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value | problem stringlengths 72 3.36k | solution stringlengths 312 10.4k | messages listlengths 2 2 |
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aops_forum | Let be a natural number $ n\ge 2 $ and $ n $ positive real numbers $ a_1,a_2,\ldots ,a_n $ whose product is $ 1. $ Prove that the function $ f:\mathbb{R}_{>0}\longrightarrow\mathbb{R} ,\quad f(x)=\prod_{i=1}^n \left( 1+a_i^x \right) , $ is nondecreasing. | 1. **Define the function and its domain:**
We are given a function \( f:\mathbb{R}_{>0} \to \mathbb{R} \) defined by
\[
f(x) = \prod_{i=1}^n (1 + a_i^x)
\]
where \( a_1, a_2, \ldots, a_n \) are positive real numbers such that their product is 1, i.e., \( \prod_{i=1}^n a_i = 1 \).
2. **Differentiate the ... | [
{
"content": "Let be a natural number $ n\\ge 2 $ and $ n $ positive real numbers $ a_1,a_2,\\ldots ,a_n $ whose product is $ 1. $ Prove that the function $ f:\\mathbb{R}_{>0}\\longrightarrow\\mathbb{R} ,\\quad f(x)=\\prod_{i=1}^n \\left( 1+a_i^x \\right) , $ is nondecreasing.",
"role": "user"
},... |
aops_forum | Let $ABCD$ be a quadrilateral inscribed in a circle $k$ . Let the lines $AC\cap BD=O$ , $AD\cap BC=P$ , and $AB\cap CD=Q$ . Line $QO$ intersects $k$ in points $M$ and $N$ . Prove that $PM$ and $PN$ are tangent to $k$ . | 1. **Define the problem and setup:**
- Let \(ABCD\) be a quadrilateral inscribed in a circle \(k\).
- Let the lines \(AC\) and \(BD\) intersect at point \(O\).
- Let the lines \(AD\) and \(BC\) intersect at point \(P\).
- Let the lines \(AB\) and \(CD\) intersect at point \(Q\).
- The line \(QO\) interse... | [
{
"content": "Let $ABCD$ be a quadrilateral inscribed in a circle $k$ . Let the lines $AC\\cap BD=O$ , $AD\\cap BC=P$ , and $AB\\cap CD=Q$ . Line $QO$ intersects $k$ in points $M$ and $N$ . Prove that $PM$ and $PN$ are tangent to $k$ .",
"role": "user"
},
{
"content": "1. **Define ... |
aops_forum | Let $a, b, c$ be positive integers whose greatest common divisor is $1$ . Determine whether there always exists a positive integer $n$ such that, for every positive integer $k$ , the number $2^n$ is not a divisor of $a^k+b^k+c^k$ . | 1. **Given**: \(a, b, c\) are positive integers with \(\gcd(a, b, c) = 1\). We need to determine if there always exists a positive integer \(n\) such that for every positive integer \(k\), \(2^n\) is not a divisor of \(a^k + b^k + c^k\).
2. **Objective**: Prove that there exists such an \(n\).
3. **Approach**: We wil... | [
{
"content": "Let $a, b, c$ be positive integers whose greatest common divisor is $1$ . Determine whether there always exists a positive integer $n$ such that, for every positive integer $k$ , the number $2^n$ is not a divisor of $a^k+b^k+c^k$ .",
"role": "user"
},
{
"content": "1. **Given*... |
aops_forum | Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that
(1) For all integer $m$ , $f(m) \ne 0$ .
(2) For all positive odd integer $n$ , there exist positive integer $k$ such that $f(k)$ is divided by $n$ . | We need to prove that there exists a monic polynomial \( f(x) \) of degree 6 with integer coefficients such that:
1. For all integers \( m \), \( f(m) \neq 0 \).
2. For all positive odd integers \( n \), there exists a positive integer \( k \) such that \( f(k) \) is divisible by \( n \).
We propose the polynomial \( ... | [
{
"content": "Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that\n(1) For all integer $m$ , $f(m) \\ne 0$ .\n(2) For all positive odd integer $n$ , there exist positive integer $k$ such that $f(k)$ is divided by $n$ .",
"role": "user"
},
... |
aops_forum | Let $ABCD$ be a tetrahedron and let $S$ be its center of gravity. A line through $S$ intersects the surface of $ABCD$ in the points $K$ and $L$ . Prove that \[\frac{1}{3}\leq \frac{KS}{LS}\leq 3\] | 1. **Identify the center of gravity \( S \) of the tetrahedron \( ABCD \):**
The center of gravity \( S \) of a tetrahedron is the point where the medians intersect. Each median of a tetrahedron is a line segment connecting a vertex to the centroid of the opposite face. The centroid of a face of a tetrahedron is the... | [
{
"content": "Let $ABCD$ be a tetrahedron and let $S$ be its center of gravity. A line through $S$ intersects the surface of $ABCD$ in the points $K$ and $L$ . Prove that \\[\\frac{1}{3}\\leq \\frac{KS}{LS}\\leq 3\\]",
"role": "user"
},
{
"content": "1. **Identify the center of gravity \\... |
aops_forum | Let $S = \{x_0, x_1,\dots , x_n\}$ be a finite set of numbers in the interval $[0, 1]$ with $x_0 = 0$ and $x_1 = 1$ . We consider pairwise distances between numbers in $S$ . If every distance that appears, except the distance $1$ , occurs at least twice, prove that all the $x_i$ are rational. | 1. **Define the set and distances:**
Let \( S = \{x_0, x_1, \dots, x_n\} \) be a finite set of numbers in the interval \([0, 1]\) with \( x_0 = 0 \) and \( x_1 = 1 \). We consider the pairwise distances between the numbers in \( S \).
2. **Condition on distances:**
Every distance that appears, except the distan... | [
{
"content": "Let $S = \\{x_0, x_1,\\dots , x_n\\}$ be a finite set of numbers in the interval $[0, 1]$ with $x_0 = 0$ and $x_1 = 1$ . We consider pairwise distances between numbers in $S$ . If every distance that appears, except the distance $1$ , occurs at least twice, prove that all the $x_i$ are ... |
aops_forum | The sequence $a_1 = 1$ , $a_2, a_3, \cdots$ is defined as follows: if $a_n - 2$ is a natural number not already occurring on the board, then $a_{n+1} = a_n-2$ ; otherwise, $a_{n+1} = a_n + 3$ . Prove that every nonzero perfect square occurs in the sequence as the previous term increased by $3$ . | 1. **Define the sequence and the problem statement:**
The sequence \( \{a_n\} \) is defined as follows:
- \( a_1 = 1 \)
- For \( n \geq 1 \), if \( a_n - 2 \) is a natural number not already in the sequence, then \( a_{n+1} = a_n - 2 \); otherwise, \( a_{n+1} = a_n + 3 \).
We need to prove that every nonze... | [
{
"content": "The sequence $a_1 = 1$ , $a_2, a_3, \\cdots$ is defined as follows: if $a_n - 2$ is a natural number not already occurring on the board, then $a_{n+1} = a_n-2$ ; otherwise, $a_{n+1} = a_n + 3$ . Prove that every nonzero perfect square occurs in the sequence as the previous term increased ... |
aops_forum | The square $ABCD$ is inscribed in a circle with center $O$ . Let $E$ be the midpoint of $AD$ . The line $CE$ meets the circle again at $F$ . The lines $FB$ and $AD$ meet at $H$ . Prove $HD = 2AH$ | 1. **Define the problem geometrically:**
- Let \(ABCD\) be a square inscribed in a circle with center \(O\).
- \(E\) is the midpoint of \(AD\).
- The line \(CE\) intersects the circle again at \(F\).
- The lines \(FB\) and \(AD\) intersect at \(H\).
2. **Establish key properties and relationships:**
- S... | [
{
"content": "The square $ABCD$ is inscribed in a circle with center $O$ . Let $E$ be the midpoint of $AD$ . The line $CE$ meets the circle again at $F$ . The lines $FB$ and $AD$ meet at $H$ . Prove $HD = 2AH$ ",
"role": "user"
},
{
"content": "1. **Define the problem geometrically:*... |
aops_forum | A group of $4046$ friends will play a videogame tournament. For that, $2023$ of them will go to one room which the computers are labeled with $a_1,a_2,\dots,a_{2023}$ and the other $2023$ friends go to another room which the computers are labeled with $b_1,b_2,\dots,b_{2023}$ . The player of computer $a_i$ *... | 1. Define \( i_a \) and \( j_b \) as the \( i \)-th player from room \( a \) and the \( j \)-th player from room \( b \) respectively.
2. Assume that player \( 1_a \) does not switch computers. We need to show that no other player switches computers.
3. Since player \( 1_a \) does not switch, they remain at computer \(... | [
{
"content": "A group of $4046$ friends will play a videogame tournament. For that, $2023$ of them will go to one room which the computers are labeled with $a_1,a_2,\\dots,a_{2023}$ and the other $2023$ friends go to another room which the computers are labeled with $b_1,b_2,\\dots,b_{2023}$ . The play... |
aops_forum | Problem 3.
Let $f:\left[ 0,\frac{\pi }{2} \right]\to \left[ 0,\infty \right)$ an increasing function .Prove that:
(a) $\int_{0}^{\frac{\pi }{2}}{\left( f\left( x \right)-f\left( \frac{\pi }{4} \right) \right)}\left( \sin x-\cos x \right)dx\ge 0.$ (b) Exist $a\in \left[ \frac{\pi }{4},\frac{\pi }{2} \right]$ suc... | ### Part (a)
We need to prove that:
\[
\int_{0}^{\frac{\pi}{2}} \left( f(x) - f\left( \frac{\pi}{4} \right) \right) \left( \sin x - \cos x \right) dx \geq 0
\]
1. **Substitute \( u = \frac{\pi}{2} - x \):**
Let \( u = \frac{\pi}{2} - x \). Then \( du = -dx \) and the limits of integration change as follows:
- ... | [
{
"content": "Problem 3.\nLet $f:\\left[ 0,\\frac{\\pi }{2} \\right]\\to \\left[ 0,\\infty \\right)$ an increasing function .Prove that:\n(a) $\\int_{0}^{\\frac{\\pi }{2}}{\\left( f\\left( x \\right)-f\\left( \\frac{\\pi }{4} \\right) \\right)}\\left( \\sin x-\\cos x \\right)dx\\ge 0.$ (b) Exist $a\\in \\... |
aops_forum | **Q.** Show that for any given positive integers $k, l$ , there exists infinitely many positive integers $m$ , such that $i) m \geqslant k$ $ii) \text{gcd}\left(\binom{m}{k}, l\right)=1$ *Suggested by pigeon_in_a_hole* | 1. **Define the sequence \( m_n \):**
Consider the arithmetic sequence \( m_n = k + nl^a \) where \( n \in \mathbb{N} \) and \( a \) is sufficiently large such that \( l^a > k \). This ensures that \( m_n > k \) for all \( n \in \mathbb{N} \).
2. **Expression for the binomial coefficient:**
The binomial coeffici... | [
{
"content": "**Q.** Show that for any given positive integers $k, l$ , there exists infinitely many positive integers $m$ , such that $i) m \\geqslant k$ $ii) \\text{gcd}\\left(\\binom{m}{k}, l\\right)=1$ *Suggested by pigeon_in_a_hole*",
"role": "user"
},
{
"content": "1. **Define the sequence ... |
aops_forum | Let $L$ be a subset in the coordinate plane defined by $L = \{(41x + 2y, 59x + 15y) | x, y \in \mathbb Z \}$ , where $\mathbb Z$ is set of integers. Prove that for any parallelogram with center in the origin of coordinate and area $1990$ , there exist at least two points of $L$ located in it. | 1. **Define the set \( L \) and the problem context:**
\[
L = \{(41x + 2y, 59x + 15y) \mid x, y \in \mathbb{Z}\}
\]
We need to prove that for any parallelogram centered at the origin with area 1990, there exist at least two points of \( L \) located in it.
2. **Consider a specific parallelogram \( P \) wit... | [
{
"content": "Let $L$ be a subset in the coordinate plane defined by $L = \\{(41x + 2y, 59x + 15y) | x, y \\in \\mathbb Z \\}$ , where $\\mathbb Z$ is set of integers. Prove that for any parallelogram with center in the origin of coordinate and area $1990$ , there exist at least two points of $L$ locate... |
aops_forum | Let $ABCD$ be a trapezoid, with $AB \parallel CD$ (the vertices are listed in cyclic order). The diagonals of this trapezoid are perpendicular to one another and intersect at $O$ . The base angles $\angle DAB$ and $\angle CBA$ are both acute. A point $M$ on the line sgement $OA$ is such that $\angle BMD =... | 1. **Given Information and Setup:**
- Trapezoid \(ABCD\) with \(AB \parallel CD\).
- Diagonals \(AC\) and \(BD\) are perpendicular and intersect at \(O\).
- Base angles \(\angle DAB\) and \(\angle CBA\) are acute.
- Point \(M\) on \(OA\) such that \(\angle BMD = 90^\circ\).
- Point \(N\) on \(OB\) such t... | [
{
"content": "Let $ABCD$ be a trapezoid, with $AB \\parallel CD$ (the vertices are listed in cyclic order). The diagonals of this trapezoid are perpendicular to one another and intersect at $O$ . The base angles $\\angle DAB$ and $\\angle CBA$ are both acute. A point $M$ on the line sgement $OA$ is... |
aops_forum | We have three circles $w_1$ , $w_2$ and $\Gamma$ at the same side of line $l$ such that $w_1$ and $w_2$ are tangent to $l$ at $K$ and $L$ and to $\Gamma$ at $M$ and $N$ , respectively. We know that $w_1$ and $w_2$ do not intersect and they are not in the same size. A circle passing through $K... | 1. **Define the centers and radii:**
Let \( O_1 \), \( O_2 \), and \( O \) be the centers of circles \( w_1 \), \( w_2 \), and \( \Gamma \) respectively. Let the radii of \( w_1 \), \( w_2 \), and \( \Gamma \) be \( r_1 \), \( r_2 \), and \( R \) respectively.
2. **Angles involving tangents:**
Since \( w_1 \) an... | [
{
"content": "We have three circles $w_1$ , $w_2$ and $\\Gamma$ at the same side of line $l$ such that $w_1$ and $w_2$ are tangent to $l$ at $K$ and $L$ and to $\\Gamma$ at $M$ and $N$ , respectively. We know that $w_1$ and $w_2$ do not intersect and they are not in the same size. A ci... |
aops_forum | Let $ABC$ be a triangle. The ex-circles touch sides $BC, CA$ and $AB$ at points $U, V$ and $W$ , respectively. Be $r_u$ a straight line that passes through $U$ and is perpendicular to $BC$ , $r_v$ the straight line that passes through $V$ and is perpendicular to $AC$ and $r_w$ the straight line th... | 1. **Lemma:** Let \(ABC\) be a triangle with the orthic triangle \(DEF\). Let \(A', B', C'\) be the feet of perpendiculars from the points \(A, B, C\) to \(FE, FD, DE\) respectively. Then the lines \(AA', BB',\) and \(CC'\) are concurrent.
**Proof of Lemma:**
- The triangles \(ABC\) and \(AEF\) are similar, so ... | [
{
"content": "Let $ABC$ be a triangle. The ex-circles touch sides $BC, CA$ and $AB$ at points $U, V$ and $W$ , respectively. Be $r_u$ a straight line that passes through $U$ and is perpendicular to $BC$ , $r_v$ the straight line that passes through $V$ and is perpendicular to $AC$ and $r_w$... |
aops_forum | Let $ n\geq 3 $ be an integer and let $ \mathcal{S} \subset \{1,2,\ldots, n^3\} $ be a set with $ 3n^2 $ elements. Prove that there exist nine distinct numbers $ a_1,a_2,\ldots,a_9 \in \mathcal{S} $ such that the following system has a solution in nonzero integers:
\begin{eqnarray*} a_1x + a_2y +a_3 z &=& 0 ... | 1. **Define the sets and function:**
Let \( \mathcal{S} \subset \{1, 2, \ldots, n^3\} \) be a set with \( 3n^2 \) elements. Suppose \( x_1 < x_2 < \ldots < x_{3n^2} \) are the elements of \( \mathcal{S} \). Define the subsets:
\[
X_1 = \{x_1, x_2, \ldots, x_{n^2}\}, \quad X_2 = \{x_{n^2+1}, x_{n^2+2}, \ldots, ... | [
{
"content": "Let $ n\\geq 3 $ be an integer and let $ \\mathcal{S} \\subset \\{1,2,\\ldots, n^3\\} $ be a set with $ 3n^2 $ elements. Prove that there exist nine distinct numbers $ a_1,a_2,\\ldots,a_9 \\in \\mathcal{S} $ such that the following system has a solution in nonzero integers:\r\n\\begin{eqn... |
aops_forum | The extension of the median $AM$ of the triangle $ABC$ intersects its circumcircle at $D$ . The circumcircle of triangle $CMD$ intersects the line $AC$ at $C$ and $E$ .The circumcircle of triangle $AME$ intersects the line $AB$ at $A$ and $F$ . Prove that $CF$ is the altitude of triangle $ABC$ . | 1. **Identify the given elements and relationships:**
- $AM$ is the median of $\triangle ABC$.
- $D$ is the point where the extension of $AM$ intersects the circumcircle of $\triangle ABC$.
- The circumcircle of $\triangle CMD$ intersects $AC$ at $C$ and $E$.
- The circumcircle of $\triangle AME$ intersects... | [
{
"content": "The extension of the median $AM$ of the triangle $ABC$ intersects its circumcircle at $D$ . The circumcircle of triangle $CMD$ intersects the line $AC$ at $C$ and $E$ .The circumcircle of triangle $AME$ intersects the line $AB$ at $A$ and $F$ . Prove that $CF$ is the altitude ... |
aops_forum | Let $p$ and $q$ be given prime numbers and $S$ be a subset of ${1,2,3,\dots ,p-2,p-1}$ . Prove that the number of elements in the set $A=\{ (x_1,x_2,…,x_q ):x_i\in S,\sum_{i=1}^q x_i \equiv 0(mod\: p)\}$ is multiple of $q$ .
| 1. **Base Case: \( |S| = 1 \)**
If \( S = \{a\} \) for some \( 1 \le a \le p-1 \), then the set \( A \) is defined as:
\[
A = \{ (x_1, x_2, \ldots, x_q) : x_i \in S, \sum_{i=1}^q x_i \equiv 0 \pmod{p} \}
\]
Since \( S = \{a\} \), every \( x_i = a \). Therefore, the sum \( \sum_{i=1}^q x_i = qa \). For \... | [
{
"content": "Let $p$ and $q$ be given prime numbers and $S$ be a subset of ${1,2,3,\\dots ,p-2,p-1}$ . Prove that the number of elements in the set $A=\\{ (x_1,x_2,…,x_q ):x_i\\in S,\\sum_{i=1}^q x_i \\equiv 0(mod\\: p)\\}$ is multiple of $q$ .\n",
"role": "user"
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"content": "1. **Base ... |
aops_forum | Let $l, l'$ be two lines in $3$ -space and let $A,B,C$ be three points taken on $l$ with $B$ as midpoint of the segment $AC$ . If $a, b, c$ are the distances of $A,B,C$ from $l'$ , respectively, show that $b \leq \sqrt{ \frac{a^2+c^2}{2}}$ , equality holding if $l, l'$ are parallel. | 1. Let $l$ and $l'$ be two lines in 3-space, and let $A, B, C$ be three points on $l$ such that $B$ is the midpoint of the segment $AC$. This implies that $B$ divides $AC$ into two equal segments, i.e., $AB = BC$.
2. Let $a, b, c$ be the perpendicular distances from points $A, B, C$ to the line $l'$, respectively. We n... | [
{
"content": "Let $l, l'$ be two lines in $3$ -space and let $A,B,C$ be three points taken on $l$ with $B$ as midpoint of the segment $AC$ . If $a, b, c$ are the distances of $A,B,C$ from $l'$ , respectively, show that $b \\leq \\sqrt{ \\frac{a^2+c^2}{2}}$ , equality holding if $l, l'$ are par... |
aops_forum | A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figu... | 1. **Identify the Midpoints of Diagonals:**
Consider the midpoints of the diagonals of length $\sqrt{2}$ drawn as the rectangle is divided into the described figures.
2. **Connect Midpoints with Red Segments:**
Connect the midpoints of the diagonals as follows:
- The midpoints of the two equal sides of lengt... | [
{
"content": "A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove t... |
aops_forum | The point $O$ is situated inside the parallelogram $ABCD$ such that $\angle AOB+\angle COD=180^{\circ}$ . Prove that $\angle OBC=\angle ODC$ . | 1. **Construct Point \( O' \)**:
Construct point \( O' \) such that \( DAO'O \) is a parallelogram. This means \( O' \) is the reflection of \( O \) across the line \( DA \).
2. **Properties of Parallelograms**:
Since \( AB \parallel CD \) and \( AD \parallel BC \) (properties of parallelogram \( ABCD \)), and ... | [
{
"content": "The point $O$ is situated inside the parallelogram $ABCD$ such that $\\angle AOB+\\angle COD=180^{\\circ}$ . Prove that $\\angle OBC=\\angle ODC$ .",
"role": "user"
},
{
"content": "1. **Construct Point \\( O' \\)**: \n Construct point \\( O' \\) such that \\( DAO'O \\) is a pa... |
aops_forum | A real valued function $f$ is defined on the interval $(-1,2)$ . A point $x_0$ is said to be a fixed point of $f$ if $f(x_0)=x_0$ . Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$ . Show that if $f'(1)>1$ , then $f$ has a fixed point in the interval $(0,1)$ . | To show that the function \( f \) has a fixed point in the interval \( (0,1) \), we will use the Intermediate Value Theorem (IVT) and the given conditions.
1. **Given Conditions:**
- \( f \) is differentiable on \( (-1,2) \).
- \( f(0) > 0 \).
- \( f(1) = 1 \).
- \( f'(1) > 1 \).
2. **Intermediate Value T... | [
{
"content": "A real valued function $f$ is defined on the interval $(-1,2)$ . A point $x_0$ is said to be a fixed point of $f$ if $f(x_0)=x_0$ . Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$ . Show that if $f'(1)>1$ , then $f$ has a fixed point in the interval $... |
aops_forum | Let $n$ be an integer greater than or equal to 3. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with a rational area. | 1. **Base Case: \( n = 3 \)**
We need to find three points such that the distance between any two points is irrational and the area of the triangle formed by any three points is rational. Consider the points \((0,0)\), \((1,1)\), and \((2,4)\).
- Distance between \((0,0)\) and \((1,1)\):
\[
\sqrt{(1-0... | [
{
"content": "Let $n$ be an integer greater than or equal to 3. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with a rational area.",
"role": "user"
},
{
"content... |
aops_forum | Let $n>k \geq 1$ be integers and let $p$ be a prime dividing $\tbinom{n}{k}$ . Prove that the $k$ -element subsets of $\{1,\ldots,n\}$ can be split into $p$ classes of equal size, such that any two subsets with the same sum of elements belong to the same class.
*Ankan Bhattacharya* | 1. **Generating Function Setup**:
Consider the generating function:
\[
F(x,y) := \prod_{j=1}^n (1 + x^j y).
\]
The coefficient of \(x^s y^k\) in this expansion counts the number of subsets of \([n] := \{1, \ldots, n\}\) with \(k\) elements and sum \(s\).
2. **Lucas' Theorem Application**:
By Lucas' t... | [
{
"content": "Let $n>k \\geq 1$ be integers and let $p$ be a prime dividing $\\tbinom{n}{k}$ . Prove that the $k$ -element subsets of $\\{1,\\ldots,n\\}$ can be split into $p$ classes of equal size, such that any two subsets with the same sum of elements belong to the same class.\n\n*Ankan Bhattachary... |
aops_forum | Let $A,B$ be two points on a give circle, and $M$ be the midpoint of one of the arcs $AB$ . Point $C$ is the orthogonal projection of $B$ onto the tangent $l$ to the circle at $A$ . The tangent at $M$ to the circle meets $AC,BC$ at $A',B'$ respectively. Prove that if $\hat{BAC}<\frac{\pi}{8}$ then ... | 1. **Define the given elements and their relationships:**
- Let \( A \) and \( B \) be points on a circle with center \( O \) and radius \( R \).
- Let \( M \) be the midpoint of the arc \( AB \) that does not contain the other arc \( AB \).
- Let \( l \) be the tangent to the circle at \( A \).
- Let \( C ... | [
{
"content": "Let $A,B$ be two points on a give circle, and $M$ be the midpoint of one of the arcs $AB$ . Point $C$ is the orthogonal projection of $B$ onto the tangent $l$ to the circle at $A$ . The tangent at $M$ to the circle meets $AC,BC$ at $A',B'$ respectively. Prove that if $\\hat{BAC... |
aops_forum | The sequence ${a_1, a_2, ..., a_{2019}}$ satisfies the following condition. $a_1=1, a_{n+1}=2019a_{n}+1$ Now let $x_1, x_2, ..., x_{2019}$ real numbers such that $x_1=a_{2019}, x_{2019}=a_1$ (The others are arbitary.)
Prove that $\sum_{k=1}^{2018} (x_{k+1}-2019x_k-1)^2 \ge \sum_{k=1}^{2018} (a_{2019-k}-2019a_{20... | 1. **Define the sequence \(a_n\):**
Given the sequence \( \{a_n\} \) defined by:
\[
a_1 = 1, \quad a_{n+1} = 2019a_n + 1
\]
We need to find a general form for \(a_n\).
2. **Find the general form of \(a_n\):**
Let's compute the first few terms to identify a pattern:
\[
a_1 = 1
\]
\[
a_2... | [
{
"content": "The sequence ${a_1, a_2, ..., a_{2019}}$ satisfies the following condition. $a_1=1, a_{n+1}=2019a_{n}+1$ Now let $x_1, x_2, ..., x_{2019}$ real numbers such that $x_1=a_{2019}, x_{2019}=a_1$ (The others are arbitary.)\nProve that $\\sum_{k=1}^{2018} (x_{k+1}-2019x_k-1)^2 \\ge \\sum_{k=1}^{2... |
aops_forum | Three real numbers are given. Fractional part of the product of every two of them is $ 1\over 2$ . Prove that these numbers are irrational.
*Proposed by A. Golovanov* | 1. Let \( a, b, c \) be the three given real numbers. According to the problem, the fractional part of the product of every two of them is \( \frac{1}{2} \). This can be written as:
\[
\{ab\} = \frac{1}{2}, \quad \{bc\} = \frac{1}{2}, \quad \{ca\} = \frac{1}{2}
\]
where \(\{x\}\) denotes the fractional part... | [
{
"content": "Three real numbers are given. Fractional part of the product of every two of them is $ 1\\over 2$ . Prove that these numbers are irrational.\r\n\r\n*Proposed by A. Golovanov*",
"role": "user"
},
{
"content": "1. Let \\( a, b, c \\) be the three given real numbers. According to the pro... |
aops_forum | Let $ x,y,z$ be real numbers greater than or equal to $ 1.$ Prove that
\[ \prod(x^{2} \minus{} 2x \plus{} 2)\le (xyz)^{2} \minus{} 2xyz \plus{} 2.\] | 1. Let \( x = a + 1 \), \( y = b + 1 \), and \( z = c + 1 \) where \( a, b, c \geq 0 \). Then the inequality transforms into:
\[
\prod (x^2 - 2x + 2) \leq (xyz)^2 - 2xyz + 2
\]
Substituting \( x = a + 1 \), \( y = b + 1 \), and \( z = c + 1 \), we get:
\[
(a^2 + 1)(b^2 + 1)(c^2 + 1) \leq ((a + 1)(b + ... | [
{
"content": "Let $ x,y,z$ be real numbers greater than or equal to $ 1.$ Prove that\r\n\\[ \\prod(x^{2} \\minus{} 2x \\plus{} 2)\\le (xyz)^{2} \\minus{} 2xyz \\plus{} 2.\\]",
"role": "user"
},
{
"content": "1. Let \\( x = a + 1 \\), \\( y = b + 1 \\), and \\( z = c + 1 \\) where \\( a, b, c \\g... |
aops_forum | Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2\pi]$ and $f''(x) \geq 0$ for all $x$ in $[0,2\pi]$ . Show that
\[\int_{0}^{2\pi} f(x)\cos x dx \geq 0\] | 1. **Given Conditions and Definitions:**
- Let \( f(x) \) be a continuous function on \([0, 2\pi]\).
- The first and second derivatives of \( f(x) \) are continuous on \([0, 2\pi]\).
- \( f''(x) \geq 0 \) for all \( x \) in \([0, 2\pi]\), which implies that \( f(x) \) is convex on \([0, 2\pi]\).
2. **Lemma 1:... | [
{
"content": "Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2\\pi]$ and $f''(x) \\geq 0$ for all $x$ in $[0,2\\pi]$ . Show that\n\\[\\int_{0}^{2\\pi} f(x)\\cos x dx \\geq 0\\]",
"role": "user"
},
{
"content": "1. **Given Conditions and Definit... |
aops_forum | Let $p(n)\geq 0$ for all positive integers $n$ . Furthermore, $x(0)=0, v(0)=1$ , and \[x(n)=x(n-1)+v(n-1), \qquad v(n)=v(n-1)-p(n)x(n) \qquad (n=1,2,\dots).\]
Assume that $v(n)\to 0$ in a decreasing manner as $n \to \infty$ . Prove that the sequence $x(n)$ is bounded if and only if $\sum_{n=1}^{\infty}n\cdot ... | 1. **Notation and Assumptions**:
- Let \( p_n, v_n, x_n \) denote \( p(n), v(n), x(n) \) respectively.
- Assume \( v_n \) is decreasing and positive.
- Let \( S_n \) denote the sequence of partial sums of \( \sum_{n=1}^{\infty} v_n \).
2. **Expressing \( x_n \) in terms of \( S_n \)**:
- Given \( x(n) = x(... | [
{
"content": "Let $p(n)\\geq 0$ for all positive integers $n$ . Furthermore, $x(0)=0, v(0)=1$ , and \\[x(n)=x(n-1)+v(n-1), \\qquad v(n)=v(n-1)-p(n)x(n) \\qquad (n=1,2,\\dots).\\]\nAssume that $v(n)\\to 0$ in a decreasing manner as $n \\to \\infty$ . Prove that the sequence $x(n)$ is bounded if and only... |
aops_forum | Let $a, b, c$ be naturals. The equation $ax^2-bx + c = 0$ has two roots at $[0, 1]$ . Prove that $a\ge 5$ and $b\ge 5$ . | To prove that \(a \geq 5\) and \(b \geq 5\) for the quadratic equation \(ax^2 - bx + c = 0\) with roots in the interval \([0, 1]\), we will use the properties of quadratic equations and the conditions given.
1. **Identify the roots and their properties:**
Let the roots of the quadratic equation \(ax^2 - bx + c = 0\... | [
{
"content": "Let $a, b, c$ be naturals. The equation $ax^2-bx + c = 0$ has two roots at $[0, 1]$ . Prove that $a\\ge 5$ and $b\\ge 5$ .",
"role": "user"
},
{
"content": "To prove that \\(a \\geq 5\\) and \\(b \\geq 5\\) for the quadratic equation \\(ax^2 - bx + c = 0\\) with roots in the in... |
aops_forum | $a,b$ are odd numbers. Prove, that exists natural $k$ that $b^k-a^2$ or $a^k-b^2$ is divided by $2^{2018}$ . | 1. **Base Case:**
- Given that \(a\) and \(b\) are odd numbers, we know that \(a^2\) and \(b^2\) are also odd.
- Let \(t = \nu_2(b^2 - 1)\), where \(\nu_2(x)\) denotes the highest power of 2 dividing \(x\).
- We need to show that there exists a positive integer \(k_t\) such that \(2^t \mid b^{k_t} - a^2\).
... | [
{
"content": " $a,b$ are odd numbers. Prove, that exists natural $k$ that $b^k-a^2$ or $a^k-b^2$ is divided by $2^{2018}$ .",
"role": "user"
},
{
"content": "1. **Base Case:**\n - Given that \\(a\\) and \\(b\\) are odd numbers, we know that \\(a^2\\) and \\(b^2\\) are also odd.\n - Let \... |
aops_forum | Let $a, b, c, d$ be positive real numbers. Prove the following inequality and determine all cases in which the equality holds : $$ \frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a}+\frac{d - a}{a + b} \ge 0. $$ | 1. **Introduce the inequality and the variables:**
Let \( a, b, c, d \) be positive real numbers. We need to prove the inequality:
\[
\frac{a - b}{b + c} + \frac{b - c}{c + d} + \frac{c - d}{d + a} + \frac{d - a}{a + b} \geq 0.
\]
2. **Consider the cyclic nature of the terms:**
Notice that the terms in ... | [
{
"content": "Let $a, b, c, d$ be positive real numbers. Prove the following inequality and determine all cases in which the equality holds : $$ \\frac{a - b}{b + c}+\\frac{b - c}{c + d}+\\frac{c - d}{d + a}+\\frac{d - a}{a + b} \\ge 0. $$ ",
"role": "user"
},
{
"content": "1. **Introduce the i... |
aops_forum | Let be given a circle with center $O$ and a point $P$ outside the circle. A line $l$ passes through $P$ and cuts the circle at $A$ and $B$ . Let $C$ be the point symmetric to $A$ with respect to $OP$ , and let $m$ be the line $BC$ . Prove that all lines $m$ have a common point as $l$ varies. | 1. **Define the problem and setup:**
- Let \( O \) be the center of the circle.
- Let \( P \) be a point outside the circle.
- Let \( l \) be a line passing through \( P \) and intersecting the circle at points \( A \) and \( B \).
- Let \( C \) be the point symmetric to \( A \) with respect to \( OP \).
... | [
{
"content": "Let be given a circle with center $O$ and a point $P$ outside the circle. A line $l$ passes through $P$ and cuts the circle at $A$ and $B$ . Let $C$ be the point symmetric to $A$ with respect to $OP$ , and let $m$ be the line $BC$ . Prove that all lines $m$ have a common point... |
aops_forum | A finite set of unit circles is given in a plane such that the area of their union $U$ is $S$ . Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that $\frac{2S}{9}.$ | 1. **Constructing the Collection Greedily**:
- We start by constructing a collection \( T \) of mutually disjoint circles. We do this by iteratively selecting the largest circle that does not intersect any of the circles already chosen in \( T \). This ensures that all circles in \( T \) are disjoint.
2. **Forming ... | [
{
"content": "A finite set of unit circles is given in a plane such that the area of their union $U$ is $S$ . Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that $\\frac{2S}{9}.$ ",
"role": "user"
},
{
"content": "1. **Constructing the ... |
aops_forum | A palindrome is a number that remains the same when its digits are reversed. Let $n$ be a product of distinct primes not divisible by $10$ . Prove that infinitely many multiples of $n$ are palindromes. | 1. **Define the problem and the given conditions:**
- A palindrome is a number that reads the same forwards and backwards.
- Let \( n \) be a product of distinct primes not divisible by 10.
- We need to prove that there are infinitely many multiples of \( n \) that are palindromes.
2. **Express \( n \) as a p... | [
{
"content": "A palindrome is a number that remains the same when its digits are reversed. Let $n$ be a product of distinct primes not divisible by $10$ . Prove that infinitely many multiples of $n$ are palindromes.",
"role": "user"
},
{
"content": "1. **Define the problem and the given conditi... |
aops_forum | In three piles there are $51, 49$ , and $5$ stones, respectively. You can combine any two piles into one pile or divide a pile consisting of an even number of stones into two equal piles. Is it possible to get $105$ piles with one stone in each? | 1. **Understanding the operations**:
- You can combine any two piles into one pile.
- You can divide a pile consisting of an even number of stones into two equal piles.
2. **Analyzing the initial piles**:
- The initial piles have 51, 49, and 5 stones respectively.
3. **Divisibility by 2**:
- Notice that o... | [
{
"content": "In three piles there are $51, 49$ , and $5$ stones, respectively. You can combine any two piles into one pile or divide a pile consisting of an even number of stones into two equal piles. Is it possible to get $105$ piles with one stone in each?",
"role": "user"
},
{
"content": "1... |
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