) For each $n \\ge 2$ , find the maximum constant $c_{n}$ such that\r $\\frac 1{a_{1}+1}+\\frac 1{a_{2}+1}+\\ldots+\\frac 1{a_{n}+1}\\ge c_{n}$ \r\nfor all positive reals $a_{1},a_{2},\\ldots,a_{n}$ such that $a_{1}a_{2}\\cdots a_{n}= 1$ .\r\n\r\nb) For each $n \\ge 2$ , find the maximum c... |
aops_forum | a) Prove that for $n = 2,3,4,...$ holds: $$ \sin a + \sin 2a + ...+ \sin (n-1)a=\frac{\cos a \left(\frac{a}{2}\right) - \cos \left(n-\frac{1}{2}\right) a}{2 \sin \left(\frac{a}{2}\right)} $$ b) A point on the circumference of a wheel, which, remaining in a vertical plane, rolls along a horizontal path, describes, a... | ### Part (a)
To prove the identity:
\[ \sin a + \sin 2a + \cdots + \sin (n-1)a = \frac{\cos \left(\frac{a}{2}\right) - \cos \left(n-\frac{1}{2}\right) a}{2 \sin \left(\frac{a}{2}\right)} \]
1. **Sum of Sines Using Complex Exponentials:**
We can use the identity for the sum of sines in terms of complex exponentials.... | [
{
"content": "a) Prove that for $n = 2,3,4,...$ holds: $$ \\sin a + \\sin 2a + ...+ \\sin (n-1)a=\\frac{\\cos a \\left(\\frac{a}{2}\\right) - \\cos \\left(n-\\frac{1}{2}\\right) a}{2 \\sin \\left(\\frac{a}{2}\\right)} $$ b) A point on the circumference of a wheel, which, remaining in a vertical plane, rolls... |
aops_forum | a) Find the minimum positive integer $k$ so that for every positive integers $(x, y) $ , for which $x/y^2$ and $y/x^2$ , then $xy/(x+y) ^k$ b) Find the minimum positive integer $l$ so that for every positive integers $(x, y, z) $ , for which $x/y^2$ , $y/z^2$ and $z/x^2$ , then $xyz/(x+y+z)^l$ | ### Part (a)
1. We need to find the minimum positive integer \( k \) such that for every pair of positive integers \( (x, y) \), if \( x \mid y^2 \) and \( y \mid x^2 \), then \( \frac{xy}{(x+y)^k} \) is an integer.
2. Given \( x \mid y^2 \) and \( y \mid x^2 \), we can write \( y = x^a \) and \( x = y^b \) for some in... | [
{
"content": "a) Find the minimum positive integer $k$ so that for every positive integers $(x, y) $ , for which $x/y^2$ and $y/x^2$ , then $xy/(x+y) ^k$ b) Find the minimum positive integer $l$ so that for every positive integers $(x, y, z) $ , for which $x/y^2$ , $y/z^2$ and $z/x^2$ , then $xyz... |
aops_forum | a) Determine the largest real number $A$ with the following property: For all non-negative real numbers $x,y,z$ , one has
\[\frac{1+yz}{1+x^2}+\frac{1+zx}{1+y^2}+\frac{1+xy}{1+z^2} \ge A.\]
b) For this real number $A$ , find all triples $(x,y,z)$ of non-negative real numbers for which equality holds in the above ... | 1. **Determine the largest real number \( A \):**
We start by analyzing the given inequality:
\[
\frac{1+yz}{1+x^2}+\frac{1+zx}{1+y^2}+\frac{1+xy}{1+z^2} \ge A.
\]
To find an upper bound for \( A \), consider the case when \( y = z = 0 \) and \( x \to \infty \):
\[
\frac{1+0}{1+x^2} + \frac{1+0}{... | [
{
"content": "a) Determine the largest real number $A$ with the following property: For all non-negative real numbers $x,y,z$ , one has\n\\[\\frac{1+yz}{1+x^2}+\\frac{1+zx}{1+y^2}+\\frac{1+xy}{1+z^2} \\ge A.\\]\nb) For this real number $A$ , find all triples $(x,y,z)$ of non-negative real numbers for whic... |
aops_forum | a) Calculate the product $$ \left(1+\frac{1}{2}\right) \left(1+\frac{1}{3}\right) \left(1+\frac{1}{4}\right)... \left(1+\frac{1}{2006}\right) \left(1+\frac{1}{2007}\right) $$ b) Let the set $$ A =\left\{\frac{1}{2}, \frac{1}{3},\frac{1}{4}, ...,\frac{1}{2006}, \frac{1}{2007}\right\} $$ Determine the sum of... | ### Part (a)
1. Consider the product:
\[
\left(1 + \frac{1}{2}\right) \left(1 + \frac{1}{3}\right) \left(1 + \frac{1}{4}\right) \cdots \left(1 + \frac{1}{2006}\right) \left(1 + \frac{1}{2007}\right)
\]
2. Each term in the product can be rewritten as:
\[
1 + \frac{1}{k} = \frac{k+1}{k}
\]
Therefore... | [
{
"content": "a) Calculate the product $$ \\left(1+\\frac{1}{2}\\right) \\left(1+\\frac{1}{3}\\right) \\left(1+\\frac{1}{4}\\right)... \\left(1+\\frac{1}{2006}\\right) \\left(1+\\frac{1}{2007}\\right) $$ b) Let the set $$ A =\\left\\{\\frac{1}{2}, \\frac{1}{3},\\frac{1}{4}, ...,\\frac{1}{2006}, \\fr... |
aops_forum | a) Prove that for any nonnegative real $x$ , holds $$ x^{\frac32} + 6x^{\frac54} + 8x^{\frac34}\ge 15x. $$ b) Determine all x for which the equality holds | 1. **Substitution**: Let \( x = Y^4 \). Then, we have:
\[
x^{\frac{3}{2}} = (Y^4)^{\frac{3}{2}} = Y^6, \quad x^{\frac{5}{4}} = (Y^4)^{\frac{5}{4}} = Y^5, \quad x^{\frac{3}{4}} = (Y^4)^{\frac{3}{4}} = Y^3
\]
Substituting these into the original inequality, we get:
\[
Y^6 + 6Y^5 + 8Y^3 \ge 15Y^4
\]
... | [
{
"content": "a) Prove that for any nonnegative real $x$ , holds $$ x^{\\frac32} + 6x^{\\frac54} + 8x^{\\frac34}\\ge 15x. $$ b) Determine all x for which the equality holds",
"role": "user"
},
{
"content": "1. **Substitution**: Let \\( x = Y^4 \\). Then, we have:\n \\[\n x^{\\frac{3}{2}} =... |
aops_forum | a) Prove that in an infinite sequence ${a_k}$ of integers, pairwise distinct and each member greater than $1$ , one can find $100$ members for which $a_k > k$ .
b) Prove that in an infinite sequence ${a_k}$ of integers, pairwise distinct and each member greater than $1$ there are infinitely many such number... | ### Part (a)
1. **Assume the contrary**: Suppose there are not 100 members in the sequence such that \( a_k > k \). This means there are at most 99 such members.
2. **Define sets**: Let \( A_t = \{a_1, a_2, \ldots, a_t\} \) and \( N_j = \{2, 3, \ldots, j\} \).
3. **Boundedness**: Since \( a_k \) are pairwise distinc... | [
{
"content": "a) Prove that in an infinite sequence ${a_k}$ of integers, pairwise distinct and each member greater than $1$ , one can find $100$ members for which $a_k > k$ . \nb) Prove that in an infinite sequence ${a_k}$ of integers, pairwise distinct and each member greater than $1$ there are infi... |
aops_forum | a) Show that the product of all differences of possible couples of six given positive integers is divisible by $960$ b) Show that the product of all differences of possible couples of six given positive integers is divisible by $34560$
PS. a) original from Albania
b) modified by problem selecting committee | ### Part (a): Divisibility by 960
1. **Prime Factorization of 960:**
\[
960 = 2^6 \times 3 \times 5
\]
2. **Divisibility by 5:**
- Given six integers, by the pigeonhole principle, at least two of them will have the same remainder when divided by 5. Therefore, their difference is a multiple of 5.
3. **Div... | [
{
"content": "a) Show that the product of all differences of possible couples of six given positive integers is divisible by $960$ b) Show that the product of all differences of possible couples of six given positive integers is divisible by $34560$ \n\nPS. a) original from Albania\nb) modified by problem se... |
aops_forum | a) Consider the polynomial $P(X)=X^5\in \mathbb{R}[X]$ . Show that for every $\alpha\in\mathbb{R}^*$ , the polynomial $P(X+\alpha )-P(X)$ has no real roots.
b) Let $P(X)\in\mathbb{R}[X]$ be a polynomial of degree $n\ge 2$ , with real and distinct roots. Show that there exists $\alpha\in\mathbb{Q}^*$ such that... | ### Part (a)
1. Consider the polynomial \( P(X) = X^5 \). We need to show that for every \( \alpha \in \mathbb{R}^* \), the polynomial \( P(X + \alpha) - P(X) \) has no real roots.
2. Compute \( P(X + \alpha) \):
\[
P(X + \alpha) = (X + \alpha)^5
\]
3. Expand \( (X + \alpha)^5 \) using the binomial theorem:
... | [
{
"content": "a) Consider the polynomial $P(X)=X^5\\in \\mathbb{R}[X]$ . Show that for every $\\alpha\\in\\mathbb{R}^*$ , the polynomial $P(X+\\alpha )-P(X)$ has no real roots.\n\nb) Let $P(X)\\in\\mathbb{R}[X]$ be a polynomial of degree $n\\ge 2$ , with real and distinct roots. Show that there exists $... |
aops_forum | a) Find an integer $a$ for which $(x - a)(x - 10) + 1$ factors in the product $(x + b)(x + c)$ with integers $b$ and $c$ .
b) Find nonzero and nonequal integers $a, b, c$ so that $x(x - a)(x - b)(x - c) + 1$ factors into the product of two polynomials with integer coefficients. | ### Part (a)
1. **Expand the given expression:**
\[
(x - a)(x - 10) + 1
\]
Expanding the product:
\[
(x - a)(x - 10) = x^2 - (a + 10)x + 10a
\]
Adding 1:
\[
x^2 - (a + 10)x + 10a + 1
\]
2. **Compare with the target form:**
We want this to be equal to:
\[
(x + b)(x + c) = x^2 +... | [
{
"content": "a) Find an integer $a$ for which $(x - a)(x - 10) + 1$ factors in the product $(x + b)(x + c)$ with integers $b$ and $c$ .\nb) Find nonzero and nonequal integers $a, b, c$ so that $x(x - a)(x - b)(x - c) + 1$ factors into the product of two polynomials with integer coefficients.",
... |
aops_forum | a) Prove that there are infinitely many pairs $(m, n)$ of positive integers satisfying the following equality $[(4 + 2\sqrt3)m] = [(4 -2\sqrt3)n]$ b) Prove that if $(m, n)$ satisfies the equality, then the number $(n + m)$ is odd.
(I. Voronovich) | ### Part (a)
We need to prove that there are infinitely many pairs \((m, n)\) of positive integers satisfying the equality \(\lfloor (4 + 2\sqrt{3})m \rfloor = \lfloor (4 - 2\sqrt{3})n \rfloor\).
1. Define the functions:
\[
f(m) = (4 + 2\sqrt{3})m \quad \text{and} \quad g(n) = (4 - 2\sqrt{3})n
\]
We need t... | [
{
"content": "a) Prove that there are infinitely many pairs $(m, n)$ of positive integers satisfying the following equality $[(4 + 2\\sqrt3)m] = [(4 -2\\sqrt3)n]$ b) Prove that if $(m, n)$ satisfies the equality, then the number $(n + m)$ is odd.\n\n(I. Voronovich)",
"role": "user"
},
{
"cont... |
aops_forum | a) Prove that there are two polynomials in $ \mathbb Z[x]$ with at least one coefficient larger than 1387 such that coefficients of their product is in the set $ \{\minus{}1,0,1\}$ .
b) Does there exist a multiple of $ x^2\minus{}3x\plus{}1$ such that all of its coefficient are in the set $ \{\minus{}1,0,1\}$ | ### Part (a)
1. Consider the polynomial:
\[
f(x) = (x - 1)(x^2 - 1)(x^4 - 1) \cdots (x^{2^n} - 1)
\]
The coefficients of this polynomial are in the set \(\{-1, 0, 1\}\).
2. We can write \(f(x)\) as a product of two polynomials:
\[
f(x) = g(x)h(x)
\]
where:
\[
g(x) = (x - 1)^n
\]
and... | [
{
"content": "a) Prove that there are two polynomials in $ \\mathbb Z[x]$ with at least one coefficient larger than 1387 such that coefficients of their product is in the set $ \\{\\minus{}1,0,1\\}$ .\r\nb) Does there exist a multiple of $ x^2\\minus{}3x\\plus{}1$ such that all of its coefficient are in th... |
aops_forum | a) Prove that there is not an infinte sequence $(x_n)$ , $n=1,2,...$ of positive real numbers satisfying the relation $x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_{n}}$ , $\forall n \in N$ (*)
b) Do there exist sequences satisfying (*) and containing arbitrary many terms?
I.Voronovich | 1. We start by analyzing the given recurrence relation:
\[
x_{n+2} = \sqrt{x_{n+1}} - \sqrt{x_n}
\]
for all \( n \in \mathbb{N} \).
2. Since \( x_n \) are positive real numbers, we have \( x_{n+2} > 0 \). This implies:
\[
\sqrt{x_{n+1}} - \sqrt{x_n} > 0 \implies \sqrt{x_{n+1}} > \sqrt{x_n} \implies ... | [
{
"content": "a) Prove that there is not an infinte sequence $(x_n)$ , $n=1,2,...$ of positive real numbers satisfying the relation $x_{n+2}=\\sqrt{x_{n+1}}-\\sqrt{x_{n}}$ , $\\forall n \\in N$ (*)\nb) Do there exist sequences satisfying (*) and containing arbitrary many terms? \n\nI.Voronovich",
"rol... |
aops_forum | a) 2 white and 2 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 4, to the other black cat is 8. The sum of distances from the black cats to one white cat is 3, to the other white cat is 9. What cats are sitting on the edges?
b) 2 white and 3 black cats are sitting on t... | a) Let \( |W_1B_1| = a_1 \), \( |W_2B_1| = a_2 \), \( |W_1B_2| = b_1 \), and \( |W_2B_2| = b_2 \).
We are given the following conditions:
1. \( a_1 + a_2 = 4 \)
2. \( b_1 + b_2 = 8 \)
3. \( b_1 + a_1 = 9 \)
4. \( b_2 + a_2 = 3 \)
From these conditions, we can derive the following steps:
1. Subtract equation (4) from... | [
{
"content": "a) 2 white and 2 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 4, to the other black cat is 8. The sum of distances from the black cats to one white cat is 3, to the other white cat is 9. What cats are sitting on the edges?\n\nb) 2 white and 3 bla... |
aops_forum | a) Prove that there are no one-to-one (injective) functions $f: \mathbb{N} \to \mathbb{N}\cup \{0\}$ such that
\[ f(mn) = f(m)+f(n) , \ \forall \ m,n \in \mathbb{N}. \]
b) Prove that for all positive integers $k$ there exist one-to-one functions $f: \{1,2,\ldots,k\}\to\mathbb{N}\cup \{0\}$ such that $f(mn... | ### Part (a)
We need to prove that there are no injective functions \( f: \mathbb{N} \to \mathbb{N} \cup \{0\} \) such that
\[ f(mn) = f(m) + f(n) \quad \forall \, m,n \in \mathbb{N}. \]
1. **Initial Conditions:**
- Let \( m = 1 \). Then \( f(mn) = f(n) \) implies \( f(1 \cdot n) = f(1) + f(n) \). Hence, \( f(n) ... | [
{
"content": "a) Prove that there are no one-to-one (injective) functions $f: \\mathbb{N} \\to \\mathbb{N}\\cup \\{0\\}$ such that \r\n\\[ f(mn) = f(m)+f(n) , \\ \\forall \\ m,n \\in \\mathbb{N}. \\] \r\n\r\nb) Prove that for all positive integers $k$ there exist one-to-one functions $f: \\{1,2,\\ldots,k\... |
aops_forum | a)Find one natural $k$ , for which exist natural $a,b,c$ and $$ a^2+k^2=b^2+(k+1)^2=c^2+(k+2)^2(*) $$ b)Prove, that there are infinitely many such $k$ , for which condition (*) is true.
c)Prove, that if for some $k$ there are such $a,b,c$ that condition (*) is true, than $144| abc$ d)Prove, that there are ... | ### Part (a)
1. We need to find a natural number \( k \) such that there exist natural numbers \( a, b, \) and \( c \) satisfying the equation:
\[
a^2 + k^2 = b^2 + (k+1)^2 = c^2 + (k+2)^2
\]
2. Let's check the given example: \( (a, b, c, k) = (12, 9, 4, 31) \).
\[
12^2 + 31^2 = 144 + 961 = 1105
\]
... | [
{
"content": "a)Find one natural $k$ , for which exist natural $a,b,c$ and $$ a^2+k^2=b^2+(k+1)^2=c^2+(k+2)^2(*) $$ b)Prove, that there are infinitely many such $k$ , for which condition (*) is true.\nc)Prove, that if for some $k$ there are such $a,b,c$ that condition (*) is true, than $144| abc$ d... |
aops_forum | a) The real number $a$ and the continuous function $f : [a, \infty) \rightarrow [a, \infty)$ are such that $|f(x)-f(y)| < |x–y|$ for every two different $x, y \in [a, \infty)$ . Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, \infty)$ ?
b) The real numbers $a$ an... | ### Part (a)
1. **Given:**
- A real number \( a \).
- A continuous function \( f : [a, \infty) \rightarrow [a, \infty) \).
- The condition \( |f(x) - f(y)| < |x - y| \) for every two different \( x, y \in [a, \infty) \).
2. **Objective:**
- To determine if the equation \( f(x) = x \) has only one solution... | [
{
"content": "a) The real number $a$ and the continuous function $f : [a, \\infty) \\rightarrow [a, \\infty)$ are such that $|f(x)-f(y)| < |x–y|$ for every two different $x, y \\in [a, \\infty)$ . Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, \\infty)$ ? \n\... |
aops_forum | a) Consider\[f\text{:}\left[ \text{0,}\infty \right)\to \left[ \text{0,}\infty \right)\] a differentiable and convex function .Show that $f\left( x \right)\le x$ , for every $x\ge 0$ , than ${f}'\left( x \right)\le 1$ ,for every $x\ge 0$ b) Determine \[f\text{:}\left[ \text{0,}\infty \right)\to \left[ \text{0,... | ### Part (a)
1. **Assume for contradiction**: Suppose there exists some \( a \geq 0 \) such that \( f'(a) > 1 \).
2. **Convexity implies**: Since \( f \) is convex, \( f' \) is non-decreasing. Therefore, for all \( x \in [a, \infty) \), we have \( f'(x) \geq f'(a) \).
3. **Define a new function**: Let \( g(x) = f(x) - ... | [
{
"content": "a) Consider\\[f\\text{:}\\left[ \\text{0,}\\infty \\right)\\to \\left[ \\text{0,}\\infty \\right)\\] a differentiable and convex function .Show that $f\\left( x \\right)\\le x$ , for every $x\\ge 0$ , than ${f}'\\left( x \\right)\\le 1$ ,for every $x\\ge 0$ b) Determine \\[f\\text{:}\\left... |
aops_forum | a) prove that every discrete subgroup of $(\mathbb R^2,+)$ is in one of these forms:
i- $\{0\}$ .
ii- $\{mv|m\in \mathbb Z\}$ for a vector $v$ in $\mathbb R^2$ .
iii- $\{mv+nw|m,n\in \mathbb Z\}$ for tho linearly independent vectors $v$ and $w$ in $\mathbb R^2$ .(lattice $L$ )
b) prove that every finite gr... | ### Part (a)
To prove that every discrete subgroup of $(\mathbb{R}^2, +)$ is in one of the given forms, we will consider the structure of discrete subgroups in $\mathbb{R}^2$.
1. **Case 1: The trivial subgroup $\{0\}$**
The set $\{0\}$ is clearly a discrete subgroup of $(\mathbb{R}^2, +)$.
2. **Case 2: Subgroups ... | [
{
"content": "a) prove that every discrete subgroup of $(\\mathbb R^2,+)$ is in one of these forms:\ni- $\\{0\\}$ .\nii- $\\{mv|m\\in \\mathbb Z\\}$ for a vector $v$ in $\\mathbb R^2$ .\niii- $\\{mv+nw|m,n\\in \\mathbb Z\\}$ for tho linearly independent vectors $v$ and $w$ in $\\mathbb R^2$ .(lattic... |
aops_forum | a) Given any positive integer $n$ , show that there exist distint positive integers $x$ and $y$ such that $x + j$ divides $y + j$ for $j = 1 , 2, 3, \ldots, n$ ;
b) If for some positive integers $x$ and $y$ , $x+j$ divides $y+j$ for all positive integers $j$ , prove that $x = y$ . | ### Part (a)
Given any positive integer \( n \), we need to show that there exist distinct positive integers \( x \) and \( y \) such that \( x + j \) divides \( y + j \) for \( j = 1, 2, 3, \ldots, n \).
1. **Choose \( x \) and \( y \):**
Let \( x = 1 \) and \( y = k \cdot n! + 1 \) for some integer \( k \geq 2 \)... | [
{
"content": "a) Given any positive integer $n$ , show that there exist distint positive integers $x$ and $y$ such that $x + j$ divides $y + j$ for $j = 1 , 2, 3, \\ldots, n$ ;\r\n\r\nb) If for some positive integers $x$ and $y$ , $x+j$ divides $y+j$ for all positive integers $j$ , prove that ... |
aops_forum | a) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $x_n-2x_{n+1}+x_{n+2} \le 0$ for any $n$ . Moreover $x_o=1,x_{20}=9,x_{200}=6$ . What is the maximal value of $x_{2009}$ can be?
b) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $2x_n-3x_{n+1}+x... | ### Part (a)
Given the sequence \( x_0, x_1, x_2, \ldots \) satisfies the inequality:
\[ x_n - 2x_{n+1} + x_{n+2} \le 0 \]
for any \( n \). Additionally, we know:
\[ x_0 = 1, \quad x_{20} = 9, \quad x_{200} = 6 \]
We need to find the maximal value of \( x_{2009} \).
1. **Rewrite the inequality:**
\[ x_{n+2} - x_{... | [
{
"content": "a) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $x_n-2x_{n+1}+x_{n+2} \\le 0$ for any $n$ . Moreover $x_o=1,x_{20}=9,x_{200}=6$ . What is the maximal value of $x_{2009}$ can be?\nb) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the ineq... |
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