Let $ABC$ be a scalene acute triangle with incentre $I{}$ and circumcentre $O{}$. Let $AI$ cross $BC$ at $D$. On circle $ABC$, let $X$ and $Y$ be the mid-arc points of $ABC$ and $BCA$, respectively. Let $DX{}$ cross $CI{}$ at $E$ and let $DY{}$ cross $BI{}$ at $F{}$. Prove that the lines $FX, EY$ and $IO$ are concurrent on the external bisector of $\angle BAC$. [i]David-Andrei Anghel[/i]

There are 22 black and 3 blue balls in a bag. Ahmet chooses an integer $ n$ in between 1 and 25. Betül draws $ n$ balls from the bag one by one such that no ball is put back to the bag after it is drawn. If exactly 2 of the $ n$ balls are blue and the second blue ball is drawn at $ n^{th}$ order, Ahmet wins, otherwise Betül wins. To increase the possibility to win, Ahmet must choose $\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ 23$

Susan wants to place 35.5 kg of sugar in small bags. If each bag holds 0.5 kg, how many bags are needed? $\textbf{(A)}\ 36 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 53 \qquad \textbf{(D)}\ 70 \qquad \textbf{(E)}\ 71$

Consider the function $$f(x)=\sum_{k=1}^{m}(x-k)^{4}~, \qquad~ x \in \mathbb{R}$$ where $m>1$ is an integer. Show that $f$ has a unique minimum and find the point where the minimum is attained.

Let $\mathbb{R}^{+}$ be the set of all positive real numbers. Find all the functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that for all $x, y \in \mathbb{R}^{+}$, \[ f(x)f(y) = f(y)f(xf(y)) + \frac{1}{xy}. \]

A subset $S$ of the natural numbers is called [i]dense [/i] for every $7$ consecutive natural numbers, at least $5$ of them are in $S$. Show that there exists a dense subset for which the equation $a^2+b^2=c^2$ has no solution for $a,b,c \in S$.

Sequence $\{a_n\}$ defined by recurrence relation $a_{n+1} = 1+\frac{n^2}{a_n}$. Given $a_1>1$, find the value of $\lim\limits_{n\to\infty} \frac{a_n}{n}$ with proof.

[b](a)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ ends with the string $2004$, followed by a number of digits from the set $\left\{0;\;4\right\}$ ? [b](b)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ starts with the string $2004$ ?

For each prime $p$ of the form $4k+3$ with $k \in \mathbb{Z}^+$, consider the polynomial $$Q(x)=px^{2p} - x^{2p-1} + p^2x^{\frac{3p+1}{2}} - px^{p+1} +2(p^2+1)x^p -px^{p-1}+ p^2 x^{\frac{p-1}{2}} -x + p.$$ Determine all ordered pairs of polynomials $f, g$ with integer coefficients such that $Q(x)=f(x)g(x)$.

In how many ways can we change the sign $ *$ with $ \plus{}$ or $ \minus{}$, such that the following equation is true? \[ 1 *2*3*4*5*6*7*8*9*10\equal{}29\]

A nonempty word is called pronounceable if it alternates in vowels (A, E, I, O, U) and consonants (all other letters) and it has at least one vowel. How many pronounceable words can be formed using the letters P, U, M, A, C at most once each? Words of length shorter than $5$ are allowed.

Let $ x,y\in\mathbb{R}$ , and $ x,y \in $ $ \left(0,\frac{\pi}{2}\right) $, and $ m \in \left(2,+\infty\right) $ such that $ \tan x * \tan y = m $ . Find the minimum value of the expression $ E(x,y) = \cos x + \cos y $.

Show that $$ \frac{a^4}{(a+b)\left( a^2+b^2 \right)} +\frac{b^4}{(b+c)\left( b^2+c^2 \right)} +\frac{c^4}{(c+a)\left( c^2+a^2 \right)}\ge \frac{a+b+c}{4} , $$ for any positive real numbers $ a,b,c. $ [i]Costică Ambrinoc[/i]

Magicman and his helper want to do some magic trick. They have special card desk. Back of all cards is common color and face is one of $2017$ colors. Magic trick: magicman go away from scene. Then viewers should put on the table $n>1$ cards in the row face up. Helper looks at these cards, then he turn all cards face down, except one, without changing order in row. Then magicman returns on the scene, looks at cards, then show on the one card, that lays face down and names it face color. What is minimal $n$ such that magicman and his helper can has strategy to make magic trick successfully?

For positive integers $m, n$, denote $$S_m(n)=\sum_{1\le k \le n} \left[ \sqrt[k^2]{k^m}\right]$$ Prove that $S_m(n) \le n + m (\sqrt[4]{2^m}-1)$

In quadrilateral $ABCD$ holds $AB=6$, $AD=4$, $\angle DAB=\angle ABC = 60^{\circ}$ and $\angle ADC = 90^{\circ}$. Find length of diagonals and area of the quadrilateral

In fig. the bisectors of the angles $\angle DAC$, $ \angle EBD$, $\angle ACE$, $\angle BDA$ and $\angle CEB$ intersect at one point. Prove that the bisectors of the angles $\angle TPQ$, $\angle PQR$, $\angle QRS$, $\angle RST$ and $\angle STP$ also intersect at one point. [img]https://cdn.artofproblemsolving.com/attachments/6/e/870e4f20bc7fdcb37534f04541c45b1cd5034a.png[/img]

The figure shows a network of roads bounding $12$ blocks. Person $P$ goes from $A$ to $B,$ and person $Q$ goes from $B$ to $A,$ each going by a shortest path (along roads). The persons start simultaneously and go at the same constant speed. At each point with two possible directions to take, both have the same probability. Find the probability that the persons meet. [asy] import graph; size(150); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; draw((0,3)--(4,3),linewidth(1.2pt)); draw((4,3)--(4,0),linewidth(1.2pt)); draw((4,0)--(0,0),linewidth(1.2pt)); draw((0,0)--(0,3),linewidth(1.2pt)); draw((1,3)--(1,0),linewidth(1.2pt)); draw((2,3)--(2,0),linewidth(1.2pt)); draw((3,3)--(3,0),linewidth(1.2pt)); draw((0,1)--(4,1),linewidth(1.2pt)); draw((4,2)--(0,2),linewidth(1.2pt)); dot((0,0),ds); label("$A$", (-0.3,-0.36),NE*lsf); dot((4,3),ds); label("$B$", (4.16,3.1),NE*lsf); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle); [/asy]

An $8'\text{ X }10'$ table sits in the corner of a square room, as in Figure 1 below. The owners desire to move the table to the position shown in Figure 2. The side of the room is $S$ feet. What is the smallest integer value of $S$ for which the table can be moved as desired without tilting it or taking it apart? [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=(0,0), B=(16,0), C=(16,16), D=(0,16), E=(32,0), F=(48,0), G=(48,16), H=(32,16), I=(0,8), J=(10,8), K=(10,16), L=(32,6), M=(40,6), N=(40,16); draw(A--B--C--D--A^^E--F--G--H--E^^I--J--K^^L--M--N); label("S", (18,8)); label("S", (50,8)); label("Figure 1", (A+B)/2, 2*S); label("Figure 2", (E+F)/2, 2*S); label("10'", (I+J)/2, S); label("8'", (12,12)); label("8'", (L+M)/2, S); label("10'", (42,11)); label("table", (5,12)); label("table", (36,11)); [/asy] $ \textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15 $

Find all pairs $(a,b)$ of positive integers, such that for [b]every[/b] $n$ positive integer, the equality $a^n+b^n=c_n^{n+1}$ is true, for some $c_n$ positive integer.

A convex quadrilateral $ABCD$ has no parallel sides. The angles between the diagonal $AC$ and the four sides are $55^o, 55^o, 19^o$ and $16^o$ in some order. Determine all possible values of the acute angle between $AC$ and $BD$.

$p(n) $ is a product of all digits of n.Calculate: $ p(1001) + p(1002) + ... + p(2011) $

If $f(x)=\dfrac{x^4+x^2}{x+1}$, then $f(i)$, where $i=\sqrt{-1}$, is equal to: $\textbf{(A) }1+i\qquad\textbf{(B) }1\qquad\textbf{(C) }-1\qquad\textbf{(D) }0\qquad \textbf{(E) }-1-i$

Find all integers $a,b,c$ such that $a^2 = bc + 1$ and $b^2 = ac + 1$

At the vertices of a cube are written eight pairwise distinct natural numbers, and on each of its edges is written the greatest common divisor of the numbers at the endpoints of the edge. Can the sum of the numbers written at the vertices be the same as the sum of the numbers written at the edges? [i]A. Shapovalov[/i]