Let $ A\equal{}A_1A_2A_3A_4$ be a tetrahedron, and suppose that for each $ j \not\equal{} k, [A_j,A_{jk}]$ is a segment of length $ \rho$ extending from $ A_j$ in the direction of $ A_k$. Let $ p_j$ be the intersection line of the planes $ [A_{jk}A_{jl}A_{jm}]$ and $ [A_kA_lA_m]$. Show that there are infinitely many straight lines that intersect the straight lines $ p_1,p_2,p_3,p_4$ simultaneously.

Let $ABC$ be a triangle with $AB<AC$ and $\omega$ be its circumcircle. The tangent line to $\omega$ at $A$ intersects line $BC$ at $D$ and let $E$ be a point on $\omega$ such that $BE$ is parallel to $AD$. $DE$ intersects segment $AB$ and $\omega$ at $F$ and $G$, respectively. The circumcircle of $BGF$ intersects $BE$ at $N$. The line $NF$ intersects lines $AD$ and $EA$ at $S$ and $T$, respectively. Prove that $DGST$ is cyclic.

Let $A,B,C$ be points on the sides $B_1C_1, C_1A_1,A_1B_1$ of a triangle $A_1B_1C_1$ such that $A_1A,B_1B,C_1C$ are the bisectors of angles of the triangle. We have that $AC = BC$ and $A_1C_1 \neq B_1C_1.$ $(a)$ Prove that $C_1$ lies on the circumcircle of the triangle $ABC$. $(b)$ Suppose that $\angle BAC_1 =\frac{\pi}{6};$ find the form of triangle $ABC$.

Cassandra sets her watch to the correct time at noon. At the actual time of $ \text{1: \!\,00 PM}$, she notices that her watch reads $ \text{12: \!\,57}$ and $ 36$ seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch first reads $ \text{10: \!\,00 PM}$? $ \textbf{(A)}\ \text{10: \!\,22 PM and 24 seconds} \qquad\textbf{(B)}\ \text{10: \!\,24 PM}\qquad\textbf{(C)}\ \text{10: \!\,25 PM}$ $ \textbf{(D)}\ \text{10: \!\,27 PM}\qquad\textbf{(E)}\ \text{10: \!\,30 PM}$

In the expression $ xy^2$, the values of $ x$ and $ y$ are each decreased $ 25\%$; the value of the expression is: $ \textbf{(A)}\ \text{decreased } 50\% \qquad\textbf{(B)}\ \text{decreased }75\%\\ \textbf{(C)}\ \text{decreased }\frac{37}{64}\text{ of its value} \qquad\textbf{(D)}\ \text{decreased }\frac{27}{64}\text{ of its value}\\ \textbf{(E)}\ \text{none of these}$

Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.

Consider triangles $ABC$ in space. (a) What condition must the angles $\angle A, \angle B , \angle C$ of $\triangle ABC$ fulfill in order that there is a point $P$ in space such that $\angle APB, \angle BPC, \angle CPA$ are right angles? (b) Let $d$ be the longest of the edges $PA,PB,PC$ and let $h$ be the longest altitude of $\triangle ABC$. Show that $\frac{1}{3}\sqrt6 h \le d \le h$.

Solve the following equation $$\sqrt{\frac{x^2}3-ax+a^2}+\sqrt{\frac{x^2}3-bx+b^2}=\sqrt{a^2-ab+b^2}$$ where $a;b\in\mathbb{R^+}$

We are given an infinite deck of cards, each with a real number on it. For every real number $x$, there is exactly one card in the deck that has $x$ written on it. Now two players draw disjoint sets $A$ and $B$ of $100$ cards each from this deck. We would like to define a rule that declares one of them a winner. This rule should satisfy the following conditions: 1. The winner only depends on the relative order of the $200$ cards: if the cards are laid down in increasing order face down and we are told which card belongs to which player, but not what numbers are written on them, we can still decide the winner. 2. If we write the elements of both sets in increasing order as $A =\{ a_1 , a_2 , \ldots, a_{100} \}$ and $B= \{ b_1 , b_2 , \ldots , b_{100} \}$, and $a_i > b_i$ for all $i$, then $A$ beats $B$. 3. If three players draw three disjoint sets $A, B, C$ from the deck, $A$ beats $B$ and $B$ beats $C$ then $A$ also beats $C$. How many ways are there to define such a rule? Here, we consider two rules as different if there exist two sets $A$ and $B$ such that $A$ beats $B$ according to one rule, but $B$ beats $A$ according to the other. [i]Proposed by Ilya Bogdanov, Russia[/i]

Let $x$ and $y$ be positive integers such that $\frac{x^2-1}{y+1}+\frac{y^2-1}{x+1}$ is integer. Prove that numbers $\frac{x^2-1}{y+1}$ and $\frac{y^2-1}{x+1}$ are integers

Let $F(n)$ be the set of polynomials $P(x) = a_0+a_1x+\cdots+a_nx^n$, with $a_0, a_1, . . . , a_n \in \mathbb R$ and $0 \leq a_0 = a_n \leq a_1 = a_{n-1 } \leq \cdots \leq a_{[n/2] }= a_{[(n+1)/2]}.$ Prove that if $f \in F(m)$ and $g \in F(n)$, then $fg \in F(m + n).$

Let $ABC$ be an acute-angled triangle, let $M$ be the midpoint of $BC$ and let $H$ be the foot of the $B$-altitude. Let $Q$ be the circumcenter of $ABM$ and let $X$ be the intersection point between $BH$ and the axis of $BC$. Show that the circumcircles of the two triangles $ACM$, $AXH$ and the line $CQ$ pass through a same point if and only if $BQ$ is perpendicular to $CQ$.

11.4 Let $AA_1$ and $BB_1$ are altitudes of an acute non-isosceles triangle $ABC$, $A'$ is a midpoint of $BC$ and $B'$ is a midpoint of $AC$. A segement $A_1B_1$ intersects $A'B'$ at point $C'$. Prove that $CC'\perp HO$, where $H$ is a orthocenter and $O$ is a circumcenter of $ABC$. ([i]L. Emel'yanov[/i])

Given a positive integer $N$, determine all positive integers $n$, satisfying the following condition: for any list $d_1,d_2,\ldots,d_k$ of (not necessarily distinct) divisors of $n$ such that $\frac{1}{d_1} + \frac{1}{d_2} + \ldots + \frac{1}{d_k} > N$, some of the fractions $\frac{1}{d_1}, \frac{1}{d_2}, \ldots, \frac{1}{d_k}$ add up to exactly $N$.

For real numbers $a_0,a_1,\cdots,a_n(a_0\neq a_1)$, we have$a_{i-1}+a_{i+1}=2a_i$ for $i=1,2,\cdots,n-1$. Prove that $P(x)=a_0\text{C}_n^0(1-x)^n+a_1\text{C}_n^1x(1-x)^{n-1}+\cdots+a_n\text{C}_n^nx^n$ is a linear polynomial.

Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $P$ be a point in the plane such that $AP \perp BC$. Let $Q$ and $R$ be the reflections of $P$ in the lines $CA$ and $AB$, respectively. Let $Y$ be the orthogonal projection of $R$ onto $CA$. Let $Z$ be the orthogonal projection of $Q$ onto $AB$. Assume that $H \neq O$ and $Y \neq Z$. Prove that $YZ \perp HO$. [asy] import olympiad; unitsize(30); pair A,B,C,H,O,P,Q,R,Y,Z,Q2,R2,P2; A = (-14.8, -6.6); B = (-10.9, 0.3); C = (-3.1, -7.1); O = circumcenter(A,B,C); H = orthocenter(A,B,C); P = 1.2 * H - 0.2 * A; Q = reflect(A, C) * P; R = reflect(A, B) * P; Y = foot(R, C, A); Z = foot(Q, A, B); P2 = foot(A, B, C); Q2 = foot(P, C, A); R2 = foot(P, A, B); draw(B--(1.6*A-0.6*B)); draw(B--C--A); draw(P--R, blue); draw(R--Y, red); draw(P--Q, blue); draw(Q--Z, red); draw(A--P2, blue); draw(O--H, darkgreen+linewidth(1.2)); draw((1.4*Z-0.4*Y)--(4.6*Y-3.6*Z), red+linewidth(1.2)); draw(rightanglemark(R,Y,A,10), red); draw(rightanglemark(Q,Z,B,10), red); draw(rightanglemark(C,Q2,P,10), blue); draw(rightanglemark(A,R2,P,10), blue); draw(rightanglemark(B,P2,H,10), blue); label("$\textcolor{blue}{H}$",H,NW); label("$\textcolor{blue}{P}$",P,N); label("$A$",A,W); label("$B$",B,N); label("$C$",C,S); label("$O$",O,S); label("$\textcolor{blue}{Q}$",Q,E); label("$\textcolor{blue}{R}$",R,W); label("$\textcolor{red}{Y}$",Y,S); label("$\textcolor{red}{Z}$",Z,NW); dot(A, filltype=FillDraw(black)); dot(B, filltype=FillDraw(black)); dot(C, filltype=FillDraw(black)); dot(H, filltype=FillDraw(blue)); dot(P, filltype=FillDraw(blue)); dot(Q, filltype=FillDraw(blue)); dot(R, filltype=FillDraw(blue)); dot(Y, filltype=FillDraw(red)); dot(Z, filltype=FillDraw(red)); dot(O, filltype=FillDraw(black)); [/asy]

Let $ u, v, w$ be positive real numbers such that $ u\sqrt {vw} \plus{} v\sqrt {wu} \plus{} w\sqrt {uv} \geq 1$. Find the smallest value of $ u \plus{} v \plus{} w$.

$P$ and $Q$ are points on the longest side $AB$ of triangle $ABC$ such that $AQ = AC$ and $BP = BC$. Prove that the circumcentre of triangle $CPQ$ coincides with the incentre of triangle $ABC$.

Let $a,b,n$ be positive integers, $b > 1$ and $b^n-1\mid a.$ Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.

A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.

Pedro and Juan are playing the following game: $-$ There are $2$ piles of rocks, with $X$ rocks in one pile and $Y$ rocks in the other pile ($X < 12, Y < 11$). $-$ Each player can draw: -- $1$ rock from one of the piles, or -- $2$ rocks from one of the piles, or -- $1$ rock from each pile, or -- $2$ rock from one pile and $1$ from the other pile. Each player must perform one of these four operations in their turns. The looser is the one who takes the last rock. Pedro plays first and has a winning strategy. What are the three maximum possible values of ($X+Y$)?

Let $ ABC$ be a triangle and $ P$ an arbitrary point in the plane. Let $ \alpha, \beta, \gamma$ be interior angles of the triangle and its area is denoted by $ F.$ Prove: \[ \ov{AP}^2 \cdot \sin 2\alpha + \ov{BP}^2 \cdot \sin 2\beta + \ov{CP}^2 \cdot \sin 2\gamma \geq 2F \] When does equality occur?

Martha writes down a random mathematical expression consisting of 3 single-digit positive integers with an addition sign "$+$" or a multiplication sign "$\times$" between each pair of adjacent digits. (For example, her expression could be $4 + 3\times 3$, with value 13.) Each positive digit is equally likely, each arithmetic sign ("$+$" or "$\times$") is equally likely, and all choices are independent. What is the expected value (average value) of her expression?

In a scalene triangle $ ABC, H$ and $ M$ are the orthocenter an centroid respectively. Consider the triangle formed by the lines through $ A,B$ and $ C$ perpendicular to $ AM,BM$ and $ CM$ respectively. Prove that the centroid of this triangle lies on the line $ MH$.

Let $a$, $b$, $c$, $x$, $y$, and $z$ be positive integers such that \[ \frac{a^2-2}{x} = \frac{b^2-37}{y} = \frac{c^2-41}{z} = a+b+c. \] Let $S=a+b+c+x+y+z$. Compute the sum of all possible values of $S$. [i]Proposed by Luke Robitaille[/i]