Let $ x,y\in \mathbb{R} $. Show that if the set $ A_{x,y}=\{ \cos {(n\pi x)}+\cos {(n\pi y)} \mid n\in \mathbb{N}\} $ is finite then $ x,y \in \mathbb{Q} $. [i]Vasile Pop[/i]

Show that for any positive integers $a$ and $b$, the number \[n=\mathrm{LCM}(a,b)+\mathrm{GCD}(a,b)-a-b\] is an even non-negative integer. [i]Proposer: Nanang Susyanto[/i]

Let $a$ and $b$ be positive integers such that $(2a+b)(2b+a)=4752$. Find the value of $ab$. [i]Proposed by James Lin[/i]

Carson and Emily attend different schools. Emily’s school has four times as many students as Carson’s school. The total number of students in both schools combined is $10105$. How many students go to Carson’s school?

If $x_k >0$ ($k=1, 2, \cdots , n$), then $$\sum \limits_{k=1}^n \left ( \frac{x_k}{1+x_1^2+x_2^2+\cdots +x_k^2} \right )^2 \leq \frac{\sum \limits_{k=1}^n x_k^2}{1+\sum \limits_{k=1}^n x_k^2} $$

Knowing that the system \[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\] has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.

Find the maximum value of $\frac{1}{x^{2}}+\frac{2}{y^{2}}+\frac{3}{z^{2}}$, where $x, y, z$ are positive reals satisfying $\frac{1}{\sqrt{2}}\leq z <\frac{ \min(x\sqrt{2}, y\sqrt{3})}{2}, x+z\sqrt{3}\geq\sqrt{6}, y\sqrt{3}+z\sqrt{10}\geq 2\sqrt{5}.$

[b]p1.[/b] The number $100$ is written as a sum of distinct positive integers. Determine, with proof, the maximum number of terms that can occur in the sum. [b]p2.[/b] Inside an equilateral triangle of side length $s$ are three mutually tangent circles of radius $1$, each one of which is also tangent to two sides of the triangle, as depicted below. Find $s$. [img]https://cdn.artofproblemsolving.com/attachments/4/3/3b68d42e96717c83bd7fa64a2c3b0bf47301d4.png[/img] [b]p3.[/b] Color a $4\times 7$ rectangle so that each of its $28$ unit squares is either red or green. Show that no matter how this is done, there will be two columns and two rows, so that the four squares occurring at the intersection of a selected row with a selected column all have the same color. [b]p4.[/b] (a) Show that the $y$-intercept of the line through any two distinct points of the graph of $f(x) = x^2$ is $-1$ times the product of the $x$-coordinates of the two points. (b) Find all real valued functions with the property that the $y$-intercept of the line through any two distinct points of its graph is $-1$ times the product of the $x$-coordinates. Prove that you have found all such functions and that all functions you have found have this property. [b]p5.[/b] Let $n$ be a positive integer. We consider sets $A \subseteq \{1, 2,..., n\}$ with the property that the equation $x+y=z$ has no solution with $x\in A$, $y \in A$, $z \in A$. (a) Show that there is a set $A$ as described above that contains $[(n + l)/2]$ members where $[x]$ denotes the largest integer less than or equal to $x$. (b) Show that if $A$ has the property described above, then the number of members of $A$ is less than or equal to $[(n + l)/2]$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that the equation $$z^4 + 4(i + 1)z + 1 = 0$$ has a root in each quadrant of the complex plane.

The fourth-degree equation $x^4-x-504=0$ has $4$ roots $r_1$, $r_2$, $r_3$, $r_4$. If $S_x$ denotes the value of ${r_1}^4+{r_2}^4+{r_3}^4+{r_4}^4$, compute $S_4$. [i]2015 CCA Math Bonanza Individual Round #10[/i]

Let $\Gamma$ be a circunference and $O$ its center. $AE$ is a diameter of $\Gamma$ and $B$ the midpoint of one of the arcs $AE$ of $\Gamma$. The point $D \ne E$ in on the segment $OE$. The point $C$ is such that the quadrilateral $ABCD$ is a parallelogram, with $AB$ parallel to $CD$ and $BC$ parallel to $AD$. The lines $EB$ and $CD$ meets at point $F$. The line $OF$ cuts the minor arc $EB$ of $\Gamma$ at $I$. Prove that the line $EI$ is the angle bissector of $\angle BEC$.

The number $r$ can be expressed as a four-place decimal $0.abcd,$ where $a, b, c,$ and $d$ represent digits, any of which could be zero. It is desired to approximate $r$ by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to $r$ is $\frac 27.$ What is the number of possible values for $r$?

Define the sequence $s_0$, $s_1$, $s_2$,$ . . .$ by $s_0 = 0$ and $s_n = 3s_{n-1}+2$ for $n \ge 1$. The monic polynomial $f(x)$ defined as $$f(x) =\frac{1}{s_{2023}} \sum^{32}_{k=0} s_{2023+k}x^{32-k}$$ can be factored uniquely (up to permutation) as the product of $16$ monic quadratic polynomials $p_1$, $p_2$, $....$, $p_{16}$ with real coefficients, where $p_i(x) = x^2 + a_ix + b_i$ for $1\le i \le 16$. Compute the integer $N$ that minimizes $$\left|N - \sum^{16}_{k=1} (a_k + b_k)\right|.$$

[b]p1.[/b] Prove that no matter what digits are placed in the four empty boxes, the eight-digit number $9999\Box\Box\Box\Box$ is not a perfect square. [b]p2.[/b] Prove that the number $m/3+m^2/2+m^3/6$ is integral for all integral values of $m$. [b]p3.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p4.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/4/b/ca707bf274ed54c1b22c4f65d3d0b0a5cfdc56.png[/img] [b]p5.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $7$ rocks in the first pile and $9$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p6.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying \[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\ 1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\ + f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\ + f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases} \] for all nonnegative integers $ p$, $ q$, $ r$.

$A_1, A_2, ..., A_n$ are the subsets of $|S|=2019$ such that union of any three of them gives $S$ but if we combine two of subsets it doesn't give us $S$. Find the maximum value of $n$.

An equilateral triangle of side $x$ has its vertices on the sides of a square side $1$. What are the possible values of $x$?

In a chess tournament there are $n \geq 5$ players, and they have already played $\left[ \frac{n^2}{4} \right] +2$ games (each pair have played each other at most once). [b](a)[/b] Prove that there are five players $a, b, c, d, e$ for which the pairs $ab, ac, bc, ad, ae, de$ have already played. [b](b)[/b] Is the statement also valid for the $\left[ \frac{n^2}{4} \right] +1$ games played? Make the proof by induction over $n.$

Given a cyclic $ABCD$ with $AB=AD$. Points $M$ and $N$ are marked on the sides $CD$ and $BC$, respectively, so that $DM+BN=MN$. Prove that the circumcenter of the triangle $AMN$ belongs to the segment $AC$. N.Sedrakian

$6$ open disks in the plane are such that the center of no disk lies inside another. Show that no point lies inside all $6$ disks.

Prove the following assertion: The four altitudes of a tetrahedron $ABCD$ intersect in a point if and only if \[AB^2 + CD^2 = BC^2 + AD^2 = CA^2 + BD^2.\]

Show that among the square roots of the first $ 2015 $ natural numbers, we cannot choose an arithmetic sequence composed of $ 45 $ elements.

Determine if there exist non-constant polynomials $P(x)$, $Q(x)$ and $R(x)$ with real coefficients and leading coefficient $1$, such that each of the polynomials \[ P(Q(x)), \quad Q(R(x)), \quad R(P(x)) \] has at least one real root, while each of the polynomials \[ Q(P(x)), \quad R(Q(x)), \quad P(R(x)) \] has no real roots.

Let $f : \left[ 0, 1 \right] \to \mathbb R$ be a continuous function and $g : \left[ 0, 1 \right] \to \left( 0, \infty \right)$. Prove that if $f$ is increasing, then \[\int_{0}^{t}f(x) g(x) \, dx \cdot \int_{0}^{1}g(x) \, dx \leq \int_{0}^{t}g(x) \, dx \cdot \int_{0}^{1}f(x) g(x) \, dx .\]

Let $\langle x_n\rangle$ be a sequence of real numbers defined as \[x_1=1; x_n=\dfrac{2n}{(n-1)^2}\sum_{i=1}^{n-1}x_i\] Show that the sequence $y_n=x_{n+1}-x_n$ has finite limits as $n\to \infty.$