Given a function $p(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$. Each coefficient $a, b, c, d, e$, and$ f$ is equal to either $ 1$ or $-1$. If $p(2) = 11$, what is the value of $p(3)$?

Albert writes $2025$ numbers $a_1, \ldots, a_{2025}$ in a circle on a blackboard. Initially, each of the numbers is uniformly and independently sampled at random from the interval $[0,1].$ Then, each second, he [i]simultaneously[/i] replaces $a_i$ with $\max(a_{i-1},a_i,a_{i+1})$ for all $i = 1, 2, \ldots, 2025$ (where $a_0 = a_{2025}$ and $a_{2026} = a_1$). Compute the expected value of the number of distinct values remaining after $100$ seconds.

Let $x,y,z,t$ be real numbers with $x,y,z$ not all equal such that \[x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}=t.\] Find all possible values of $ t$ such that $xyz+t=0$.

Quadrilateral $ABCD$ is inscribed on circle $O$. $AC\cap BD=P$. Circumcenters of $\triangle ABP,\triangle BCP,\triangle CDP,\triangle DAP$ are $O_1,O_2,O_3,O_4$. Prove that $OP,O_1O_3,O_2O_4$ share one point.

Evaluate $$\frac{1}{1!21!} + \frac{1}{3!19!} + \frac{1}{5!16!} + ... + \frac{1}{21!1!}$$

In triangle $ABC$, choose point $A_1$ on side $BC$, point $B_1$ on side $CA$, and point $C_1$ on side $AB$ in such a way that the three segments $AA_1, BB_1$, and $CC_1$ intersect in one point $P$. Prove that $P$ is the centroid of triangle $ABC$ if and only if $P$ is the centroid of triangle $A_1B_1C_1$. Note: A median in a triangle is a segment connecting a vertex of the triangle with the midpoint of the opposite side. The centroid of a triangle is the intersection point of the three medians of the triangle. The centroid of a triangle is also known by the names ”center of mass” and ”medicenter” of the triangle.

$21$ distinct numbers are chosen from the set $\{1,2,3,\ldots,2046\}.$ Prove that we can choose three distinct numbers $a,b,c$ among those $21$ numbers such that \[bc<2a^2<4bc\]

The diagram below shows some small squares each with area $3$ enclosed inside a larger square. Squares that touch each other do so with the corner of one square coinciding with the midpoint of a side of the other square. Find integer $n$ such that the area of the shaded region inside the larger square but outside the smaller squares is $\sqrt{n}$. [asy] size(150); real r=1/(2sqrt(2)+1); path square=(0,1)--(r,1)--(r,1-r)--(0,1-r)--cycle; path square2=(0,.5)--(r/sqrt(2),.5+r/sqrt(2))--(r*sqrt(2),.5)--(r/sqrt(2),.5-r/sqrt(2))--cycle; defaultpen(linewidth(0.8)); filldraw(unitsquare,gray); filldraw(square2,white); filldraw(shift((0.5-r/sqrt(2),0.5-r/sqrt(2)))*square2,white); filldraw(shift(1-r*sqrt(2),0)*square2,white); filldraw(shift((0.5-r/sqrt(2),-0.5+r/sqrt(2)))*square2,white); filldraw(shift(0.5-r/sqrt(2)-r,-(0.5-r/sqrt(2)-r))*square,white); filldraw(shift(0.5-r/sqrt(2)-r,-(0.5+r/sqrt(2)))*square,white); filldraw(shift(0.5+r/sqrt(2),-(0.5+r/sqrt(2)))*square,white); filldraw(shift(0.5+r/sqrt(2),-(0.5-r/sqrt(2)-r))*square,white); filldraw(shift(0.5-r/2,-0.5+r/2)*square,white); [/asy]

If $f(x + y) = f(xy)$ for all real numbers $x$ and $y$, and $f(2019) = 17$, what is the value of $f(17)$?

A function f : $R$ $\rightarrow$ $R$ satisfies the property $f(x^2) - f^2(x) \geq 1/4$ for all x. Verify if the function is one-one.

We say that the real numbers $a$ and $b$ have property $P$ if: $a^2+b \in Q$ and $b^2 + a \in Q$.Prove that: a) The numbers $a= \frac{1+\sqrt2}{2}$ and $b= \frac{1-\sqrt2}{2}$ are irrational and have property $P$ b) If $a, b$ have property $P$ and $a+b \in Q -\{1\}$, then $a$ and $b$ are rational numbers c) If $a, b$ have property $P$ and $\frac{a}{b} \in Q$, then $a$ and $b$ are rational numbers.

Does there exist non-zero complex numbers $a, b, c$ and natural number $h$ such that if integers $k, l, m$ satisfy $|k| + |l| + |m| \geq 1996$, then $|ka + lb + mc| > \frac {1}{h}$ is true?

Find the pairs of functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ with $ f $ continuous, $ g $ differentiable and satisfying: $$ -\sin g(x) + \int \cos f(x)dx =\cos g(x) +\int \sin f(x)dx $$

We are given $2001$ lines in the plane, no two of which are parallel and no three of which are concurrent. These lines partition the plane into regions (not necessarily finite) bounded by segments of these lines. These segments are called [i]sides[/i], and the collection of the regions is called a [i]map[/i]. Intersection points of the lines are called [i]vertices[/i]. Two regions are [i]neighbors [/i]if they share a side, and two vertices are neighbors if they lie on the same side. A [i]legal coloring[/i] of the regions (resp. vertices) is a coloring in which each region (resp. vertex) receives one color, such that any two neighboring regions (vertices) have different colors. (a) What is the minimum number of colors required for a legal coloring of the regions? (b) What is the minimum number of colors required for a legal coloring of the vertices?

Given a cube and two colours. Two players paint in turn a triple of arbitrary unpainted edges with his colour. (Everyone makes two moves.) The first wins if he has painted all the edges of some face with his colour. Can he always win?

Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.

Consider a triangle $ABC$ with $BC = 3$. Choose a point $D$ on $BC$ such that $BD = 2$. Find the value of \[AB^2 + 2AC^2 - 3AD^2.\]

Suppose $S= \{1,2,\dots,n\}$ and $n \geq 3$. There is $f:S^k \longmapsto S$ that if $a,b \in S^k$ and $a$ and $b$ differ in all of elements then $f(a) \neq f(b)$. Prove that $f$ is a function of one of its elements.

In the hexagon $ABCDEF$, all angles are equally large. The side lengths satisfy $AB = CD = EF = 3$ and $BC = DE = F A = 2$. The diagonals $AD$ and $CF$ intersect each other in the point $G$. The point $H$ lies on the side $CD$ so that $DH = 1$. Prove that triangle $EGH$ is equilateral.

Is it true that for all natural $n$, it is always possible to give each of the $n$ children a hat painted in one of $100$ colors so that if a girl is known to more than $2015$ boys, then not all of these boys have hats of the same color, and, if a boy is acquainted with more than $2015$ girls, don't all these girls have hats of the same color? [hide=original wording]Vai tiesa, ka visiem naturāliem n vienmēr iespējams katram no n bērniem iedot pa cepurei, kas nokrāsota vienā no 100 krāsām tā, ka, ja kāda meitene ir pazīstama ar vairāk nekā 2015 zēniem, tad ne visiem šiem zēniem cepures ir vienā krāsā, un, ja kāds zēns ir pazīštams ar vairāk nekā 2015 meitenēm, tad ne visām šīm meitenēm cepures ir vienā krāsā?[/hide]

Assume $c$ is a real number. If there exists $x\in[1,2]$ such that $\max\left\{\left |x+\frac cx\right |, \left |x+\frac cx + 2\right |\right\}\geq 5$, please find the value range of $c$.

Let $x,y,z$ be complex numbers such that\\ $\hspace{ 2cm} \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=9$\\ $\hspace{ 2cm} \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=64$\\ $\hspace{ 2cm} \frac{x^3}{y+z}+\frac{y^3}{z+x}+\frac{z^3}{x+y}=488$\\ \\ If $\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}=\frac{m}{n}$ where $m,n$ are positive integers with $GCD(m,n)=1$, find $m+n$.

What is the units digit of $19^{19} + 99^{99}$? $\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9$

Let $N \geq 2$ be an integer, and let $\mathbf a$ $= (a_1, \ldots, a_N)$ and $\mathbf b$ $= (b_1, \ldots b_N)$ be sequences of non-negative integers. For each integer $i \not \in \{1, \ldots, N\}$, let $a_i = a_k$ and $b_i = b_k$, where $k \in \{1, \ldots, N\}$ is the integer such that $i-k$ is divisible by $n$. We say $\mathbf a$ is $\mathbf b$-[i]harmonic[/i] if each $a_i$ equals the following arithmetic mean: \[a_i = \frac{1}{2b_i+1} \sum_{s=-b_i}^{b_i} a_{i+s}.\] Suppose that neither $\mathbf a $ nor $\mathbf b$ is a constant sequence, and that both $\mathbf a$ is $\mathbf b$-[i]harmonic[/i] and $\mathbf b$ is $\mathbf a$-[i]harmonic[/i]. Prove that at least $N+1$ of the numbers $a_1, \ldots, a_N,b_1, \ldots, b_N$ are zero.

In order to determine $\frac{1}{A}$ for $A>0$, one can use the iteration $X_{k+1}=X_{k}(2-AX_{k}),$ where $X_0$ is a selected starting value. Find the limitation, if any, on the starting value $X_0$ so that the above iteration converges to $\frac{1}{A}.$