Prove that if $f(x)$ is a polynomial with integer coefficients and there exists an integer $k$ such that none of $f(1),\ldots,f(k)$ is divisible by $k$, then $f(x)$ has no integral root.

Let $n$ be a fixed odd positive integer. For each odd prime $p$, define $$a_p=\frac{1}{p-1}\sum_{k=1}^{\frac{p-1}{2}}\bigg\{\frac{k^{2n}}{p}\bigg\}.$$ Prove that there is a real number $c$ such that $a_p = c$ for infinitely many primes $p$. [i]Note: $\left\{x\right\} = x - \left\lfloor x\right\rfloor$ is the fractional part of $x$.[/i]

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

Given a pentagon of area $1993$ and $995$ points inside the pentagon, let $S$ be the set containing the vertices of the pentagon and the $995$ points. Show that we can find three points of $S$ which form a triangle of area $\le 1$.

A mathematical competition was attended by 120 participants from several contingents. At the closing ceremony, each participant gave 1 souvenir each to every other participants from the same contingent, and 1 souvenir to any person from every other contingents. It is known that there are 3840 souvenirs whom were exchanged. Find the maximum possible contingents such that the above condition still holds? [i]Raymond Christopher Sitorus, Singapore[/i]

Denote by $d(n)$ be the biggest prime divisor of $|n|>1$. Find all polynomials with integer coefficients satisfy; $$P(n+d(n))=n+d(P(n)) $$ for the all $|n|>1$ integers such that $P(n)>1$ and $d(P(n))$ can be defined.

Let $ a,b,c $ be positive real numbers, prove the inequality: $ (a+b+c)^2+ab+bc+ac\geq 6\sqrt{abc(a+b+c)} $

Every phone number in an area consists of eight digits and starts with digit $ 8$. Mr Edy, who has just moved to the area, apply for a new phone number. What is the chance that Mr Edy gets a phone number which consists of at most five different digits?

A given line passes through the center $O$ of a circle. The line intersects the circle at points $A$ and $B$. Point $P$ lies in the exterior of the circle and does not lie on the line $AB$. Using only an unmarked straightedge, construct a line through $P$, perpendicular to the line $AB$. Give complete instructions for the construction and prove that it works.

A ladder style tournament is held with $2016$ participants. The players begin seeded $1,2,\cdots 2016$. Each round, the lowest remaining seeded player plays the second lowest remaining seeded player, and the loser of the game gets eliminated from the tournament. After $2015$ rounds, one player remains who wins the tournament. If each player has probability of $\tfrac{1}{2}$ to win any game, then the probability that the winner of the tournament began with an even seed can be expressed has $\tfrac{p}{q}$ for coprime positive integers $p$ and $q$. Find the remainder when $p$ is divided by $1000$. [i]Proposed by Nathan Ramesh

Let $ A_n $ be the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that every two neighbouring terms of each subsequence have different parity,and $ B_n $ the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that all the terms of each subsequence have the same parity ( for example,the partition $ {(1,4,5,8),(2,3),(6,9),(7)} $ is an element of $ A_9 $,and the partition $ {(1,3,5),(2,4),(6)} $ is an element of $ B_6 $ ). Prove that for every positive integer $ n $ the sets $ A_n $ and $ B_{n+1} $ contain the same number of elements.

In $\triangle ABC$, $D,E,F$ are respectively the midpoints of $AB, BC, and CA$. Futhermore $AB=10$, $CD=9$, $CD\perp AE$. Find $BF$.

Let $n$ be a fixed positive integer. Compute over $\mathbb{R}$ the minimum of the following polynomial: \[f(x)=\sum_{t=0}^{2n}(2n+1-t)x^t.\]

Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n ?$ $\textbf{(A)} ~47 \qquad\textbf{(B)} ~94 \qquad\textbf{(C)} ~227 \qquad\textbf{(D)} ~471 \qquad\textbf{(E)} ~542$

Let $a_1 , a_2 , ..., a_n$ be non-zero integers that satisfy the equation $$a_1 +\dfrac{1}{a_2+\dfrac{1}{a_3+ ... \dfrac{1}{a_n+\dfrac{1}{x}} } } = x$$ for all values of $x$ for which the lefthand side of the equation makes sense. (a) Prove that $n$ is even. (b) What is the smallest n for which such numbers $a_1 , a_2 , ..., a_n$ exist? (M Skopenko)

Given an acute-angled triangle $ABC$, in which $P$, $M$, $N$ are the midpoints of the sides $AB$, $BC$, $AC$, respectively. A point $H$ is taken inside the triangle and perpendiculars $HK$, $HS$, $HQ$ are lowered from it to the sides $AB$, $BC$, $AC$, respectively ($K \in AB$, $S \in BC$, $Q \in AC$). It turned out that $MK = MQ$, $NS = NK$, $PS=PQ$. Prove that $H$ is the point of intersection of the altitudes of triangle $ABC$.

Let $ABC$ be a triangle with $AB = 3$, $BC = 5$, $AC = 7$, and let $ P$ be a point in its interior. If $G_A$, $G_B$, $G_C$ are the centroids of $\vartriangle PBC$, $\vartriangle PAC$, $\vartriangle PAB$, respectively, find the maximum possible area of $\vartriangle G_AG_BG_C$.

A positive integer is a [i]palindrome [/i] if it is written identically forwards and backwards. For example, $285582$ is a palindrome. A six digit number $ABCDEF$, where $A, B, C, D, E, F$ are digits, is called [i]cozy [/i] if $AB$ divides $CD$ and $CD$ divides $EF$. For example, $164896$ is cozy. Determine all cozy palindromes.

Let $k\geqslant 2$ be an integer, and $a,b$ be real numbers. prove that $a-b$ is an integer divisible by $k$ if and only if for every positive integer $n$ $$\lfloor an \rfloor \equiv \lfloor bn \rfloor \ (mod \ k)$$ Proposed by Navid Safaei

In a class of seven students, a poll is conducted. The poll asks $~$(a) Were you born after $2021$? $~$(b) Were you born after $2019$? $~$(c) Were you born in $2020$? Five students respond "yes" to question (a), and four students respond "no" to question (b). If everyone truthfully answers all questions, how many students responded "yes" to (c)? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

The circumference of a circle is divided into eight arcs by a convex quadrilateral $ABCD$ with four arcs lying inside the quadrilateral and the remaining four lying outside it. The lengths of the arcs lying inside the quadrilateral are denoted by $p,q,r,s$ in counter-clockwise direction. Suppose $p+r = q+s$. Prove that $ABCD$ is cyclic.

Solve in the set of real numbers the equation $ a^{[ x ]} +\log_a\{ x \} =x , $ where $ a $ is a real number from the interval $ (0,1). $ $ [] $ and $ \{\} $ [i]denote the floor, respectively, the fractional part.[/i]

$ABCD$ is a square of center $O$. On the sides $DC$ and $AD$ the equilateral triangles DAF and DCE have been constructed. Decide if the area of the $EDF$ triangle is greater, less or equal to the area of the $DOC$ triangle. [img]https://4.bp.blogspot.com/-o0lhdRfRxl0/XNYtJgpJMmI/AAAAAAAAKKg/lmj7KofAJosBZBJcLNH0JKjW3o17CEMkACK4BGAYYCw/s1600/may4_2.gif[/img]

Problems 15, 16, and 17 all refer to the following: In the very center of the Irenic Sea lie the beautiful Nisos Isles. In 1998 the number of people on these islands is only 200, but the population triples every 25 years. Queen Irene has decreed that there must be at least 1.5 square miles for every person living in the Isles. The total area of the Nisos Isles is 24,900 square miles. 17. In how many years, approximately, from 1998 will the population of Nisos be as much as Queen Irene has proclaimed that the islands can support? $ \text{(A)}\ 50\text{ yrs.}\qquad\text{(B)}\ 75\text{ yrs.}\qquad\text{(C)}\ 100\text{ yrs.}\qquad\text{(D)}\ 125\text{ yrs.}\qquad\text{(E)}\ 150\text{ yrs.} $

Find all pairs of natural numbers $(m, n)$ for which $2^m3^n +1$ is the square of some integer.