Show that any real coefficient polynomial $f(x,y)$ is a linear combination of polynomials of the form $(x+ay)^k$. ($a$ is a real number and $k$ is a non-negative integer.)

Find the maximum possible value of the inradius of a triangle whose vertices lie in the interior, or on the boundary, of a unit square.

A farmer wants to create a rectangular plot along the side of a barn where the barn forms one side of the rectangle and a fence forms the other three sides. The farmer will build the fence by tting together $75$ straight sections of fence which are each $4$ feet long. The farmer will build the fence to maximize the area of the rectangular plot. Find the length in feet along the side of the barn of this rectangular plot.

Let $ ABC$ be an equilateral triangle with side length 2, and let $ \Gamma$ be a circle with radius $ \frac {1}{2}$ centered at the center of the equilateral triangle. Determine the length of the shortest path that starts somewhere on $ \Gamma$, visits all three sides of $ ABC$, and ends somewhere on $ \Gamma$ (not necessarily at the starting point). Express your answer in the form of $ \sqrt p \minus{} q$, where $ p$ and $ q$ are rational numbers written as reduced fractions.

Is there any polynomial $P(x)$ with degree $n$ such that $ \underbrace{P(...(P(x))...)}_{m\,\, times \,\, P}$ has all roots from $1,2,..., mn$ ?

[b]6.[/b] Let $\{ n_k \}_{k=1}^{\infty}$ be a stricly increasing sequence of positive integers such that $\lim_{k \to \infty} n_k^{\frac {1}{2^k}}= \infty$ Show that the sum of the series $\sum_{k=1}^{\infty} \frac {1}{n_k} $ is an irrational number. [b](N. 19)[/b]

A participant is given a deck of thirteen cards numerated from 1 to 13, from which he chooses seven and gives them to the assistant. Then the assistant chooses three of these seven cards and the participant – one of the remaining six in his hand. The magician then takes the chosen four cards (arranged by the participant) and guesses which one is chosen from the participant. What should the magician and assistant do so that the magic trick always happens?

Compute the sum of all real solutions to $4^x - 2021 \cdot 2^x + 1024 = 0$.

Find all integer sequences of the form $ x_i, 1 \le i \le 1997$, that satisfy $ \sum_{k\equal{}1}^{1997} 2^{k\minus{}1} x_{k}^{1997}\equal{}1996\prod_{k\equal{}1}^{1997}x_k$.

Let $\triangle ABC$ be an acute-angled triangle with $AB \not= AC$. Let $H$ be the orthocenter of triangle $ABC$, and let $M$ be the midpoint of the side $BC$. Let $D$ be a point on the side $AB$ and $E$ a point on the side $AC$ such that $AE=AD$ and the points $D$, $H$, $E$ are on the same line. Prove that the line $HM$ is perpendicular to the common chord of the circumscribed circles of triangle $\triangle ABC$ and triangle $\triangle ADE$.

Let $ABC$ be an acute triangle and $M$ be the midpoint of $AB$. A circle through the points $B$ and $C$ intersects the segments $CM$ and $BM$ at points $P$ and $Q$ respectively. Point $K$ is symmetric to $P$ with respect to point $M$. The circumcircles of $\triangle AKM$ and $\triangle CQM$ intersect for the second time at $X$. The circumcircles of $\triangle AMC$ and $\triangle KMQ$ intersect for the second time at $Y$. The segments $BP$ and $CQ$ intersect at point $T$. Prove that the line $MT$ is tangent to the circumcircle of $\triangle MXY$.

Let $k$ be a positive integer and $P$ a point in the plane. We wish to draw lines, none passing through $P$, in such a way that any ray starting from $P$ intersects at least $k$ of these lines. Determine the smallest number of lines needed.

Show that the polynomial $x^{8} +98 x^{4}+1$ can be expressed as the product of two nonconstant polynomials with integer coefficients.

$\boxed{A2}$ Find the maximum value of $z+x$ if $x,y,z$ are satisfying the given conditions.$x^2+y^2=4$ $z^2+t^2=9$ $xt+yz\geq 6$

Find all reals $x$ satisfying $0 \le x \le 5$ and $\lfloor x^2-2x \rfloor = \lfloor x \rfloor ^2 - 2 \lfloor x \rfloor$.

In the usual coordinate system $ xOy$ a line $ d$ intersect the circles $ C_1:$ $ (x\plus{}1)^2\plus{}y^2\equal{}1$ and $ C_2:$ $ (x\minus{}2)^2\plus{}y^2\equal{}4$ in the points $ A,B,C$ and $ D$ (in this order). It is known that $ A\left(\minus{}\frac32,\frac{\sqrt3}2\right)$ and $ \angle{BOC}\equal{}60^{\circ}$. All the $ Oy$ coordinates of these $ 4$ points are positive. Find the slope of $ d$.

Let $x, y, z$ be real numbers that are greater than or equal to $0$ and less than or equal to $\frac{1}{2}$ [list] [*] (a) Determine the minimum possible value of \[x+y+z-xy-yz-zx\] and determine all triples $(x,y,z)$ for which this minimum is obtained. [*] (b) Determine the maximum possible value of \[x+y+z-xy-yz-zx\] and determine all triples $(x,y,z)$ for which this maximum is obtained.[/list]

A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of \[ \sum_{i = 1}^{2012} | a_i - i |, \] then compute the sum of the prime factors of $S$. [i]Proposed by Aaron Lin[/i]

Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

Let $ABC$ be an acute, scalene triangle with circumcenter $O$ and symmedian point $K$. Let $X$ be the point on the circumcircle of triangle $BOC$ such that $\angle AXO = 90^\circ$. Assume that $X\neq K$. The hyperbola passing through $B$, $C$, $O$, $K$, and $X$ intersects the circumcircle of triangle $ABC$ at points $U$ and $V$, distinct from $B$ and $C$. Prove that $UV$ is the perpendicular bisector of $AX$. [i]The symmedian point of triangle $ABC$ is the intersection of the reflections of $B$-median and $C$-median across the angle bisectors of $\angle ABC$ and $\angle ACB$, respectively.[/i] [i]Pitchayut Saengrungkongka[/i]

Given a point $O$ and lengths $x, y, z$, prove that there exists an equilateral triangle $ABC$ for which $OA = x, OB = y, OC = z$, if and only if $x+y \geq z, y+z \geq x, z+x \geq y$ (the points $O,A,B,C$ are coplanar).

Consider an array of numbers of size $8 \times 8$. Each of the numbers in the array equals 1 or -1. "Doing a move" means that you pick any number in the array and you change the sign of all numbers which are in the same row or column as the number you picked. (This includes changing the sign of the "chosen" number itself.) Prove that one can transform any given array into an array containing numbers +1 only by performing this kind of moves repeatedly.

Suppose all the pairs of a positive integers from a finite collection \[A=\{a_{1}, a_{2}, \cdots \}\] are added together to form a new collection \[A^{*}=\{a_{i}+a_{j}\;\; \vert \; 1 \le i < j \le n \}.\] For example, $A=\{ 2, 3, 4, 7 \}$ would yield $A^{*}=\{ 5, 6, 7, 9, 10, 11 \}$ and $B=\{ 1, 4, 5, 6 \}$ would give $B^{*}=\{ 5, 6, 7, 9, 10, 11 \}$. These examples show that it's possible for different collections $A$ and $B$ to generate the same collections $A^{*}$ and $B^{*}$. Show that if $A^{*}=B^{*}$ for different sets $A$ and $B$, then $|A|=|B|$ and $|A|=|B|$ must be a power of $2$.

Vasya conceived a two-digit number $a$, and Petya is trying to guess it. To do this, he tells Vasya a natural number $k$, and Vasya tells Petya the sum of the digits of the number $ka$. What is the smallest number of questions that Petya has to ask so that he can certainly be able to determine Vasya’s number?

Real function $f$ [b]generates[/b] real function $g$ if there exists a natural $k$ such that $f^k=g$ and we show this by $f \rightarrow g$. In this question we are trying to find some properties for relation $\rightarrow$, for example it's trivial that if $f \rightarrow g$ and $g \rightarrow h$ then $f \rightarrow h$.(transitivity) (a) Give an example of two real functions $f,g$ such that $f\not = g$ ,$f\rightarrow g$ and $g\rightarrow f$. (b) Prove that for each real function $f$ there exists a finite number of real functions $g$ such that $f \rightarrow g$ and $g \rightarrow f$. (c) Does there exist a real function $g$ such that no function generates it, except for $g$ itself? (d) Does there exist a real function which generates both $x^3$ and $x^5$? (e) Prove that if a function generates two polynomials of degree 1 $P,Q$ then there exists a polynomial $R$ of degree 1 which generates $P$ and $Q$. Time allowed for this problem was 75 minutes.