The prime number $p$ and a positive integer $k$ are given. Assume that $P(x)\in \mathbb Z[X]$ is a polynomial with coefficients in the set $\{0,1,\cdots,p-1\}$ with least degree which satisfies the following property: There exists a permutaion of numbers $1,2,\cdots,p-1$ around a circle such that for any $k$ consecutive numbers $a_1,a_2,\cdots,a_k$ one has $$ p | P(a_1)+P(a_2)+\cdots+ P(a_k). $$ Prove that $P(x)$ is of the form $ax^d+b$. Proposed by [i]Yahya Motevassel[/i]

There are $n\geq 2$ people at a meeting. Show that there exist two people at the meeting who have the same number of friends among the persons at the meeting. (It is assumed that if $A$ is a friend of $B,$ then $B$ is a friend of $A;$ moreover, nobody is his own friend.)

Let $a$, $b$ and $m$ be three positive real numbers and $a>b$. Which of the numbers $A=\sqrt{a+m}-\sqrt{a}$ and $B=\sqrt{b+m}-\sqrt{b}$ is bigger:

The Bell Zoo has the same number of rhinoceroses as the Carlton Zoo has lions. The Bell Zoo has three more elephants than the Carlton Zoo has lions. The Bell Zoo has the same number of elephants as the Carlton Zoo has rhinoceroses. The Carlton Zoo has two more elephants than rhinoceroses. The Carlton Zoo has twice as many monkeys as it has rhinoceroses, elephants, and lions combined, and it has two more penguins than monkeys. The Bell Zoo has two-thirds as many monkeys as the Carlton Zoo has penguins. The Bell Zoo has two more penguins than monkeys but only half as many lions as penguins. The total of the numbers of rhinoceroses, elephants, lions, monkeys, and penguins in the Bell Zoo is $48$. Find the total of the numbers of rhinoceroses, elephants, lions, monkeys, and penguins in the Carlton Zoo.

Let $n$ be a positive integer. At most how many distinct unit vectors can be selected in $\mathbb{R}^n$ such that from any three of them, at least two are orthogonal?

Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}$, $BC = CD = 43$, and $\overline{AD} \perp \overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. GIven that $OP = 11$, the length $AD$ can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m + n$? $\textbf{(A)}\: 65\qquad\textbf{(B)}\: 132\qquad\textbf{(C)}\: 157\qquad\textbf{(D)}\: 194\qquad\textbf{(E)}\: 215$

a) Points $A_1, A_2, A_3, A_4, A_5, A_6$ divide a circle of radius $1$ into six equal arcs. Ray $\ell_1$ from $A_1$ connects $A_1$ with $A_2$, ray $\ell_2$ from $A_2$ connects $A_2$ with $A_3$, and so on, ray $\ell_6$ from $A_6$ connects $A_6$ with $A_1$. From a point $B_1$ on $\ell_1$ the perpendicular is drawn on $\ell_6$, from the foot of this perpendicular another perpendicular is drawn on $\ell_5$, and so on. Let the foot of the $6$-th perpendicular coincide with $B_1$. Find the length of segment $A_1B_1$. b) Find points $B_1, B_2,... , B_n$ on the extensions of sides $A_1A_2, A_2A_3,... , A_nA_1$ of a regular $n$-gon $A_1A_2...A_n$ such that $B_1B_2 \perp A_1A_2$, $B_2B_3 \perp A_2A_3$,$ . . . $, $B_nB_1 \perp A_nA_1$.

Let $n \in N$ such that $1 + 2 + ... + n$ is divisible by $3$. Integers $a_1\ge a_2\ge a_3\ge 2$ have sum $n$ and they satisfy $1 + 2 + ... + a_1\le \frac{1}{3}( 1 + 2 + ... + n ) $ and $1 + 2 + ... + (a_1+ a_2) \le \frac{2}{3}( 1 + 2 + ... + n )$. Prove that there is a partition of $\{ 1 , 2 , ... , n\}$ in three subsets $A_1, A_2, A_3$ with cardinals $| A_i| = a_i, i = 1 , 2 , 3$, and with equal sums of their elements .

Suppose that $ ABCD$ is a cyclic quadrilateral and $ CD\equal{}DA$. Points $ E$ and $ F$ belong to the segments $ AB$ and $ BC$ respectively, and $ \angle ADC\equal{}2\angle EDF$. Segments $ DK$ and $ DM$ are height and median of triangle $ DEF$, respectively. $ L$ is the point symmetric to $ K$ with respect to $ M$. Prove that the lines $ DM$ and $ BL$ are parallel.

Prove that there exist infinitely many positive integers which can't be represented as sum of less than $10$ odd positive integers' perfect squares.

For how many permutations $\pi$ of $\{1,2,\ldots,9\}$ does there exist an integer $N$ such that \[N\equiv \pi(i)\pmod{i}\text{ for all integers }1\leq i\leq 9?\]

Allen and Brian are playing a game in which they roll a $6$-sided die until one of them wins. Allen wins if two consecutive rolls are equal and at most 3. Brian wins if two consecutive rolls add up to $7$ and the latter is at most $3$. What is the probability that Allen wins

A positive integer $N$ has the digits $1, 2, 3, 4, 5, 6$ and $7$, so that each digit $i$, $i \in \{1, 2, 3, 4, 5, 6, 7\}$ occurs $4i$ times in the decimal representation of $N$. Prove that $N$ is not a perfect square.

A triangle $ABC$ is given with barycentre $G$ and circumcentre $O$. The perpendicular bisectors of $GA, GB$ meet at $C_1$,of $GB,GC$ meet at $A _1$, and $GC,GA$ meet at $B_1$. Prove that $O$ is the barycenter of the triangle $A_1B_1C_1$.

Prove that for all $a,b>0$ with $a+b=2$, we have $$\left(a^n+1\right)\left(b^n+1\right)\ge4$$ for all $n\in\mathbb N_{\ge2}$.

There are unique integers $ a_2, a_3, a_4, a_5, a_6, a_7$ such that \[ \frac {5}{7} \equal{} \frac {a_2}{2!} \plus{} \frac {a_3}{3!} \plus{} \frac {a_4}{4!} \plus{} \frac {a_5}{5!} \plus{} \frac {a_6}{6!} \plus{} \frac {a_7}{7!},\] where $ 0 \le a_i < i$ for $ i \equal{} 2,3...,7$. Find $ a_2 \plus{} a_3 \plus{} a_4 \plus{} a_5 \plus{} a_6 \plus{} a_7$. $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 12$

Find the units digit of the decimal expansion of \[(15 \plus{} \sqrt{220})^{19} \plus{} (15 \plus{} \sqrt{220})^{82}.\] $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ \text{none of these}$

Let $\triangle ABC$ be a triangle such that $AB=40$ and $AC=30.$ Points $X$ and $Y$ are on the segment $AB$ and $BC,$ respectively such that $AX:BX=3:2$ and $BY:CY=1:4.$ Given that $XY=12,$ the area of $\triangle ABC$ can be written as $a\sqrt{b}$ where $a$ and $b$ are positive integers and $b$ is squarefree. Compute $a+b.$

Let $ABCD$ be a square, and let $E$ be a point external to $ABCD$ such that $AE = CE = 9$ and $BE = 8$. Compute the side length of $ABCD$.

For positive integral $k>1$, we let $p(k)$ be its smallest prime divisor. Given an integer $a_1>2$, we define an infinite sequence $a_n$ by $a_{n+1}=a_n^n-1$ for each $n\geq 1$. For which values of $a_1$ is the sequence $p(a_n)$ bounded?

Let a constant $\alpha$ as $0<\alpha<1$, prove that: $(1)$ There exist a constant $C(\alpha)$ which is only depend on $\alpha$ such that for every $x\ge 0$, $\ln(1+x)\le C(\alpha)x^\alpha$. $(2)$ For every two complex numbers $z_1,z_2$, $|\ln|\frac{z_1}{z_2}||\le C(\alpha)\left(|\frac{z_1-z_2}{z_2}|^\alpha+|\frac{z_2-z_1}{z_1}|^\alpha\right)$.

Let $n>1$ be an integer. Hippo chooses a list of $n$ points in the plane $P_1, \dots, P_n$; some of these points may coincide, but not all of them can be identical. After this, Wombat picks a point from the list $X$ and measures the distances from it to the other $n-1$ points in the list. The average of the resulting $n-1$ numbers will be denoted $m(X)$. Find all values of $n$ for which Hippo can prepare the list in such a way, that for any point $X$ Wombat may pick, he can point to a point $Y$ from the list such that $XY=m(X)$.

Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $\$105$, Dorothy paid $\$125$, and Sammy paid $\$175$. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$? $ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35 $

Let $ABC$ a triangle inscribed in a circle $(O)$ with orthocenter $H$. Two lines $d_1$ and $d_2$ are mutually perpendicular at $H$. Let $d_1$ meet $BC,CA,AB$ at $X_1, Y_1,Z_1$ respectively. Let $A_1B_1C_1$ be a triangle formed by the line through $X_1$ perpendicular to $BC$, the line through $Y_1$ perpendicular to CA, the line through $Z_1$ perpendicular perpendicular to $AB$. Triangle $A_2B_2C_2$ is defined in the same manner. Prove that the circumcircles of triangles $A_1B_1C_1$ and $A_2B_2C_2$ touch each other at a point on $(O)$. Nguyễn Văn Linh

In his bag, Salman has a number of stones. The weight of each stone is not greater than $0.5$ kg and the total weight of the stones is not greater than $2.5$ kg. Prove that Salman can divide his stones into $4$ groups, each group has a total weight not greater than $1$ kg Trần Nam Dũng