Let $D$ be the midpoint of the hypotenuse $AB$ of a right triangle $ABC$. Let $O_1$ and $O_2$ be the circumcenters of the $ADC$ and $DBC$ triangles, respectively. a) Prove that $\angle O_1DO_2$ is right. b) Prove that $AB$ is tangent to the circle of diameter $O_1O_2$ .

Describe how to place the vertices of a triangle in the faces of a cube in such a way that the shortest side of the triangle is the biggest possible.

Given $n>4$ points in the plane, no three collinear. Prove that there are at least $\frac{(n-3)(n-4)}{2}$ convex quadrilaterals with vertices amongst the $n$ points.

Let $A_{11}$ denote the answer to problem $11$. Determine the smallest prime $p$ such that the arithmetic sequence $p,p+A_{11},p+2A_{11},\cdots$ begins with the largest number of primes. There is just one triple of possible $(A_{10},A_{11},A_{12})$ of answers to these three problems. Your team will receive credit only for answers matching these. (So, for example, submitting a wrong answer for problem $11$ will not alter the correctness of your answer to problem $12$.)

Let $ABCD$ be a rhombus of sides $AB = BC = CD= DA = 13$. On the side $AB$ construct the rhombus $BAFE$ outside $ABCD$ and such that the side $AF$ is parallel to the diagonal $BD$ of $ABCD$. If the area of $BAFE$ is equal to $65$, calculate the area of $ABCD$.

On a circle are arranged $100$ baskets, each containing at least one candy. The total number of candies is $780$. Asad and Sevinch make moves alternatingly, with Asad going first. On one move, Asad takes all the candies from $9$ consecutive non-empty baskets, while Sevinch takes all the candies from a single non-empty basket that has at least one empty neighboring basket. Prove that Asad can take overall at least $700$ candies, regardless of the initial distribution of candies and Sevinch's actions. [i] Shubin Yakov, Russia [/i]

Let $ f: \mathbb{R}^2\to\mathbb{R}$ be a function such that $ f(x,y)\plus{}f(y,z)\plus{}f(z,x)\equal{}0$ for real numbers $ x,y,$ and $ z.$ Prove that there exists a function $ g: \mathbb{R}\to\mathbb{R}$ such that $ f(x,y)\equal{}g(x)\minus{}g(y)$ for all real numbers $ x$ and $ y.$

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Let $ABC$ be a triangle and let $BB_1,CC_1$ be respectively the bisectors of $\angle{B},\angle{C}$ with $B_1$ on $AC$ and $C_1$ on $AB$, Let $E,F$ be the feet of perpendiculars drawn from $A$ onto $BB_1,CC_1$ respectively. Suppose $D$ is the point at which the incircle of $ABC$ touches $AB$. Prove that $AD=EF$

Prove that the second last digit of each power of three is even . (V . I . Plachkos)

Let $a>0$. Evaluate $\int_0^a x^2\left(1-\frac{x}{a}\right)^adx$. [i]2011 Keio University entrance exam/Science and Technology[/i]

In a family there are four children of different ages, each age being a positive integer not less than $2$ and not greater than $16$. A year ago the square of the age of the eldest child was equal to the sum of the squares of the ages of the remaining children. One year from now the sum of the squares of the youngest and the oldest will be equal to the sum of the squares of the other two. How old is each child?

For a positive integer $n$, denote by $\tau (n)$ the number of its positive divisors. For a positive integer $n$, if $\tau (m) < \tau (n)$ for all $m < n$, we call $n$ a good number. Prove that for any positive integer $k$, there are only finitely many good numbers not divisible by $k$.

Rhombus $ ABCD$ is similar to rhombus $ BFDE$. The area of rhombus $ ABCD$ is 24, and $ \angle BAD \equal{} 60^\circ$. What is the area of rhombus $ BFDE$? [asy] size(180); defaultpen(linewidth(0.7)+fontsize(11)); pair A=origin, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/sqrt(3)); pair point=(3/2, sqrt(3)/2); draw(B--C--D--A--B--F--D--E--B); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F));[/asy] $ \textbf{(A) } 6 \qquad \textbf{(B) } 4\sqrt {3} \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 6\sqrt {3}$

The bisectors of angles $ B$ and $ C$ of triangle $ ABC$ intersect the opposite sides in points $ B'$ and $ C'$ respectively. Show that the line $ B'C'$ intersects the incircle of the triangle.

Determine all integers $n$ that can be written in the form \[ n = \frac{a^2 - b^2}{b}, \] where $a$ and $b$ are positive integers. [i](Walther Janous)[/i]

Define the sequence $a_n$ by the rule $$a_{n+1} =\left \lfloor \frac{a_n} 2 \right \rfloor + \left \lfloor \frac{a_n}3 \right \rfloor$$ for $n \in \{1, 2, 3, 4, 5, 6, 7\}$, where $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$. If $a_8=8$, how many possible values are there for $a_1$ given that it is a positive integer? [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 10)[/i]

On each edge of a graph is written a real number,such that for every even tour of this graph,sum the edges with signs alternatively positive and negative is zero.prove that one can assign to each of the vertices of the graph a real number such that sum of the numbers on two adjacent vertices is the number on the edge between them.(tour is a closed path from the edges of the graph that may have repeated edges or vertices)

Let $f: [0,\infty)\to\mathbb R$ be a function such that for any $x>0$ the sequence $\{f(nx)\}_{n\geq 0}$ is increasing. a) If the function is also continuous on $[0,1]$ is it true that $f$ is increasing? b) The same question if the function is continuous on $\mathbb Q \cap [0, \infty)$.

Rick has $7$ books on his shelf: three identical red books, two identical blue books, a yellow book, and a green book. Dave accidentally knocks over the shelf and has to put the books back on in the same order. He knows that none of the red books were next to each other and that the yellow book was one of the first four books on the shelf, counting from the left. If Dave puts back the books according to the rules, but otherwise randomly, what is the probability that he puts the books back correctly?

A sequence of natural numbers is written according to the following rule: [i] the first two numbers are chosen and thereafter, in order to write a new number, the sum of the last numbers is calculated using the two written numbers, we find the greatest odd divisor of their sum and the sum of this greatest odd divisor plus one is the following written number. [/i]The first numbers are $25$ and $126$ (in that order), and the sequence has $2015$ numbers. Find the last number written.

How many multiples of $17$ are there between $23$ and $227$?

Let $ABCDEF$ be a convex hexagon such that $\angle B+\angle D+\angle F=360^{\circ }$ and \[ \frac{AB}{BC} \cdot \frac{CD}{DE} \cdot \frac{EF}{FA} = 1. \] Prove that \[ \frac{BC}{CA} \cdot \frac{AE}{EF} \cdot \frac{FD}{DB} = 1. \]

For a positive integer $n$, consider a square cake which is divided into $n \times n$ pieces with at most one strawberry on each piece. We say that such a cake is [i]delicious[/i] if both diagonals are fully occupied, and each row and each column has an odd number of strawberries. Find all positive integers $n$ such that there is an $n \times n$ delicious cake with exactly $\left\lceil\frac{n^2}{2}\right\rceil$ strawberries on it.

In a row the numbers $1,2,...,2010$ have been written. Two players, taking turns, write $+$ or $\times$ between two consecutive numbers whenever possible. The first player wins if the algebraic sum obtained is divisible by $3$; otherwise, the second player wins. Find a winning strategy for one of the players.