Let $ABC$ and $AMN$ be two similar triangles with the same orientation, such that $AB=AC$, $AM=AN$ and having disjoint interiors. Let $O$ be the circumcenter of the triangle $MAB$. Prove that the points $O$, $C$, $N$, $A$ lie on the same circle if and only if the triangle $ABC$ is equilateral. [i]Valentin Vornicu[/i]

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

Let $ n $ be an integer greater than or equal to $ 2 $. Prove that if the real numbers $ a_1 , a_2 , \cdots , a_n $ satisfy $ a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n $, then \[\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2} \] must hold.

Find the necessary and sufficient conditions for two conics that every tangent to one of them contains a real point of the other.

Let $q<50$ be a prime number. Call a sequence of polynomials $P_0(x), P_1(x), P_2(x), ..., P_{q^2}(x)$ [i]tasty[/i] if it satisfies the following conditions: [list] [*] $P_i$ has degree $i$ for each $i$ (where we consider constant polynomials, including the $0$ polynomial, to have degree $0$) [*] The coefficients of $P_i$ are integers between $0$ and $q-1$ for each $i$. [*] For any $0\le i,j\le q^2$, the polynomial $P_i(P_j(x)) - P_j(P_i(x))$ has all its coefficients divisible by $q$. [/list] As $q$ varies over all such prime numbers, determine the total number of tasty sequences of polynomials. [i]Proposed by Vincent Huang[/i]

Pictured below is part of a large circle with radius $30$. There is a chain of three circles with radius $3$, each internally tangent to the large circle and each tangent to its neighbors in the chain. There are two circles with radius $2$ each tangent to two of the radius $3$ circles. The distance between the centers of the two circles with radius $2$ can be written as $\textstyle\frac{a\sqrt b-c}d$, where $a,b,c,$ and $d$ are positive integers, $c$ and $d$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a+b+c+d$. [asy] size(200); defaultpen(linewidth(0.5)); real r=aCos(79/81); pair x=dir(270+r)*27,y=dir(270-r)*27; draw(arc(origin,30,210,330)); draw(circle(x,3)^^circle(y,3)^^circle((0,-27),3)); path arcl=arc(y,5,0,180), arcc=arc((0,-27),5,0,180), arcr=arc(x,5,0,180); pair centl=intersectionpoint(arcl,arcc), centr=intersectionpoint(arcc,arcr); draw(circle(centl,2)^^circle(centr,2)); dot(x^^y^^(0,-27)^^centl^^centr,linewidth(2)); [/asy]

The king assembled 300 wizards and gave them the following challenge. For this challenge, 25 colors can be used, and they are known to the wizards. Each of the wizards receives a hat of one of those 25 colors. If for each color the number of used hats would be written down then all these number would be different, and the wizards know this. Each wizard sees what hat was given to each other wizard but does not see his own hat. Simultaneously each wizard reports the color of his own hat. Is it possible for the wizards to coordinate their actions beforehand so that at least 150 of them would report correctly?

Solve the following equation for $x, y \in \mathbb{N}$ \[x^3+y^3=101xy+101.\]

Does there exist a positive integer $n$ such that all its digits (in the decimal system) are greather than 5, while all the digits of $n^2$ are less than 5?

Knowing that $1 < y < 2$ and $x - y + 1 = 0,$ calculate the value of the expression: $$E = \sqrt{4x^2 +4y-3} + 2\sqrt{y^2 - 6x - 2y +10}.$$

Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.

Two circles intersect at $A$ and $B$. A line through $A$ meets the first circle again at $C$ and the second circle again at $D$. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ not containing $A$, and let $K$ be the midpoint of the segment $CD$. Show that $\angle MKN =\pi/2$. (You may assume that $C$ and $D$ lie on opposite sides of $A$.) [i]D. Tereshin[/i]

Let $p$ be a prime number and $\mathcal{A}$ be a finite set of integers, with at least $p^k$ elements. Denote by $N_{\text{even}}$ the number of subsets of $\mathcal{A}$ with even cardinality and sum of elements divisible by $p^k$. Define $N_{\text{odd}}$ similarly. Prove that $N_{\text{even}}\equiv N_{\text{odd}}\bmod{p}.$

In cube $ABCD-A_1B_1C_1D_1$, $E$ is midpoint of $BC$, $F\in AA_1$, and $A_1F:FA=1:2$. Calculate the dihedral angle between plane $B_1EF$ and plane $A_1B_1C_1D_1$.

The circle centered at point $A$ with radius $19$ and the circle centered at point $B$ with radius $32$ are both internally tangent to a circle centered at point $C$ with radius $100$ such that point $C$ lies on segment $\overline{AB}$. Point $M$ is on the circle centered at $A$ and point $N$ is on the circle centered at $B$ such that line $MN$ is a common internal tangent of those two circles. Find the distance $MN$. [img]https://cdn.artofproblemsolving.com/attachments/3/d/1933ce259c229d49e21b9a2dcadddea2a6b404.png[/img]

Let $\{ a_n \}_{n=0}^{\infty}$, $\{ b_n \}_{n=0}^{\infty}$, and $\{ c_n \}_{n=0}^{\infty}$ be sequences of real numbers such that for all $k\geq 1$, \begin{align*} a_k&=\left\lfloor \sqrt{2}+\frac{k-1}{2024} \right\rfloor+a_{k-1} \\ b_k+c_k&=1 \\ a_{k-1}b_k&=a_kc_k. \end{align*} Suppose that $a_0=1$, $b_0=2$, and $c_0=3$. Given that $\sqrt2\approx1.4142$, compute \[ \sum_{k=1}^{2024}(a_kb_k-a_{k-1}c_k). \]

Let $n$ blue lines, no two of which are parallel and no three concurrent, be drawn on a plane. An intersection of two blue lines is called a blue point. Through any two blue points that have not already been joined by a blue line, a red line is drawn. An intersection of two red lines is called a red point, and an intersection of red line and a blue line is called a purple point. What is the maximum possible number of purple points?

Let $ n \ge 2009$ be an integer and define the set: \[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}. \] Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that \[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}. \]

Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]2+\sqrt[3]4)=0$. (A polynomial is called $\textit{monic}$ if its leading coefficient is $1$.)

Let $x_1,..., x_n$ be non-negative real numbers, not all of which are zeros. (i) Prove that $$1 \le \frac{\left(x_1+\frac{x_2}{2}+\frac{x_3}{3}+...+\frac{x_n}{n}\right)(x_1+2x_2+3x_3+...+nx_n)}{(x_1+x_2+x_3+...+x_n)^2} \le \frac{(n+1)^2}{4n}$$ (ii) Show that, for each $n > 1$, both inequalities can hold as equalities.

A function $S(m, n)$ satisfies the initial conditions $S(1, n) = n$, $S(m, 1) = 1$, and the recurrence $S(m, n) = S(m - 1, n)S(m, n - 1)$ for $m\geq 2, n\geq 2$. Find the largest integer $k$ such that $2^k$ divides $S(7, 7)$.

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?