Let $n$ and $k$ be integers with $k\leq n-2$. The absolute value of the sum of elements of any $k$-element subset of $\{a_1,a_2,\cdots,a_n\}$ is less than or equal to 1. Show that: If $|a_1|\geq1$, then for any $2\leq i \leq n$, we have: $$|a_1|+|a_i|\leq2$$

Let $ABC$ be a triangle. The internal bisector of $\angle B$ meets $AC$ in $P$ and $I$ is the incenter of $ABC$. Prove that if $AP+AB = CB$, then $API$ is an isosceles triangle.

Two dueling wizards are at an altitude of $100$ above the sea. They cast spells in turn, and each spell is of the form "decrease the altitude by $a$ for me and by $b$ for my rival" where $a$ and $b$ are real numbers such that $0 < a < b$. Different spells have different values for $a$ and $b$. The set of spells is the same for both wizards, the spells may be cast in any order, and the same spell may be cast many times. A wizard wins if after some spell, he is still above water but his rival is not. Does there exist a set of spells such that the second wizard has a guaranteed win, if the number of spells is $(a)$ finite; $(b)$ infinite?

Let $I$ and $H$ be the incentre and orthocentre of triangle $ABC$ respectively. Let $P,Q$ be the midpoints of $AB,AC$. The rays $PI,QI$ intersect $AC,AB$ at $R,S$ respectively. Suppose that $T$ is the circumcentre of triangle $BHC$. Let $RS$ intersect $BC$ at $K$. Prove that $A,I$ and $T$ are collinear if and only if $[BKS]=[CKR]$. [i]Shen Wunxuan[/i]

Shiori places seven books, numbered from $1$ to $7$, on a bookshelf in some order. Later, she discovers that she can place two dividers between the books, separating the books into left, middle, and right sections, such that: $\bullet$ The left section is numbered in increasing order from left to right, or has at most one book. $\bullet$ The middle section is numbered in decreasing order from left to right, or has at most one book. $\bullet$ The right section is numbered in increasing order from left to right, or has at most one book. In how many possible orderings could Shiori have placed the books? For example, $(2, 3, 5, 4, 1, 6, 7)$ and $(2, 3, 4, 1, 5, 6, 7)$ are possible orderings with the partitions $2, 3, 5|4, 1|6, 7$ and $2, 3, 4|1|5, 6, 7$, but $(5, 6, 2, 4, 1, 3, 7)$ is not.

Find the sum of all positive integers $n$ such that \[ \frac{2n+1}{n(n-1)} \] has a terminating decimal representation. [i]Proposed by Evan Chen[/i]

The integer $ 9$ can be written as a sum of two consecutive integers: $ 9 \equal{} 4\plus{}5.$ Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: $ 9 \equal{} 4\plus{}5 \equal{} 2\plus{}3\plus{}4.$ Is there an integer that can be written as a sum of $ 1990$ consecutive integers and that can be written as a sum of (more than one) consecutive positive integers in exactly $ 1990$ ways?

A convex quadrilateral $ABCD$ is inscribed into a circle $\omega$ . Suppose that there is a point $X$ on the segment $AC$ such that the $XB$ and $XD$ tangents to the circle $\omega$ . Tangent of $\omega$ at $C$, intersect $XD$ at $Q$. Let $E$ ($E\ne A$) be the intersection of the line $AQ$ with $\omega$ . Prove that $AD, BE$, and $CQ$ are concurrent.

From an equilateral triangle with side 2014 we cut off another equilateral triangle with side 214, such that both triangles have one common vertex and two of the side of the smaller triangles lie on two of the side of the bigger triangle. Is it possible to cover this figure with figures in the picture without overlapping (rotation is allowed) if all figures are made of equilateral triangles with side 1? Explain the answer! [asy] import olympiad; unitsize(20); pair A,B,C,D,E,F,G,H; A=(0,0); B=(1,0); C=rotate(60)*B; D=rotate(60)*C; E=rotate(60)*D; F=rotate(60)*E; G=rotate(60)*F; draw(A--B); draw(A--C); draw(A--D); draw(A--E); draw(A--F); draw(A--G); draw(B--C--D--E--F--G--B); A=(2,0); B=A+(1,0); C=A+rotate(60)*(B-A); D=A+rotate(60)*(C-A); E=A+rotate(120)*(D-A); F=A+rotate(60)*(E-A); G=2*F-E; H=2*C-D; draw(A--D--C--A--B--C--H--B--G--F--E--A--F--B); A=(4,0); B=A+(1,0); C=A+rotate(-60)*(B-A); D=B+rotate(60)*(C-B); E=B+rotate(60)*(D-B); F=B+rotate(60)*(E-B); G=E+rotate(60)*(D-E); H=E+rotate(60)*(G-E); draw(A--B--C--A); draw(C--D--B); draw(D--E--B); draw(B--F--E); draw(E--G--D); draw(E--H--G); A=(8.5,0.5); B=A+(1,0); C=A+rotate(60)*(B-A); D=A+rotate(60)*(C-A); E=A+rotate(60)*(D-A); F=A+rotate(60)*(E-A); G=A+rotate(60)*(F-A); H=G+rotate(60)*(F-G); draw(A--B); draw(A--C); draw(A--D); draw(A--E); draw(A--F); draw(A--G); draw(B--C); draw(D--E--F--G--B); draw(G--H--F);[/asy]

Inscribed hexagon $AB_1CA_1BC_1$ is given. Circle $\omega$ is inscribed in both triangles $ABC$ and $A_1B_1C_1$ and touches segments $AB$ and $A_1B_1$ at points $D$ and $D_1$ respectively. Prove that if $\angle ACD = \angle BCD_1$, then $\angle A_1C_1D_1 = \angle B_1C_1D$.

Suppose $a$ is a real number such that $\sin(\pi \cdot \cos a) = \cos(\pi \cdot \sin a)$. Evaluate $35 \sin^2(2a) + 84 \cos^2(4a)$.

Find all functions $f : (0, +\infty) \to \mathbb{R}$ satisfying the following conditions: $(i)$ $f(x) + f(\frac{1}{x}) = 1$ for all $x> 0$; $(ii)$ $f(xy + x + y) = f(x)f(y)$ for all $x, y> 0$.

In the Martian language every finite sequence of letters of the Latin alphabet letters is a word. The publisher “Martian Words” makes a collection of all words in many volumes. In the first volume there are only one-letter words, in the second, two-letter words, etc., and the numeration of the words in each of the volumes continues the numeration of the previous volume. Find the word whose numeration is equal to the sum of numerations of the words Prague, Olympiad, Mathematics.

Does there exist a function $f:\mathbb{Z}\to\mathbb{Z}$ such that $f(x+f(y))=f(x)-y$ for all integers $x$ and $y$?

Triangles $ABC$ and $DEF$ have pairwise parallel sides: $EF \| BC, FD \| CA$, and $DE \| AB$. The line $m_A$ is the reflection of $EF$ through $BC$, similarly $m_B$ is the reflection of $FD$ through $CA$, and $m_C$ the reflection of $DE$ through $AB$. Assume that the lines $m_A, m_B$, and $m_C$ meet in a common point. What is the ratio between the areas of triangles $ABC$ and $DEF$?

Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ and $f(2^{n})$ are co-prime for all natural numbers $n$.

In triangle $ABC$, $I$ is the incenter. We have chosen points $P,Q,R$ on segments $IA,IB,IC$ respectively such that $IP\cdot IA=IQ \cdot IB=IR\cdot IC$. Prove that the points $I$ and $O$ belong to Euler line of triangle $PQR$ where $O$ is circumcenter of $ABC$.

A positive number $\dfrac{m}{n}$ has the property that it is equal to the ratio of $7$ plus the number’s reciprocal and $65$ minus the number’s reciprocal. Given that $m$ and $n$ are relatively prime positive integers, find $2m + n$.

Find all continuous functions $f\colon [0;1] \to R$ such that \[\int_{0}^{1}f(x)dx = 2\int_{0}^{1}(f(x^{4}))^{2}dx+\frac{2}{7}\]

Let $0<x_1<1$ and $x_{n+1}=x_n(1-x_n), n=1,2,3, \dots$. Show that $$\lim_{n \to \infty} nx_n=1.$$

Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points. [i]Proposed by Härmel Nestra, Estonia[/i]

Find the number of pairs of real numbers $(x,y)$ such that $x^4+y^4=4xy-2$.

Compute $$\int_0^13x^2dx.$$

Find the number of $16$-tuples $(x_1, x_2,..., x_{16})$ such that (i) $x_i = \pm 1$ for $i = 1,..., 16$, (ii) $0 \le x_1 + x_2 +... + x_r < 4$, for $r = 1, 2,... , 15$, (iii) $x_1 + x_2 +...+ x_{10} = 4$

Prove that between every $27$ different positive integers , less than $100$, there exist some two which are[color=red] NOT [/color]relative prime. [u]babis[/u]