The King called two wizards. He ordered First Wizard to write down $100$ positive integers (not necessarily distinct) on cards without revealing them to Second Wizard. Second Wizard must correctly determine all these integers, otherwise both wizards will lose their heads. First Wizard is allowed to provide Second Wizard with a list of distinct integers, each of which is either one of the integers on the cards or a sum of some of these integers. He is not allowed to tell which integers are on the cards and which integers are their sums. If Second Wizard correctly determines all $100$ integers the King tears as many hairs from each wizard's beard as the number of integers in the list given to Second Wizard. What is the minimal number of hairs each wizard should sacrice to stay alive?

Let $ P$ be any point in the interior of a $ \triangle ABC$.Prove That $ \frac{PA}{a}\plus{}\frac{PB}{b}\plus{}\frac{PC}{c}\ge \sqrt{3}$.

Consider an acute triangle $ ABC. $ The points $ D,E,F $ are the feet of the altitudes of $ ABC $ from $ A,B,C, $ respectively. $ M,N,P $ are the middlepoints of $ BC,CA,AB, $ respectively. The circumcircles of $ BDP,CDN $ cross at $ X\neq D, $ the circumcircles of $ CEM,AEP $ cross at $ Y\neq E, $ and the circumcircles of $ AFN,BFM $ cross at $ Z\neq F. $ Prove that $ AX,BY,CZ $ are concurrent.

Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square then it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a $2\times4$ rectangle instead of $2\times3$ like in chess). In this movement, it doesn't land on the squares between the beginning square and the final square it lands on. A trip of the length $n$ of the knight is a sequence of $n$ squares $C1, C2, ..., Cn$ which are all distinct such that the knight starts at the $C1$ square and for each $i$ from $1$ to $n-1$ it can use the movement described before to go from the $Ci$ square to the $C(i+1)$. Determine the greatest $N \in \mathbb{N}$ such that there exists a path of the knight with length $N$ on a $5\times5$ board.

Let $O$ be a point outside a given circle. Two lines $OAB, OCD$ through $O$ meet the circle at $A,B,C,D$, where $A,C$ are the midpoints of $OB,OD$, respectively. Additionally, the acute angle $\theta$ between the lines is equal to the acute angle at which each line cuts the circle. Find $\cos \theta$ and show that the tangents at $A,D$ to the circle meet on the line $BC.$

In the Cartesian plane let $A = (1,0)$ and $B = \left( 2, 2\sqrt{3} \right)$. Equilateral triangle $ABC$ is constructed so that $C$ lies in the first quadrant. Let $P=(x,y)$ be the center of $\triangle ABC$. Then $x \cdot y$ can be written as $\tfrac{p\sqrt{q}}{r}$, where $p$ and $r$ are relatively prime positive integers and $q$ is an integer that is not divisible by the square of any prime. Find $p+q+r$.

The segments connecting the interior point of a convex non-sided $n$-gon with its vertices divide the $n$-gon into $n$ congruent triangles. For what is the smallest $n$ that is possible?

Someone wants to unscrew a square nut with side $a$, with a wrench whose hole has the form of a regular hexagon with side $b$. What condition should the lengths $a$ and $b$ meet to make this possible?

In a trapezoid $ABCD$ with $AB // CD, E$ is the midpoint of $BC$. Prove that if the quadrilaterals $ABED$ and $AECD$ are tangent, then the sides $a = AB, b = BC, c =CD, d = DA$ of the trapezoid satisfy the equalities $a+c = \frac{b}{3} +d$ and $\frac1a +\frac1c = \frac3b$ .

Given $120$ unit squares arbitrarily situated in the $20\times 25$ rectangle. Prove that you can place a circle with the unit diameter without intersecting any of the squares.

Let $ABC$ be a triangle. Draw circles $\omega_A$, $\omega_B$, and $\omega_C$ such that $\omega_A$ is tangent to $AB$ and $AC$, and $\omega_B$ and $\omega_C$ are defined similarly. Let $P_A$ be the insimilicenter of $\omega_B$ and $\omega_C$. Define $P_B$ and $P_C$ similarly. Prove that $AP_A$, $BP_B$, and $CP_C$ are concurrent. [i]Tom Lu.[/i]

The figure shows a parallelogram and the point $P$ of intersection of its diagonals is marked. Draw a straight line through $P$ so that it breaks the parallelogram into two parts, from which you can fold a rhombus. [img]https://1.bp.blogspot.com/-Df2tIBthcmI/X2ZwIx3R4vI/AAAAAAAAMhQ/8Zkxfq30H8MSCdc66tm33n6jt-QKfGMowCLcBGAsYHQ/s0/2009%2Boral%2Bmoscow%2Bj1.png[/img]

James randomly selects $4$ distinct numbers between $3$ and $10$, inclusive. What is the probability that all $4$ numbers are prime? $\textbf{(A) }0\qquad\textbf{(B) }\dfrac1{28}\qquad\textbf{(C) }\dfrac1{14}\qquad\textbf{(D) }\dfrac17\qquad\textbf{(E) }\dfrac14$

Prove the following inequality. \[ \frac{\sqrt{2}}{4}\minus{}\frac 12\minus{}\frac 14\ln 2<\int_0^{\frac{\pi}{4}} \ln \cos x\ dx<\frac 38\pi\plus{}\frac 12\minus{}\ln \ (3\plus{}2\sqrt{2})\]

Evaluate the following: [url]http://internetolympiad.org/archive/2014/AprilFools/foreign_lang.txt[/url]

Let $p$ be a certain prime number. Find all non-negative integers $n$ for which polynomial $P(x)=x^4-2(n+p)x^2+(n-p)^2$ may be rewritten as product of two quadratic polynomials $P_1, \ P_2 \in \mathbb{Z}[X]$.

Let $X,Y,Z,T$ be four points in the plane. The segments $[XY]$ and $[ZT]$ are said to be [i]connected[/i], if there is some point $O$ in the plane such that the triangles $OXY$ and $OZT$ are right-angled at $O$ and isosceles. Let $ABCDEF$ be a convex hexagon such that the pairs of segments $[AB],[CE],$ and $[BD],[EF]$ are [i]connected[/i]. Show that the points $A,C,D$ and $F$ are the vertices of a parallelogram and $[BC]$ and $[AE]$ are [i]connected[/i].

A given polynomial $P(t) = t^3 + at^2 + bt + c$ has $3$ distinct real roots. If the equation $(x^2 +x+2013)^3 +a(x^2 +x+2013)^2 + b(x^2 + x + 2013) + c = 0$ has no real roots, prove that $P(2013) >\frac{1}{64}$

Let $ n$ and $ k$ be positive integers with $ k \geq n$ and $ k \minus{} n$ an even number. Let $ 2n$ lamps labelled $ 1$, $ 2$, ..., $ 2n$ be given, each of which can be either [i]on[/i] or [i]off[/i]. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on). Let $ N$ be the number of such sequences consisting of $ k$ steps and resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off. Let $ M$ be number of such sequences consisting of $ k$ steps, resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off, but where none of the lamps $ n \plus{} 1$ through $ 2n$ is ever switched on. Determine $ \frac {N}{M}$. [i]Author: Bruno Le Floch and Ilia Smilga, France[/i]

The polygon(s) formed by $y=3x+2$, $y=-3x+2$, and $y=-2$, is (are): $ \textbf{(A) }\text{An equilateral triangle}\qquad\textbf{(B) }\text{an isosceles triangle} \qquad\textbf{(C) }\text{a right triangle} \qquad$ $\textbf{(D) }\text{a triangle and a trapezoid}\qquad\textbf{(E) }\text{a quadrilateral} $

Let $ABC$ be a triangle and let $D, E$ and $F$ be the midpoints of the sides $AB, BC$ and $CA$ respectively. Suppose that the angle bisector of $\angle BDC$ meets $BC$ at the point $M$ and the angle bisector of $\angle ADC$ meets $AC$ at the point $N$. Let $MN$ and $CD$ intersect at $O$ and let the line $EO$ meet $AC$ at $P$ and the line $FO$ meet $BC$ at $Q$. Prove that $CD = PQ$.

Let $n\in\mathbb{N}^*$. Solve the equation $\sum_{k=0}^n C_n^k\cos2kx=\cos nx$ in $\mathbb{R}$.

Triangle $ABC$ is inscribed in circle $\omega$. Circle $\gamma$ is tangent to $AB$ and $AC$ at points $P$ and $Q$ respectively. Also circle $\gamma$ is tangent to circle $\omega$ at point $S$. Let the intesection of $AS$ and $PQ$ be $T$. Prove that $\angle{BTP}=\angle{CTQ}$.

Let $ABC$ be a triangle for which $AB \neq AC$. Points $K$, $L$, $M$ are the midpoints of the sides $BC$, $CA$, $AB$. The incircle of $ABC$ with center $I$ is tangent to $BC$ in $D$. A line passing through the midpoint of $ID$ perpendicular to $IK$ meets the line $LM$ in $P$. Prove that $\angle PIA = 90 ^{\circ}$.

$\Delta oa_1b_1$ is isosceles with $\angle a_1ob_1 = 36^\circ$. Construct $a_2,b_2,a_3,b_3,...$ as below, with $|oa_{i+1}| = |a_ib_i|$ and $\angle a_iob_i = 36^\circ$, Call the summed area of the first $k$ triangles $A_k$. Let $S$ be the area of the isocseles triangle, drawn in - - -, with top angle $108^\circ$ and $|oc|=|od|=|oa_1|$, going through the points $b_2$ and $a_2$ as shown on the picture. (yes, $cd$ is parallel to $a_1b_1$ there) Show $A_k < S$ for every positive integer $k$. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=284[/img]