A finite set $V \in Z^2$ of vectors with integer coordinates is chosen on the plane. Each of them is painted one of the $n$ colors. The color is [i]suitable[/i] for the vector if this vector may be presented as' a linear combination (with integer coefficients) of the vectors from $V$ of this color. It is known,that for any vector from $Z^2$ there exist a suitable color. Find all $n$ such that there must exist a color which is suitable for any vector from $Z^2$ . (V. Lebed)

Acute triangle $\triangle ABC$ satises $AB < AC$. Let the circumcircle of this triangle be $O$, and the midpoint of $BC,CA,AB$ be $D,E,F$. Let $P$ be the intersection of the circle with $AB$ as its diameter and line $DF$, which is in the same side of $C$ with respect to $AB$. Let $Q$ be the intersection of the circle with $AC$ as its diameter and the line $DE$, which is in the same side of $B$ with respect to $AC$. Let $PQ \cap BC = R$, and let the line passing through $R$ and perpendicular to $BC$ meet $AO$ at $X$. Prove that $AX = XR$.

Prove that for any integer $n\geq 2$ it holds that $\dbinom {2n}{n}>\frac {4^n}{2n}$.

Given a positive integer $n$, show that for no set of integers modulo $n$, whose size exceeds $1+\sqrt{n+4}$, is it possible that the pairwise sums of unordered pairs be all distinct.

Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \leq N \leq 2^{2002}$. Prove that it is possible to color every subset of $S$ either black or white so that the following conditions hold: (a) the union of any two white subsets is white; (b) the union of any two black subsets is black; (c) there are exactly $N$ white subsets.

Does there exist such a finite set of real numbers ${M}$ that has at least two distinct elements and has the property that for two numbers, ${a}$, ${b}$, belonging to ${M}$, the number ${2a - b^2}$ is also an element in ${M}$?

Let $ABC$ be a triangle with $\angle BAC \ne 45^o$ and $\angle ABC \ne 135^o$. Let $P$ be the point on the line $AB$ with $\angle CPB = 45^o$. Let $O_1$ and $O_2$ be the centers of the circumcircles of the triangles $ACP$ and $BCP$ respectively. Show that the area of the square $CO_1P O_2$ is equal to the area of the triangle $ABC$.

Given a prime number $p$, a set is said to be $p$-good if the set contains exactly three elements $a, b, c$ and $a + b \equiv c \pmod{p}$. Find all prime number $p$ such that $\{ 1, 2, \cdots, p-1 \}$ can be partitioned into several $p$-good sets. [i]Proposed by capoouo[/i]

Let $ABCD$ be a convex quadrilateral. $AC$ and $BD$ meet at $P$, with $\angle APD=60^{\circ}$. Let $E,F,G$, and $H$ be the midpoints of $AB,BC,CD$ and $DA$ respectively. Find the greatest positive real number $k$ for which \[EG+3HF\ge kd+(1-k)s \] where $s$ is the semi-perimeter of the quadrilateral $ABCD$ and $d$ is the sum of the lengths of its diagonals. When does the equality hold?

$4.$ Let there be a variable positive integer whose last two digits are $3's$. Prove that this number is divisible by a prime greater than $7$.

Calculate the sum of $n$ addends $$7 + 77 + 777 +...+ 7... 7.$$

A red unit cube $ABCDEF GH$ (with $E$ below $A$, $F$ below $B$, etc.) is pushed into the corner of a room with vertex $E$ not visible, so that faces $ABF E$ and $ADHE$ are adjacent to the wall and face $EF GH$ is adjacent to the floor. A string of length $2$ is dipped in black paint, and one of its endpoints is attached to vertex $A$. How much surface area on the three visible faces of the cube can be painted black by sweeping the string over it?

Determine all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the set \[\left \{ \frac{f(x)}{x}: x \neq 0 \textnormal{ and } x \in \mathbb{R}\right \}\] is finite, and for all $x \in \mathbb{R}$ \[f(x-1-f(x)) = f(x) - x - 1\]

Given a triangle $ABC$, let $I$ be the incenter, and $J$ be the $A$-excenter. A line $\ell$ through $A$ perpendicular to $BC$ intersect the lines $BI$, $CI$, $BJ$, $CJ$ at $P$, $Q$, $R$, $S$ respectively. Suppose the angle bisector of $\angle BAC$ meet $BC$ at $K$, and $L$ is a point such that $AL$ is a diameter in $(ABC)$. Prove that the line $KL$, $\ell$, and the line through the centers of circles $(IPQ)$ and $(JRS)$, are concurrent. [i]Proposed by Chuah Jia Herng & Ivan Chan Kai Chin[/i]

Are there positive integers $a$ and $b$ such that both $a^2 + 4b$ and $b^2 + 4a$ are perfect squares?

The escalator of the station [b]champion butcher[/b] has this property that if $m$ persons are on it, then it's speed is $m^{-\alpha}$ where $\alpha$ is a fixed positive real number. Suppose that $n$ persons want to go up by the escalator and the width of the stairs is such that all the persons can stand on a stair. If the length of the escalator is $l$, what's the least time that is needed for these persons to go up? Why? [i]proposed by Mohammad Ghiasi[/i]

Determine the largest natural number $ N $ having the following property: every $ 5\times 5 $ array consisting of pairwise distinct natural numbers from $ 1 $ to $ 25 $ contains a $ 2\times 2 $ subarray of numbers whose sum is, at least, $ N. $ [i]Demetres Christofides[/i] and [i]Silouan Brazitikos[/i]

Find $\lim_{n\to\infty} \left(\frac{_{3n}C_n}{_{2n}C_n}\right)^{\frac{1}{n}}$ where $_iC_j$ is a binominal coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.

What fraction of a $5$-dimensional cube is the volume of the inscribed sphere? What fraction is it of a $10$-dimensional cube?

What is the last three digits of base-4 representation of $10\cdot 3^{195}\cdot 49^{49}$? $ \textbf{(A)}\ 112 \qquad\textbf{(B)}\ 130 \qquad\textbf{(C)}\ 132 \qquad\textbf{(D)}\ 212 \qquad\textbf{(E)}\ 232 $

Ephram is growing $3$ different variants of radishes in a row of $13$ radishes total, but he forgot where he planted each radish variant and he can't tell what variant a radish is before he picks it. Ephram knows that he planted at least one of each radish variant, and all radishes of one variant will form a consecutive string, with all such possibilities having an equal chance of occurring. He wants to pick three radishes to bring to the farmers market, and wants them to all be of different variants. Given that he uses optimal strategy, the probability that he achieves this can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Jeff Lin[/i]

Point $A$ is located in this circle of radius $1$. An arbitrary chord is drawn through it, and then a circle of radius $2$ is drawn through the ends of this chord. Prove that all such circles touch some fixed circle, not depending from the initial choice of the chord.

Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Show that $$\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+a^2}\geq \frac{3}{2}.$$

For each real number $x$, let $\textbf{[}x\textbf{]}$ be the largest integer not exceeding $x$ (i.e., the integer $n$ such that $n\le x<n+1$). Which of the following statements is (are) true? $\textbf{I. [}x+1\textbf{]}=\textbf{[}x\textbf{]}+1\text{ for all }x$ $\textbf{II. [}x+y\textbf{]}=\textbf{[}x\textbf{]}+\textbf{[}y\textbf{]}\text{ for all }x\text{ and }y$ $\textbf{III. [}xy\textbf{]}=\textbf{[}x\textbf{]}\textbf{[}y\textbf{]}\text{ for all }x\text{ and }y$ $\textbf{(A) }\text{none}\qquad\textbf{(B) }\textbf{I }\text{only}\qquad\textbf{(C) }\textbf{I}\text{ and }\textbf{II}\text{ only}\qquad\textbf{(D) }\textbf{III }\text{only}\qquad \textbf{(E) }\text{all}$

Let $Q(z)$ and $R(z)$ be the unique polynomials such that $$z^{2021}+1=(z^2+z+1)Q(z)+R(z)$$ and the degree of $R$ is less than $2.$ What is $R(z)?$ $\textbf{(A) }-z \qquad \textbf{(B) }-1 \qquad \textbf{(C) }2021\qquad \textbf{(D) }z+1 \qquad \textbf{(E) }2z+1$