[b]p1.[/b] If $$ \begin{cases} a^2 + b^2 + c^2 = 1000 \\ (a + b + c)^2 = 100 \\ ab + bc = 10 \end{cases}$$ what is $ac$? [b]p2.[/b] If a and b are real numbers such that $a \ne 0$ and the numbers $1$, $a + b$, and $a$ are, in some order, the numbers $0$, $\frac{b}{a}$ , and $b$, what is $b - a$? [b]p3.[/b] Of the first $120$ natural numbers, how many are divisible by at least one of $3$, $4$, $5$, $12$, $15$, $20$, and $60$? [b]p4.[/b] For positive real numbers $a$, let $p_a$ and $q_a$ be the maximum and minimum values, respectively, of $\log_a(x)$ for $a \le x \le 2a$. If $p_a - q_a = \frac12$ , what is $a$? [b]p5.[/b] Let $ABC$ be an acute triangle and let $a$, $b$, and $c$ be the sides opposite the vertices $A$, $B$, and $C$, respectively. If $a = 2b \sin A$, what is the measure of angle $B$? [b]p6.[/b] How many ordered triples $(x, y, z)$ of positive integers satisfy the equation $$x^3 + 2y^3 + 4z^3 = 9?$$ [b]p7.[/b] Joe has invented a robot that travels along the sides of a regular octagon. The robot starts at a vertex of the octagon and every minute chooses one of two directions (clockwise or counterclockwise) with equal probability and moves to the next vertex in that direction. What is the probability that after $8$ minutes the robot is directly opposite the vertex it started from? [b]p8.[/b] Find the nonnegative integer $n$ such that when $$\left(x^2 -\frac{1}{x}\right)^n$$ is completely expanded the constant coefficient is $15$. [b]p9.[/b] For each positive integer $k$, let $$f_k(x) = \frac{kx + 9}{x + 3}.$$ Compute $$f_1 \circ f_2\circ ... \circ f_{13}(2).$$ [b]p10.[/b] Exactly one of the following five integers cannot be written in the form $x^2 + y^2 + 5z^2$, where $x$, $y$, and $z$ are integers. Which one is it? $$2003, 2004, 2005, 2006, 2007$$ [b]p11.[/b] Suppose that two circles $C_1$ and $C_2$ intersect at two distinct points $M$ and $N$. Suppose that $P$ is a point on the line $MN$ that is outside of both $C_1$ and $C_2$. Let $A$ and $B$ be the two distinct points on $C_1$ such that AP and BP are each tangent to $C_1$ and $B$ is inside $C_2$. Similarly, let $D$ and $E$ be the two distinct points on $C_2$ such that $DP$ and $EP$ are each tangent to $C_2$ and $D$ is inside $C_1$. If $AB = \frac{5\sqrt2}{2}$ , $AD = 2$, $BD = 2$, $EB = 1$, and $ED =\sqrt2$, find $AE$. [b]p12.[/b] How many ordered pairs $(x, y)$ of positive integers satisfy the following equation? $$\sqrt{x} +\sqrt{y} =\sqrt{2007}.$$ [b]p13.[/b] The sides $BC$, $CA$, and $CB$ of triangle $ABC$ have midpoints $K$, $L$, and $M$, respectively. If $$AB^2 + BC^2 + CA^2 = 200,$$ what is $AK^2 + BL^2 + CM^2$? [b]p14.[/b] Let $x$ and $y$ be real numbers that satisfy: $$x + \frac{4}{x}= y +\frac{4}{y}=\frac{20}{xy}.$$ Compute the maximum value of $|x - y|$. [b]p15.[/b] $30$ math meet teams receive different scores which are then shuffled around to lend an aura of mystery to the grading. What is the probability that no team receives their own score? Express your answer as a decimal accurate to the nearest hundredth. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the coordinate plane, a point $A$ is chosen on the line $y =\frac32 x$ in the first quadrant. Two perpendicular lines $\ell_1$ and $\ell_2$ intersect at A where $\ell_1$ has slope $m > 1$. Let $\ell_1$ intersect the $ x$-axis at $B$, and $\ell_2$ intersects the $ x$ and $y$ axes at $C$ and $D$, respectively. Suppose that line $BD$ has slope $-m$ and $BD = 2$. Compute the length of $CD$.

In a triangle $ABC$, $AB = AC$, and $D$ is on $BC$. A point $E$ is chosen on $AC$, and a point $F$ is chosen on $AB$, such that $DE = DC$ and $DF = DB$. It is given that $\frac{DC}{BD} = 2$ and $\frac{AF}{AE} = 5$. Determine that value of $\frac{AB}{BC}$.

A circle is inscribed in an equilateral triangle. Three nested sequences of circles are then constructed as follows: each circle touches the previous circle and has two edges of the triangle as tangents. This is represented by the figure below. [asy] import olympiad; pair A, B, C; A = dir(90); B = dir(210); C = dir(330); draw(A--B--C--cycle); draw(incircle(A,B,C)); draw(incircle(A,2/3*A+1/3*B,2/3*A+1/3*C)); draw(incircle(A,8/9*A+1/9*B,8/9*A+1/9*C)); draw(incircle(A,26/27*A+1/27*B,26/27*A+1/27*C)); draw(incircle(A,80/81*A+1/81*B,80/81*A+1/81*C)); draw(incircle(A,242/243*A+1/243*B,242/243*A+1/243*C)); draw(incircle(B,2/3*B+1/3*A,2/3*B+1/3*C)); draw(incircle(B,8/9*B+1/9*A,8/9*B+1/9*C)); draw(incircle(B,26/27*B+1/27*A,26/27*B+1/27*C)); draw(incircle(B,80/81*B+1/81*A,80/81*B+1/81*C)); draw(incircle(B,242/243*B+1/243*A,242/243*B+1/243*C)); draw(incircle(C,2/3*C+1/3*B,2/3*C+1/3*A)); draw(incircle(C,8/9*C+1/9*B,8/9*C+1/9*A)); draw(incircle(C,26/27*C+1/27*B,26/27*C+1/27*A)); draw(incircle(C,80/81*C+1/81*B,80/81*C+1/81*A)); draw(incircle(C,242/243*C+1/243*B,242/243*C+1/243*A)); [/asy] What is the ratio of the area of the largest circle to the combined area of all the other circles? $\text{(A) }\frac{8}{1}\qquad\text{(B) }\frac{8}{3}\qquad\text{(C) }\frac{9}{1}\qquad\text{(D) }\frac{9}{3}\qquad\text{(E) }\frac{10}{3}$

Triangle $ABC$ has sidelengths $AB=14,AC=13,$ and $BC=15.$ Point $D$ is chosen in the interior of $\overline{AB}$ and point $E$ is selected uniformly at random from $\overline{AD}.$ Point $F$ is then defined to be the intersection point of the perpendicular to $\overline{AB}$ at $E$ and the union of segments $\overline{AC}$ and $\overline{BC}.$ Suppose that $D$ is chosen such that the expected value of the length of $\overline{EF}$ is maximized. Find $AD.$

Let $ABC$ be a triangle and let $P$ be a point in the interior of the side $BC$. Let $I_1$ and $I_2$ be the incenters of the triangles $AP B$ and $AP C$, respectively. Let $X$ be the closest point to $A$ on the line $AP$ such that $XI_1$ is perpendicular to $XI_2$. Prove that the distance $AX$ is independent of the choice of $P$.

Consider an acute triangle $ABC$ with $|AB| > |CA| > |BC|$. The vertices $D, E$, and $F$ are the base points of the altitudes from $A, B$, and $C$, respectively. The line through F parallel to $DE$ intersects $BC$ in $M$. The angular bisector of $\angle MF E$ intersects $DE$ in $N$. Prove that $F$ is the circumcentre of $\vartriangle DMN$ if and only if $B$ is the circumcentre of $\vartriangle FMN$.

Line $\ell$ is perpendicular to one of the medians of the triangle. The median perpendiculars to the sides of this triangle intersect the line $\ell$ at three points. Prove that one of them is the midpoint of the segment formed by the other two.

The diagram below is an example of a ${\textit{rectangle tiled by squares}}$: [center][img]http://i.imgur.com/XCPQJgk.png[/img][/center] Each square has been labeled with its side length. The squares fill the rectangle without overlapping. In a similar way, a rectangle can be tiled by nine squares whose side lengths are $2,5,7,9,16,25,28,33$, and $36$. Sketch one such possible arrangement of those squares. They must fill the rectangle without overlapping. Label each square in your sketch by its side length as in the picture above.

Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB=11$ and $CD=19$. Point $P$ is on segment $AB$ with $AP=6$, and $Q$ is on segment $CD$ with $CQ=7$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ=27$, find $XY$.

$P, A, B, C,$ and $D$ are five distinct points in space such that $\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta$, where $\theta$ is a given acute angle. Determine the greatest and least values of $\angle APC + \angle BPD$.

Let $ABC$ be a non-isosceles,aqute triangle with $AB<AC$ inscribed in circle $c(O,R)$.The circle $c_{1}(B,AB)$ crosses $AC$ at $K$ and $c$ at $E$. $KE$ crosses $c$ at $F$ and $BO$ crosses $KE$ at $L$ and $AC$ at $M$ while $AE$ crosses $BF$ at $D$.Prove that: i)$D,L,M,F$ are concyclic. ii)$B,D,K,M,E$ are concyclic.

Let $ABCD$ be a parallelogram. A line through $C$ crosses the side $AB$ at an interior point $X$, and the line $AD$ at $Y$. The tangents of the circle $AXY$ at $X$ and $Y$, respectively, cross at $T$. Prove that the circumcircles of triangles $ABD$ and $TXY$ intersect at two points, one lying on the line $AT$ and the other one lying on the line $CT$.

There exists a certain right triangle with the smallest area in the $2$D coordinate plane such that all of its vertices have integer coordinates but none of its sides are parallel to the $x$- or $y$-axis. Additionally, all of its sides have distinct, integer lengths. What is the area of this triangle?

A triangle has sides of length $9$, $40$, and $41$. What is its area?

Let $\triangle ABC$ be a triangle such that the angle bisector of $\triangle BAC,$ the median from $B$ to side $AC,$ and the perpendicular bisector of $AB$ intersect at a single point $X.$ If $AX = 5$ and $AC = 12,$ compute $a+b$ where $BC^2=\tfrac{a}{b}$ and $a,b$ are coprime positive integers. .

Let $n$ be a positive integer. Consider $n^2+1$ (closed, i.e. including endpoints) segments on a single line. Show that at least one of the following statements holds: a) there are $n+1$ segments with non-empty intersection, b) there are $n+1$ segments among which two of them are disjoint.

Let $ABCD$ be a parallelogram in which angle at $B$ is obtuse and $AD>AB$. Points $K$ and $L$ on $AC$ such that $\angle ADL=\angle KBA$(the points $A, K, C, L$ are all different, with $K$ between $A$ and $L$). The line $BK$ intersects the circumcircle $\omega$ of $ABC$ at points $B$ and $E$, and the line $EL$ intersects $\omega$ at points $E$ and $F$. Prove that $BF||AC$.

Is it possible to draw a hexagon with vertices in the knots of an integer lattice so that the squares of the lengths of the sides are six consecutive positive integers?

Let $ABCDEF$ be a convex hexagon and it`s diagonals have one common point $M$. It is known that the circumcenters of triangles $MAB,MBC,MCD,MDE,MEF,MFA$ lie on a circle. Show that the quadrilaterals $ABDE,BCEF,CDFA$ have equal areas.

$ABCD$ is a parallelogram, line $DF$ is drawn bisecting $BC$ at $E$ and meeting $AB$ (extended) at $F$ from vertex $C$. Line $CH$ is drawn bisecting side $AD$ at $G$ and meeting $AB$ (extended) at $H$. Lines $DF$ and $CH$ intersect at $I$. If the area of parallelogram $ABCD$ is $x$, find the area of triangle $HFI$ in terms of $x$.

In a triangle $ABC$, with $AB \ne BC$, $E$ is a point on the line $AC$ such that $BE$ is perpendicular to $AC$. A circle passing through $A$ and touching the line $BE$ at a point $P \ne B$ intersects the line $AB$ for the second time at $X$. Let $Q$ be a point on the line $PB$ different from $P$ such that $BQ = BP$. Let $Y$ be the point of intersection of the lines $CP$ and $AQ$. Prove that the points $C, X, Y, A$ are concyclic if and only if $CX$ is perpendicular to $AB$.

There is a set of $ n$ coins with distinct integer weights $ w_1, w_2, \ldots , w_n$. It is known that if any coin with weight $ w_k$, where $ 1 \leq k \leq n$, is removed from the set, the remaining coins can be split into two groups of the same weight. (The number of coins in the two groups can be different.) Find all $ n$ for which such a set of coins exists.

A triangle $ABC$ is given. A circle $\Gamma$, passing through $A$, is tangent to side $BC$ at point $P$ and intersects sides $AB$ and $AC$ at $M$ and $N$ respectively. Prove that the smaller arcs $MP$ and $NP$ of $\Gamma$ are equal iff $\Gamma$ is tangent to the circumcircle of $\Delta ABC$ at $A$.

[b]p1.[/b] Thirty players participate in a chess tournament. Every player plays one game with every other player. What maximal number of players can get exactly $5$ points? (any game adds $1$ point to the winner’s score, $0$ points to a loser’s score, in the case of a draw each player obtains $1/2$ point.) [b]p2.[/b] A father and his son returned from a fishing trip. To make their catches equal the father gave to his son some of his fish. If, instead, the son had given his father the same number of fish, then father would have had twice as many fish as his son. What percent more is the father's catch more than his son's? [b]p3.[/b] What is the maximal number of pieces of two shapes, [img]https://cdn.artofproblemsolving.com/attachments/a/5/6c567cf6a04b0aa9e998dbae3803b6eeb24a35.png[/img] and [img]https://cdn.artofproblemsolving.com/attachments/8/a/7a7754d0f2517c93c5bb931fb7b5ae8f5e3217.png[/img], that can be used to tile a $7\times 7$ square? [b]p4.[/b] Six shooters participate in a shooting competition. Every participant has $5$ shots. Each shot adds from 1 to $10$ points to shooter’s score. Every person can score totally for all five shots from $5$ to $50$ points. Each participant gets $7$ points for at least one of his shots. The scores of all participants are different. We enumerate the shooters $1$ to $6$ according to their scores, the person with maximal score obtains number $1$, the next one obtains number $2$, the person with minimal score obtains number $6$. What score does obtain the participant number 3? The total number of all obtained points is $264$. [b]p5.[/b] There are $2014$ stones in a pile. Two players play the following game. First, player $A$ takes some number of stones (from $1$ to $30$) from the pile, then player B takes $1$ or $2$ stones, then player $A$ takes $2$ or $3$ stones, then player $B$ takes $3$ or $4$ stones, then player A takes $4$ or $5$ stones, etc. The player who gets the last stone is the winner. If no player gets the last stone (there is at least one stone in the pile but the next move is not allowed) then the game results in a draw. Who wins the game using the right strategy? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].