Spinners A and B are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even? [asy] defaultpen(linewidth(1)); draw(unitcircle); draw((1,0)--(-1,0)); draw((0,1)--(0,-1)); draw(shift(3,0)*unitcircle); draw(shift(3,0)*(origin--dir(90))); draw(shift(3,0)*(origin--dir(210))); draw(shift(3,0)*(origin--dir(330))); draw(0.7*dir(200)--0.7*dir(20), linewidth(0.7), EndArrow(7)); draw(shift(3,0)*(0.7*dir(180+65)--0.7*dir(65)), linewidth(0.7), EndArrow(7)); label("$1$", (-0.45,0.1), N); label("$4$", (-0.45,-0.1), S); label("$3$", (0.45,-0.1), S); label("$2$", (0.45,0.1), N); label("$1$", shift(3,0)*(-0.25,0.1), NW); label("$2$", shift(3,0)*(0.25,0.1), NE); label("$3$", shift(3,0)*(0,-0.3), S); label("$A$", (0,-1), S); label("$B$", (3,-1), S); [/asy] $ \textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{3}{4} $

Let $a,b\ge 0$. Prove that $$\sqrt{2}\left(\sqrt{a(a+b)^3}+b\sqrt{a^2+b^2}\right)\le 3(a^2+b^2)$$ with equality if and only if $a=b$.

Suppose $n>2$ and let $A_1,\dots,A_n$ be points on the plane such that no three are collinear. [b](a)[/b] Suppose $M_1,\dots,M_n$ be points on segments $A_1A_2,A_2A_3,\dots ,A_nA_1$ respectively. Prove that if $B_1,\dots,B_n$ are points in triangles $M_2A_2M_1,M_3A_3M_2,\dots ,M_1A_1M_n$ respectively then \[|B_1B_2|+|B_2B_3|+\dots+|B_nB_1| \leq |A_1A_2|+|A_2A_3|+\dots+|A_nA_1|\] Where $|XY|$ means the length of line segment between $X$ and $Y$. [b](b)[/b] If $X$, $Y$ and $Z$ are three points on the plane then by $H_{XYZ}$ we mean the half-plane that it's boundary is the exterior angle bisector of angle $\hat{XYZ}$ and doesn't contain $X$ and $Z$ ,having $Y$ crossed out. Prove that if $C_1,\dots ,C_n$ are points in ${H_{A_nA_1A_2},H_{A_1A_2A_3},\dots,H_{A_{n-1}A_nA_1}}$ then \[|A_1A_2|+|A_2A_3|+\dots +|A_nA_1| \leq |C_1C_2|+|C_2C_3|+\dots+|C_nC_1|\] Time allowed for this problem was 2 hours.

In triangle $ABC$, we draw the circle with center $A$ and radius $AB$. This circle intersects $AC$ at two points. Also we draw the circle with center $A$ and radius $AC$ and this circle intersects $AB$ at two points. Denote these four points by $A_1, A_2, A_3, A_4$. Find the points $B_1, B_2, B_3, B_4$ and $C_1, C_2, C_3, C_4$ similarly. Suppose that these $12$ points lie on two circles. Prove that the triangle $ABC$ is isosceles.

Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle. Suppose that $\angle ADE =\angle CBE = 75^\circ$, $BE=4$, and $DE=8$. The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime. Find $a+b+c$. [i]Proposed by Tony Kim[/i]

An isosceles trapezoid $ABCD$ ($AB = CD$) is given. A point $P$ on its circumcircle is such that segments $CP$ and $AD$ meet at point $Q$. Let $L$ be tha midpoint of$ QD$. Prove that the diagonal of the trapezoid is not greater than the sum of distances from the midpoints of the lateral sides to ana arbitrary point of line $PL$.

Let $A = (1, 0)$, $B = (0, 1)$, and $C = (0, 0)$. There are three distinct points, $P, Q, R$, such that $\{A, B, C, P\}$, $\{A, B, C, Q\}$, $\{A, B, C, R\}$ are all parallelograms (vertices unordered). Find the area of $\vartriangle PQR$.

a) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+1) - f(n)$ for all $n \in N$? b) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+2) - f(n)$ for all $n \in N$? I. Voronovich

In a cardboard box are a large number of loose socks. Some of the socks are red, the others are blue. It is stated that the total number of socks does not exceed $1993$. Furthermore, it is stated that the probability of pulling two socks from the same color when two socks are randomly drawn from the box is $1/2$. What is according to the available information, the largest number of red socks that can exist in the box?

How many ordered triples of integers $(a, b, c)$ satisfy the following system? $$ \begin{cases} ab + c &= 17 \\ a + bc &= 19 \end{cases} $$ $$ \mathrm a. ~ 2\qquad \mathrm b.~3\qquad \mathrm c. ~4 \qquad \mathrm d. ~5 \qquad \mathrm e. ~6 $$

Find all functions $f:(0,\infty) \rightarrow (0,\infty)$ such that \[f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{z}\right) + f\left(z+\frac{1}{x}\right) = 1\] for all $x,y,z >0$ with $xyz =1$.

A straight line connects City A at $(0, 0)$ to City B, 300 meters away at $(300, 0)$. At time $t=0$, a bullet train instantaneously sets out from City A to City B while another bullet train simultaneously leaves from City B to City A going on the same train track. Both trains are traveling at a constant speed of $50$ meters/second. Also, at $t=0$, a super y stationed at $(150, 0)$ and restricted to move only on the train tracks travels towards City B. The y always travels at 60 meters/second, and any time it hits a train, it instantaneously reverses its direction and travels at the same speed. At the moment the trains collide, what is the total distance that the y will have traveled? Assume each train is a point and that the trains travel at their same respective velocities before and after collisions with the y

Let $ C_1,C_2 $ be two distinct concentric circles, and $ BA $ be a diameter of $ C_1. $ Choose the points $ M,N $ on $ C_1,C_2, $ respectively, but not on the line $ BA. $ [b]a)[/b] Show that there are unique points $ P,Q $ on $ MA,MB, $ respectively, so that $ N $ is the middle of $ PQ. $ [b]b)[/b] Prove that the value $ AP^2+BQ^2 $ does not depend on $ M,N. $ [i]Mihai Piticari, Mihail Bălună[/i]

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.

The number halfway between $ \frac {1}{8}$ and $ \displaystyle \frac {1}{10}$ is $ \textbf{(A)}\ \frac {1}{80} \qquad \textbf{(B)}\ \frac {1}{40} \qquad \textbf{(C)}\ \frac {1}{18} \qquad \textbf{(D)}\ \frac {1}{9} \qquad \textbf{(E)}\ \frac {9}{80}$

Let $ x$ and $ y$ be two-digit integers such that $ y$ is obtained by reversing the digits of $ x$. The integers $ x$ and $ y$ satisfy $ x^2 \minus{} y^2 \equal{} m^2$ for some positive integer $ m$. What is $ x \plus{} y \plus{} m$? $ \textbf{(A)}\ 88\qquad \textbf{(B)}\ 112\qquad \textbf{(C)}\ 116\qquad \textbf{(D)}\ 144\qquad \textbf{(E)}\ 154$

$ S$ is a non-empty subset of the set $ \{ 1, 2, \cdots, 108 \}$, satisfying: (1) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c \in S$, such that $ \gcd(a,c)\equal{}\gcd(b,c)\equal{}1$. (2) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' \neq b$, such that $ \gcd(a, c') > 1$, $ \gcd(b,c') >1$. Find the largest possible value of $ |S|$.

Is it possible to tile $2003 \times 2003$ board by $1 \times 2$ dominoes placed horizontally and $1 \times 3$ rectangles placed vertically?

Let $P$ be a point on the circumcircle of a triangle $A_{1}A_{2}A_{3}$, and let $H$ be the orthocenter of the triangle. The feet $B_{1},B_{2},B_{3}$ of the perpendiculars from $P$ to $A_{2}A_{3},A_{3}A_{1},A_{1}A_{2}$ lie on a line. Prove that this line bisects the segment $PH$.

A [i]synogon [/i] is a convex $2n$-gon with all sides of the same length and all opposite sides are parallel. Show that every synogon can be broken down into a finite number of rhombuses.

Denmark has played an international football match against Georgia. the fight ended $5-5$, and between the first and the last goal the game has justnever stood . No country has scored three goals in a row, and Denmark scored the sixth goal. Can you use this information to determine which country scored the fifth goal?

Consider a regular heptagon ( polygon of $7$ equal sides and angles) $ABCDEFG$ as in the figure below:- $(a).$ Prove $\frac{1}{\sin\frac{\pi}{7}}=\frac{1}{\sin\frac{2\pi}{7}}+\frac{1}{\sin\frac{3\pi}{7}}$ $(b).$ Using $(a)$ or otherwise, show that $\frac{1}{AG}=\frac{1}{AF}+\frac{1}{AE}$ [asy] draw(dir(360/7)..dir(2*360/7),blue); draw(dir(2*360/7)..dir(3*360/7),blue); draw(dir(3*360/7)..dir(4*360/7),blue); draw(dir(4*360/7)..dir(5*360/7),blue); draw(dir(5*360/7)..dir(6*360/7),blue); draw(dir(6*360/7)..dir(7*360/7),blue); draw(dir(7*360/7)..dir(360/7),blue); draw(dir(2*360/7)..dir(4*360/7),blue); draw(dir(4*360/7)..dir(1*360/7),blue); label("$A$",dir(4*360/7),W); label("$B$",dir(5*360/7),S); label("$C$",dir(6*360/7),S); label("$D$",dir(7*360/7),E); label("$E$",dir(1*360/7),E); label("$F$",dir(2*360/7),N); label("$G$",dir(3*360/7),W); [/asy]

Let $n \ge 2$ be an integer and $p_1 < p_2 < ... < p_n$ prime numbers. Prove that there exists an integer $k$ relatively prime with $p_1p_2... p_n$ and such that $gcd (k + p_1p_2...p_i, p_1p_2...p_n) = 1$ for all $i = 1, 2,..., n - 1$. Malik Talbi

A ladder style tournament is held with $2016$ participants. The players begin seeded $1,2,\cdots 2016$. Each round, the lowest remaining seeded player plays the second lowest remaining seeded player, and the loser of the game gets eliminated from the tournament. After $2015$ rounds, one player remains who wins the tournament. If each player has probability of $\tfrac{1}{2}$ to win any game, then the probability that the winner of the tournament began with an even seed can be expressed has $\tfrac{p}{q}$ for coprime positive integers $p$ and $q$. Find the remainder when $p$ is divided by $1000$. [i]Proposed by Nathan Ramesh

How many positive integers $n$ satisfy$$\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$$(Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.) $\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$