What is the sum of the digits of the square of $ 111,111,111$? $ \textbf{(A)}\ 18 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 63 \qquad \textbf{(E)}\ 81$

Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20. [i]Proposed by Luxembourg[/i]

Let $ABC$ be a triangle. Points $D$, $E$, and $F$ are respectively on the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ of $\triangle ABC$. Suppose that \[ \frac{AE}{AC} = \frac{CD}{CB} = \frac{BF}{BA} = x \] for some $x$ with $\frac{1}{2} < x < 1$. Segments $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ cut the triangle into 7 nonoverlapping regions: 4 triangles and 3 quadrilaterals. The total area of the 4 triangles equals the total area of the 3 quadrilaterals. Compute the value of $x$.

Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of integers defined recursively as $ x_{n+2}=5x_{n+1}-x_n. $ Prove that $ \left( x_n\right)_{n\ge 1} $ has a subsequence whose terms are multiples of $ 22 $ if $ \left( x_n\right)_{n\ge 1} $ has a term that is multiple of $ 22. $

Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

A function $ f$ defined on the positive integers (and taking positive integers values) is given by: $ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\ f(2 \cdot n) \equal{} f(n) \\ f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\ f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$ for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$

A square is cut into red and blue rectangles. The sum of areas of red triangles is equal to the sum of areas of the blue ones. For each blue rectangle, we write the ratio of the length of its vertical side to the length of its horizontal one and for each red rectangle, the ratio of the length of its horizontal side to the length of its vertical side. Find the smallest possible value of the sum of all the written numbers.

Let $f(x) = 2x^2 + x - 1, f^{0}(x) = x$, and $f^{n+1}(x) = f(f^{n}(x))$ for all real $x>0$ and $n \ge 0$ integer (that is, $f^{n}$ is $f$ iterated $n$ times). a) Find the number of distinct real roots of the equation $f^{3}(x) = x$ b) Find, for each $n \ge 0$ integer, the number of distinct real solutions of the equation $f^{n}(x) = 0$

The [i]equatorial algebra[/i] is defined as the real numbers equipped with the three binary operations $\natural$, $\sharp$, $\flat$ such that for all $x, y\in \mathbb{R}$, we have \[x\mathbin\natural y = x + y,\quad x\mathbin\sharp y = \max\{x, y\},\quad x\mathbin\flat y = \min\{x, y\}.\] An [i]equatorial expression[/i] over three real variables $x$, $y$, $z$, along with the [i]complexity[/i] of such expression, is defined recursively by the following: [list] [*] $x$, $y$, and $z$ are equatorial expressions of complexity 0; [*] when $P$ and $Q$ are equatorial expressions with complexity $p$ and $q$ respectively, all of $P\mathbin\natural Q$, $P\mathbin\sharp Q$, $P\mathbin\flat Q$ are equatorial expressions with complexity $1+p+q$. [/list] Compute the number of distinct functions $f: \mathbb{R}^3\rightarrow \mathbb{R}$ that can be expressed as equatorial expressions of complexity at most 3. [i]Proposed by Yannick Yao[/i]

Let $n$ be a positive integer. Prove that the equation \[x+y+\frac{1}{x}+\frac{1}{y}=3n\] does not have solutions in positive rational numbers.

Found the smaller multiple of $2019$ of the form $abcabc\dots abc$, where $a,b$ and $c$ are digits.

Determine the (real) domain of a function $$y=\sqrt{1-\frac{x}{4}|x|+\sqrt{1-\frac{x}{2}|x|\,}\,}-\sqrt{1-\frac{x}{4}|x|-\sqrt{1-\frac{x}{2}|x|\,}\,}$$ and draw its graph.

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 $x$. )

Consider a regular triangular array of $n(n+1)/2$ points.Let $f(n)$ denote the number of equilateral triangles formed by taking some $3$ points in the array as vertices.Prove that $f(n)=\frac{(n-1)n(n+1)(n+2)}{24}$.

From a squared sheet of paper of size $29 \times 29, 99$ pieces, each a $2\times 2$ square, are cut off (all cutting is along the lines bounding the squares). Prove that at least one more piece of size $2\times 2$ may be cut from the remaining part of the sheet. (S Fomin, Leningrad)

Let $f(x)=\cos ^ 4 x+3\sin ^ 4 x$. Evaluate $\int_0^{\frac{\pi}{2}} |f'(x)|dx$. [i]2011 Tokyo University of Science entrance exam/Management[/i]

Let $ u,v $ be two endomorphisms of a finite vectorial space that verify the relation $ uv-vu=u. $ Calculate $ u^kv-vu^k $ and show that u is nilpotent.

Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$. a) Prove that: For all $m_1,m_2\in \mathbb{Z^+}$, we have:$$k([m_1,m_2])=[k(m_1),k(m_2)].$$(Here $[a,b]$ is the least common multiple of $a,b.$) b) Determine $k(2),k(4),k(5),k(10).$

Let $p$ be an odd prime number less than $10^5$. Granite and Pomegranate play a game. First, Granite picks a integer $c \in \{2,3,\dots,p-1\}$. Pomegranate then picks two integers $d$ and $x$, defines $f(t) = ct + d$, and writes $x$ on a sheet of paper. Next, Granite writes $f(x)$ on the paper, Pomegranate writes $f(f(x))$, Granite writes $f(f(f(x)))$, and so on, with the players taking turns writing. The game ends when two numbers appear on the paper whose difference is a multiple of $p$, and the player who wrote the most recent number wins. Find the sum of all $p$ for which Pomegranate has a winning strategy. [i]Proposed by Yang Liu[/i]

Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0. \] Determine when equality occurs.

Let the sequence $(a_{n})$ be defined by $a_{1} = t$ and $a_{n+1} = 4a_{n}(1 - a_{n})$ for $n \geq 1$. How many possible values of t are there, if $a_{1998} = 0$?

Prove that the number $\left\lfloor\left(5+\sqrt{35}\right)^{2n-1}\right\rfloor$ is divisible by $10^n$ for each $n\in\mathbb N$.

Several square-shaped papers are situated on a table such that every side of the paper is positioned parallel to the sides of the table. Each paper has a colour, and there are $n$ different coloured papers. It is known that for every $n$ papers with distinct colors, we can always find an overlapping pair of papers. Prove that, using $2n- 2$ nails, it is possible to hammer all the squares of a certain colour to the table.

In a right triangle $ABC$ with a hypotenuse $AB$, the angle $A$ is greater than the angle $B$. Point $N$ lies on the hypotenuse $AB$ , such that $BN = AC$. Construct this triangle $ABC$ given the point $N$, point $F$ on the side $AC$ and a straight line $\ell$ containing the bisector of the angle $A$ of the triangle $ABC$. (Grigory Filippovsky)

There is a $11\times 11$ square grid. We divided this in $5$ rectangles along unit squares. How many ways that one of the rectangles doesn’t have a edge on basic circumference. Note that we count as different ways that one way coincides with another way by rotating or reversing.