Given a board $3\times3$ and $9$ cards with some numbers (known to the players). Two players, in turn, put those cards on the board. The first wins if the sum of the numbers in the first and the third row is greater than in the first and the third column. Prove that it doesn't matter what numbers are on the cards, but if the first plays the best way, the second can not win.

Starting from the number $ 1$ we write down a sequence of numbers where the next number in the sequence is obtained from the previous one either by doubling it, or by rearranging its digits (not allowing the first digit of the rearranged number to be $0$). For instance we might begin: $$1, 2, 4, 8, 16, 61, 122, 212, 424,...$$ Is it possible to construct such a sequence that ends with the number $1,000,000,000$? Is it possible to construct one that ends with the number $9,876,543,210$?

Find the sum of the positive integer solutions to the equation $\left\lfloor\sqrt[3]{x}\right\rfloor+\left\lfloor\sqrt[4]{x}\right\rfloor=4.$

Let $ABC$ be a triangle with incenter $I$. The points $D$ and $E$ lie on the segments $CA$ and $BC$ respectively, such that $CD = CE$. Let $F$ be a point on the segment $CD$. Prove that the quadrilateral $ABEF$ is circumscribable if and only if the quadrilateral $DIEF$ is cyclic. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Juan writes the list of pairs $(n, 3^n)$, with $n=1, 2, 3,...$ on a chalkboard. As he writes the list, he underlines the pairs $(n, 3^n)$ when $n$ and $3^n$ have the same units digit. What is the $2013^{th}$ underlined pair?

[u]General Round[/u] [b]p1.[/b] Alice can bake a pie in $5$ minutes. Bob can bake a pie in $6$ minutes. Compute how many more pies Alice can bake than Bob in $60$ minutes. [b]p2.[/b] Ben likes long bike rides. On one ride, he goes biking for six hours. For the first hour, he bikes at a speed of $15$ miles per hour. For the next two hours, he bikes at a speed of $12$ miles per hour. He remembers biking $90$ miles over the six hours. Compute the average speed, in miles per hour, Ben biked during the last three hours of his trip. [b]p3.[/b] Compute the perimeter of a square with area $36$. [b]p4.[/b] Two ants are standing side-by-side. One ant, which is $4$ inches tall, casts a shadow that is $10$ inches long. The other ant is $6$ inches tall. Compute, in inches, the length of the shadow that the taller ant casts. [b]p5.[/b] Compute the number of distinct line segments that can be drawn inside a square such that the endpoints of the segment are on the square and the segment divides the square into two congruent triangles. [b]p6.[/b] Emily has a cylindrical water bottle that can hold $1000\pi$ cubic centimeters of water. Right now, the bottle is holding $100\pi$ cubic centimeters of water, and the height of the water is $1$ centimeter. Compute the radius of the water bottle. [b]p7.[/b] Given that $x$ and $y$ are nonnegative integers, compute the number of pairs $(x, y)$ such that $5x + y = 20$. [b]p8.[/b] A sequence an is recursively defined where $a_n = 3(a_{n-1}-1000)$ for $n > 0$. Compute the smallest integer $x$ such that when $a_0 = x$, $a_n > a_0$ for all $n > 0$. [b]p9.[/b] Compute the probability that two random integers, independently chosen and both taking on an integer value between $1$ and $10$ with equal probability, have a prime product. [b]p10.[/b] If $x$ and $y$ are nonnegative integers, both less than or equal to $2$, then we say that $(x, y)$ is a friendly point. Compute the number of unordered triples of friendly points that form triangles with positive area. [b]p11.[/b] Cindy is thinking of a number which is $4$ less than the square of a positive integer. The number has the property that it has two $2$-digit prime factors. What is the smallest possible value of Cindy's number? [b]p12.[/b] Winona can purchase a pencil and two pens for $250$ cents, or two pencils and three pens for $425$ cents. If the cost of a pencil and the cost of a pen does not change, compute the cost in cents of five pencils and six pens. [b]p13.[/b] Colin has an app on his phone that generates a random integer betwen $1$ and $10$. He generates $10$ random numbers and computes the sum. Compute the number of distinct possible sums Colin can end up with. [b]p14.[/b] A circle is inscribed in a unit square, and a diagonal of the square is drawn. Find the total length of the segments of the diagonal not contained within the circle. [b]p15.[/b] A class of six students has to split into two indistinguishable teams of three people. Compute the number of distinct team arrangements that can result. [b]p16.[/b] A unit square is subdivided into a grid composed of $9$ squares each with sidelength $\frac13$ . A circle is drawn through the centers of the $4$ squares in the outermost corners of the grid. Compute the area of this circle. [b]p17.[/b] There exists exactly one positive value of $k$ such that the line $y = kx$ intersects the parabola $y = x^2 + x + 4$ at exactly one point. Compute the intersection point. [b]p18.[/b] Given an integer $x$, let $f(x)$ be the sum of the digits of $x$. Compute the number of positive integers less than $1000$ where $f(x) = 2$. [b]p19.[/b] Let $ABCD$ be a convex quadrilateral with $BA = BC$ and $DA = DC$. Let $E$ and $F$ be the midpoints of $BC$ and $CD$ respectively, and let $BF$ and $DE$ intersect at $G$. If the area of $CEGF$ is $50$, what is the area of $ABGD$? [b]p20.[/b] Compute all real solutions to $16^x + 4^{x+1} - 96 = 0$. [b]p21.[/b] At an M&M factory, two types of M&Ms are produced, red and blue. The M&Ms are transported individually on a conveyor belt. Anna is watching the conveyor belt, and has determined that four out of every five red M&Ms are followed by a blue one, while one out of every six blue M&Ms is followed by a red one. What proportion of the M&Ms are red? [b]p22.[/b] $ABCDEFGH$ is an equiangular octagon with side lengths $2$, $4\sqrt2$, $1$, $3\sqrt2$, $2$, $3\sqrt2$, $3$, and $2\sqrt2$,in that order. Compute the area of the octagon. [b]p23.[/b] The cubic $f(x) = x^3 +bx^2 +cx+d$ satisfies $f(1) = 3$, $f(2) = 6$, and $f(4) = 12$. Compute $f(3)$. [b]p24.[/b] Given a unit square, two points are chosen uniformly at random within the square. Compute the probability that the line segment connecting those two points touches both diagonals of the square. [b]p25.[/b] Compute the remainder when: $$5\underbrace{666...6666}_{2016 \,\, sixes}5$$ is divided by $17$. [u]General Tiebreaker [/u] [b]Tie 1.[/b] Trapezoid $ABCD$ has $AB$ parallel to $CD$, with $\angle ADC = 90^o$. Given that $AD = 5$, $BC = 13$ and $DC = 18$, compute the area of the trapezoid. [b]Tie 2.[/b] The cubic $f(x) = x^3- 7x - 6$ has three distinct roots, $a$, $b$, and $c$. Compute $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ . [b]Tie 3.[/b] Ben flips a fair coin repeatedly. Given that Ben's first coin flip is heads, compute the probability Ben flips two heads in a row before Ben flips two tails in a row. PS. You should use hide for answers.

Let $ABCD$ be a parallelogram and let $E$ be a point in the plane such that $AE\perp AB$ and $BC\perp EC$. Show that either $\angle AED=\angle BEC$ or $\angle AED+\angle BEC=180^\circ$.

There exists one pair of positive integers $ a, b $ such that $ 100 > a > b > 0 $ and $ \dfrac{1}{a} + \dfrac{1}{b} = \dfrac{2}{35} $. Find $ a + b $.

There are 52 cards in the deck, 13 of each suit. Vanya draws from the deck one card at a time. Removed cards are not returned to the deck. Every time Before taking out the card, Vanya makes a wish for some suit.Prove that if Vanya makes a wish every time, , the cards of which are in the deck has no less cards left than cards of any other suit, then the hidden suit will fall with the suit of the card drawn at least 13 times.

Determine all even numbers $n$, $n \in \mathbb N$ such that \[{ \frac{1}{d_{1}}+\frac{1}{d_{2}}+ \cdots +\frac{1}{d_{k}}=\frac{1620}{1003}}, \] where $d_1, d_2, \ldots, d_k$ are all different divisors of $n$.

There exist complex numbers $z_1,z_2,\dots,z_{10}$ which satisfy$$|z_ki^k+ z_{k+1}i^{k+1}| = |z_{k+1}i^k+ z_ki^{k+1}|$$for all integers $1 \leq k \leq 9$, where $i = \sqrt{-1}$. If $|z_1|=9$, $|z_2|=29$, and for all integers $3 \leq n \leq 10$, $|z_n|=|z_{n-1} + z_{n-2}|$, find the minimum value of $|z_1|+|z_2|+\cdots+|z_{10}|$. [i]Proposed by DeToasty3[/i]

Consider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$. Suppose $\angle MAC = \angle ABC$ and $\angle BAM = 105^o$. Find the measure of $\angle ABC$.

Zack keeps cutting the interval $[0, 1]$ of the number line, each time cutting at a uniformly random point in the interval, until the interval is cut into pieces, none of which have length greater than $\frac35$ . The expected number of cuts that Zack makes can be written as $\frac{p}{q}$ for $p$ and $q$ relatively prime positive integers. Find $p + q$.

Two circles (centres $d$ apart) have radii $15,95$. The external tangents to the circles cut at $60$ degrees, find $d$. $$(A) 40$$ $$(B) 80$$ $$(C) 120$$ $$(D) 160$$

A circle centered at a point $(0, 1)$ on the coordinate plane intersects the parabola $y = x^2$ at four points: $A, B, C, D.$ Find the largest possible value of the area of ​​the quadrilateral $ABCD$.

A father and two sons went to visit their grandmother, who Raya lives $33$ km from the city. My father has a motor roller, the speed of which $25$ km/h, and with a passenger - $20$ km/h (with two passengers on a scooter It’s impossible to move). Each of the brothers walks along the road at a speed of $5$ km/h. Prove that all three can get to grandma's in $3$ hours

Denote $n$ as a natural number, and $m$ as the result of writing the digits of $n$ in reverse order. Determine, if they exist, the numbers of three digits which satisfy $2m + S = n$, $S$ being the sum of the digits of $n$.

Find the largest prime number $p<1000$ for which there exists a complex number $z$ satisfying [list] [*] the real and imaginary part of $z$ are both integers; [*] $|z|=\sqrt{p}$, and [*] there exists a triangle whose three side lengths are $p$, the real part of $z^{3}$, and the imaginary part of $z^{3}$. [/list]

Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for evert integer $n \ge 1$, $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$. How many times $111$ appears on this sequence?

$A,B,C,D$ are points on a circle in that order. Prove that $|AB-CD|+|AD-BC| \ge 2|AC-BD|$.

Solve in prime numbers $ 2p^q \minus{} q^p \equal{} 7$.

Let $n\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\left\{1;\;2;\;...;\;r\right\}$ in a way such that the following two conditions are fulfilled: [b]1.[/b] Each number from the set $\left\{1;\;2;\;...;\;r\right\}$ occurs at least once as a label. [b]2.[/b] In each triangle formed by three vertices of the $n$-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side. [b](a)[/b] Find the maximal $r$ for which such a labelling is possible. [b](b)[/b] [i]Harder version (IMO Shortlist 2005):[/i] For this maximal value of $r$, how many such labellings are there? [hide="Easier version (5th German TST 2006) - contains answer to the harder version"] [i]Easier version (5th German TST 2006):[/i] Show that, for this maximal value of $r$, there are exactly $\frac{n!\left(n-1\right)!}{2^{n-1}}$ possible labellings.[/hide] [i]Proposed by Federico Ardila, Colombia[/i]

Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 6, 7, 9\}$ is one such sequence, with $n = 5$. What is the remainder when $K$ is divided by $10$? $\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$

The board has a natural number greater than $1$. At each step, Igor writes the number $n +\frac{n}{p}$ instead of the number $n$ on the board , where $p$ is some prime divisor of $n$. Prove that if Igor continues to rewrite the number infinite times, then he will choose infinitely times the number $3$ as a prime divisor of $p$. [hide=original wording]На доске записано какое-то натуральное число, большее 1. На каждом шагу Игорь переписывает имеющееся на доске число n на число n +n/p, где p - это какой-нибудь простой делитель числа n. Доказать, что если Игорь будет продолжать переписывать число бесконечно долго, то он бесконечно много раз выберет в качестве простого делителя p число 3.[/hide]

We have two lists of numbers: One initially containing 1,6,11,...,96, and the other initially containing 4,9,14,...,99. In every turn, we erase two numbers from one of the lists, and write $\frac{1}{3}$ of their sum (not necessarily an integer) in the other list. We continue this process until there are no possible moves. [list=a] [*] Prove that at the end of the process, there is exactly one number in each list. [*] Prove that those two numbers are [u]not[/u] equal. [/list]