Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $E$ be the intersection of $BH$ and $AC$ and let $M$ and $N$ be the midpoints of $HB$ and $HO$, respectively. Let $I$ be the incenter of $AEM$ and $J$ be the intersection of $ME$ and $AI$. If $AO=20$, $AN=17$, and $\angle{ANM}=90^{\circ}$, then $\frac{AI}{AJ}=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Tristan Shin[/i]

A circle $\mathcal C$ and five distinct points $M_1,M_2,M_3,M_4$ and $M$ on $\mathcal C$ are given in the plane. Prove that the product of the distances from $M$ to lines $M_1M_2$ and $M_3M_4$ is equal to the product of the distances from $M$ to the lines $M_1M_3$ and $M_2M_4$. What can one deduce for $2n+1$ distinct points $M_1,\ldots,M_{2n},M$ on $\mathcal C$?

Let $\Gamma$ consist of all polynomials in $x$ with integer coefficients. For $f$ and $g$ in $\Gamma$ and $m$ a positive integer, let $f \equiv g \pmod{m}$ mean that every coefficient of $f-g$ is an integral multiple of $m$. Let $n$ and $p$ be positive integers with $p$ prime. Given that $f,g,h,r$ and $s$ are in $\Gamma$ with $rf+sg\equiv 1 \pmod{p}$ and $fg \equiv h \pmod{p}$, prove that there exist $F$ and $G$ in $\Gamma$ with $F \equiv f \pmod{p}$, $G \equiv g \pmod{p}$, and $FG \equiv h \pmod{p^n}$.

It is considered a regular polygon with $n$ sides, where $n(n>3)$ is an odd number that does not divide by 3. From the vertices of the polygon are arbitrarily chosen $m(0\leq m\leq n)$ vertices that are colored in red and the others in black. A triangle with the vertices at the vertices of the polygon it is considered $monocolor$ ,if all of its vertices are of the same color. Prove that the number of all $monocolor$ isosceles triangles with the vertices at the given polygon ends does not depend on the way of coloring of the vertices of the polygon. Determine the number of all these $monocolor$ isosceles triangles.

Karl starts with $n$ cards labeled $1,2,3,\dots,n$ lined up in a random order on his desk. He calls a pair $(a,b)$ of these cards [i]swapped[/i] if $a>b$ and the card labeled $a$ is to the left of the card labeled $b$. For instance, in the sequence of cards $3,1,4,2$, there are three swapped pairs of cards, $(3,1)$, $(3,2)$, and $(4,2)$. He picks up the card labeled 1 and inserts it back into the sequence in the opposite position: if the card labeled 1 had $i$ card to its left, then it now has $i$ cards to its right. He then picks up the card labeled $2$ and reinserts it in the same manner, and so on until he has picked up and put back each of the cards $1,2,\dots,n$ exactly once in that order. (For example, the process starting at $3,1,4,2$ would be $3,1,4,2\to 3,4,1,2\to 2,3,4,1\to 2,4,3,1\to 2,3,4,1$.) Show that, no matter what lineup of cards Karl started with, his final lineup has the same number of swapped pairs as the starting lineup.

Let $n \geq 2$ be a positive integer. Let $\mathcal{R}$ be a connected set of unit squares on a grid. A [i]bar[/i] is a rectangle of length or width $1$ which is fully contained in $\mathcal{R}$. A bar is [i]special[/i] if it is not fully contained within any larger bar. Given that $\mathcal{R}$ contains special bars of sizes $1 \times 2,1 \times 3,\ldots,1 \times n$, find the smallest possible number of unit squares in $\mathcal{R}$. [i]Srinivas Arun[/i]

A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\sqrt{50}$ cm, the length of $AB$ is 6 cm, and that of $BC$ is 2 cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the circle. [asy] size(150); defaultpen(linewidth(0.65)+fontsize(11)); real r=10; pair O=(0,0),A=r*dir(45),B=(A.x,A.y-r),C; path P=circle(O,r); C=intersectionpoint(B--(B.x+r,B.y),P); draw(Arc(O, r, 45, 360-17.0312)); draw(A--B--C);dot(A); dot(B); dot(C); label("$A$",A,NE); label("$B$",B,SW); label("$C$",C,SE); [/asy]

Let \(d(n)\) denote the number of positive divisors of \(n\). The sequence \(a_0\), \(a_1\), \(a_2\), \(\ldots\) is defined as follows: \(a_0=1\), and for all integers \(n\ge1\), \[a_n=d(a_{n-1})+d(d(a_{n-2}))+\cdots+ {\underbrace{d(d(\ldots d(a_0)\ldots))}_{n\text{ times}}}.\] Show that for all integers \(n\ge1\), we have \(a_n\le3n\). [i]Proposed by Karthik Vedula[/i]

A piece of graph paper is folded once so that $ (0,2)$ is matched with $ (4,0)$ and $ (7,3)$ is matched with $ (m,n)$. Find $ m \plus{} n$. $ \textbf{(A)}\ 6.7\qquad \textbf{(B)}\ 6.8\qquad \textbf{(C)}\ 6.9\qquad \textbf{(D)}\ 7.0\qquad \textbf{(E)}\ 8.0$

Triangle $ABC$ has incenter $I$ and circumcenter $O$. $AI, BI, CI$ intersect the circumcircle of $ABC$ again at $M_A, M_B, M_C$, respectively. Show that the Euler line of $BIC$ passes through the circumcenter of $OM_BM_C$. (houkai)

For which real numbers $a$ does the sequence $(u_n )$ defined by the initial condition $u_0 =a$ and the recursion $u_{n+1} =2u_n - n^2$ have $u_n >0$ for all $n \geq 0?$

Let $ABCD$ be convex quadrilateral, such that exist $M,N$ inside $ABCD$ for which $\angle NAD= \angle MAB; \angle NBC= \angle MBA; \angle MCB=\angle NCD; \angle NDA=\angle MDC$ Prove, that $S_{ABM}+S_{ABN}+S_{CDM}+S_{CDN}=S_{BCM}+S_{BCN}+S_{ADM}+S_{ADN}$, where $S_{XYZ}$-area of triangle $XYZ$

Let $ f(x)$ be a nonnegative, integrable function on $ (0,2\pi)$ whose Fourier series is $ f(x)\equal{}a_0\plus{}\sum_{k\equal{}1}^{\infty} a_k \cos (n_k x)$, where none of the positive integers $ n_k$ divides another. Prove that $ |a_k| \leq a_0$. [i]G. Halasz[/i]

The bases of a trapezoid have lengths 10 and 21, and the legs have lengths $\sqrt{34}$ and $3 \sqrt{5}$. What is the area of the trapezoid?

Find the number of nonnegative integers $n$ for which $(n^2 - 3n +1)^2 + 1$ is a prime number.

For each positive integer $n$, denote by $a_{n}$ the first digit of $2^{n}$ (base ten). Is the number $0.a_{1}a_{2}a_{3}\cdots$ rational?