Determine if there exist five consecutive positive integers such that their LCM is a perfect square.

On the sides $AB$ and $BC$ of triangle $ABC$, points $M$ and $K$ are taken, respectively, so that $S_{KMC} + S_{KAC}=S_{ABC}$. Prove that all such lines $MK$ pass through one point.

In this figure $\angle RFS = \angle FDR$, $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, $FS = 7\dfrac{1}{2}$ inches. The length of $RS$, in inches, is: [asy] import olympiad; pair F,R,S,D; F=origin; R=5*dir(aCos(9/16)); S=(7.5,0); D=4*dir(aCos(9/16)+aCos(1/8)); label("$F$",F,SW);label("$R$",R,N); label("$S$",S,SE); label("$D$",D,W); label("$7\frac{1}{2}$",(F+S)/2.5,SE); label("$4$",midpoint(F--D),SW); label("$5$",midpoint(F--R),W); label("$6$",midpoint(D--R),N); draw(F--D--R--F--S--R); markscalefactor=0.1; draw(anglemark(S,F,R)); draw(anglemark(F,D,R)); //Credit to throwaway1489 for the diagram[/asy] $\textbf{(A)}\ \text{undetermined} \qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\dfrac{1}{2} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 6\dfrac{1}{4}$

Let $N$ be the number of ways to colour each cell in a $2 \times 50$ rectangle either red or blue such that each $2 \times 2$ block contains at least one blue cell. Show that $N$ is a multiple of $3^{25}$, but not a multiple of $3^{26}$

Let $ABC$ be a triangle with incenter $I$ and let $AI$ meet $BC$ at $D$. Let $E$ be a point on the segment $AC$, such that $CD=CE$ and let $F$ be on the segment $AB$ such that $BF=BD$. Let $(CEI) \cap (DFI)=P \neq I$ and $(BFI) \cap (DEI)=Q \neq I$. Prove that $PQ \perp BC$. [i]Proposed by Leonardo Franchi, Italy[/i]

Determine all positive integer solutions $(x, y, z, t)$ of the equation \[(x+y)(y+z)(z+x)=xyzt\] for which $\gcd(x, y)=\gcd(y, z)=\gcd(z, x)=1$.

A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square? $ \textbf{(A)}\ \frac{\pi}{16}\qquad \textbf{(B)}\ \frac{\pi}{8}\qquad \textbf{(C)}\ \frac{3\pi}{16}\qquad \textbf{(D)}\ \frac{\pi}{4}\qquad \textbf{(E)}\ \frac{\pi}{2}$

Let $f(X)$ be a monic irreducible polynomial over $\mathbb{Z}$; therefore, by Gauss's Lemma, $f$ is also irreducible over $\mathbb{Q}$ (you may assume this). Moreover, assume $f(X) \mid f\left(X^2+n\right)$ where $n$ is an integer such that $n \notin\{-1,0,1\}$. Show that $n^2 \nmid f(0)$.

Let $ABC$ be a triangle with incircle $\omega$, incenter $I$ and circumcircle $\Gamma$. Let $D$ be the tangency point of $\omega$ with $BC$, let $M$ be the midpoint of $ID$, and let $A'$ be the diametral opposite of $A$ with respect to $\Gamma$. If we denote $X=A'M\cap \Gamma$, then prove that the circumcircle of triangle $AXD$ is tangent to $BC$.

$2013$ users have registered on the social network "Graph". Some users are friends, and friendship in "Graph" is mutual. It is known that among network users there are no three, each of whom would be friends. Find the biggest one possible number of pairs of friends in "Graph".

Answer the following questions : [b](a)[/b] Evaluate $~~\lim_{x\to 0^{+}} \Big(x^{x^x}-x^x\Big)$ [b](b)[/b] Let $A=\frac{2\pi}{9}$, i.e. $40$ degrees. Calculate the following $$1+\cos A+\cos 2A+\cos 4A+\cos 5A+\cos 7A+\cos 8A$$ [b](c)[/b] Find the number of solutions to $$e^x=\frac{x}{2017}+1$$

Find the sum of all positive rational numbers that are less than $10$ and that have denominator $30$ when written in lowest terms.

Sam and Jonathan play a game where they take turns flipping a weighted coin, and the game ends when one of them wins. The coin has a $\frac89$ chance of landing heads and a $\frac19$ chance of landing tails. Sam wins when he flips heads, and Jonathan wins when he flips tails. Find the probability that Samwins, given that he takes the first turn. [i]Proposed by Samuel Tsui[/i]

A man named Juan has three rectangular solids, each having volume $128$. Two of the faces of one solid have areas $4$ and $32$. Two faces of another solid have areas $64$ and $16$. Finally, two faces of the last solid have areas $8$ and $32$. What is the minimum possible exposed surface area of the tallest tower Juan can construct by stacking his solids one on top of the other, face to face? (Assume that the base of the tower is not exposed.)

If $A_i=\frac{x-a_i}{|x-a_i|}$, $i=1,2,\cdots,n$ for $n$ numbers $a_1<a_2<\cdots<a_m<\cdots<a_n,$ then $\lim \limits_{x\to a_m}\Big(A_1A_2\cdots A_n\Big)=?$ [list=1] [*] $(-1)^{m-1}$ [*] $(-1)^m$ [*] $1$ [*] None of these [/list]

Let $P$ be a point in the interior of $\triangle ABC$. The lengths of the sides of $\triangle ABC$ is $a,b,c$, and the distance from $P$ to the sides of $\triangle ABC$ is $p,q,r$. Show that the circumradius $R$ of $\triangle ABC$ satisfies \[\displaystyle R\le \frac{a^2+b^2+c^2}{18\sqrt[3]{pqr}}.\] When does equality hold?

$ \sin\frac{\pi }{4n}\ge \frac{\sqrt 2 }{2n} ,\quad \forall n\in\mathbb{N} $

An exam was taken by some students. Each problem was worth $1$ point for the correct answer, and $0$ points for an incorrect one. For each question, at least one student answered it correctly. Also, there are two students with different scores on the exam. Prove that there exists a question for which the following holds: The average score of the students who answered the question correctly is greater than the average score of the students who didn't.

Let $ a,\ b,\ c$ be real numbers such that $ a\geq b\geq c\geq 1$. Prove the following inequality: \[ \int_0^1 \{(1\minus{}ax)^3\plus{}(1\minus{}bx)^3\plus{}(1\minus{}cx)^3\minus{}3x\}\ dx\geq ab\plus{}bc\plus{}ca\minus{}\frac 32(a\plus{}b\plus{}c)\minus{}\frac 34abc.\]

At most how many positive integers less than $51$ are there such that no one is triple of another one? $ \textbf{(A)}\ 17 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 38 \qquad\textbf{(D)}\ 39 \qquad\textbf{(E)}\ \text{None of the preceding} $

Find all positive integers $n$ such that $$\lfloor \sqrt{n} \rfloor + \left\lfloor \frac{n}{\lfloor \sqrt{n} \rfloor} \right \rfloor> 2\sqrt{n}.$$ If $k$ is a real number, then $\lfloor k \rfloor$ means the floor of $k$, this is the greatest integer less than or equal to $k$.

Let $n$ be a positive integer. Prove that $$\gcd(\underbrace{11\dots 1}_{n \text{times}},n)\mid 1+10^k+10^{2k}+\dots+10^{(n-1)k}$$ for all positive integer $k$.

Let the incircle of triangle $ABC$ meet the sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, and let the $A$-excircle meet the lines $BC$, $CA$, $AB$ at $P$, $Q$, $R$. Let the line passing through $A$ and perpendicular to $BC$ meet the lines $EF$, $QR$ at $K$, $L$. Let the intersection of $LD$ and $EF$ be $S$, and the intersection of $KP$ and $QR$ be $T$. Prove that $A$, $S$, $T$ are collinear.

Given is a triangle $ABC$ with altitude $AH$ and median $AM$. The line $OH$ meets $AM$ at $D$. Let $AB \cap CD=E, AC \cap BD=F$. If $EH$ and $FH$ meet $(ABC)$ at $X, Y$, prove that $BY, CX, AH$ are concurrent.

Find the least non-negative integer $n$ such that exists a non-negative integer $k$ such that the last 2012 decimal digits of $n^k$ are all $1$'s.