Let $ \left\{x_n\right\}$, with $ n \equal{} 1, 2, 3, \ldots$, be a sequence defined by $ x_1 \equal{} 603$, $ x_2 \equal{} 102$ and $ x_{n \plus{} 2} \equal{} x_{n \plus{} 1} \plus{} x_n \plus{} 2\sqrt {x_{n \plus{} 1} \cdot x_n \minus{} 2}$ $ \forall n \geq 1$. Show that: [b](1)[/b] The number $ x_n$ is a positive integer for every $ n \geq 1$. [b](2)[/b] There are infinitely many positive integers $ n$ for which the decimal representation of $ x_n$ ends with 2003. [b](3)[/b] There exists no positive integer $ n$ for which the decimal representation of $ x_n$ ends with 2004.

Let $P(x)$ be a polynomial with integer coefficients, leading coefficient 1, and $P(0) = 3$. If the polynomial $P(x)^2 + 1$ can be factored as a product of two non-constant polynomials with integer coefficients, and the degree of $P$ is as small as possible, compute the largest possible value of $P(10)$. [i]2016 CCA Math Bonanza Individual #13[/i]

Through the point of intersection of the altitudes of an acute triangle $ABC$ three circles pass through, each of which touches one of the sides triangle at the foot of the altitude . Prove that the second intersection points of the circles are the vertices of a triangle similar to the original one.

A rectangular piece of paper whose length is $\sqrt3$ times the width has area $A$. The paper is divided into equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area $B$. What is the ratio $B:A$? [asy] import graph; size(6cm); real L = 0.05; pair A = (0,0); pair B = (sqrt(3),0); pair C = (sqrt(3),1); pair D = (0,1); pair X1 = (sqrt(3)/3,0); pair X2= (2*sqrt(3)/3,0); pair Y1 = (2*sqrt(3)/3,1); pair Y2 = (sqrt(3)/3,1); dot(X1); dot(Y1); draw(A--B--C--D--cycle, linewidth(2)); draw(X1--Y1,dashed); draw(X2--(2*sqrt(3)/3,L)); draw(Y2--(sqrt(3)/3,1-L)); [/asy] $ \textbf{(A)}\ 1:2\qquad\textbf{(B)}\ 3:5\qquad\textbf{(C)}\ 2:3\qquad\textbf{(D)}\ 3:4\qquad\textbf{(E)}\ 4:5 $

Let $f{}$ and $g{}$ be polynomials with integers coefficients. The leading coefficient of $g{}$ is equal to 1. It is known that for infinitely many natural numbers $n{}$ the number $f(n)$ is divisible by $g(n)$ . Prove that $f(n)$ is divisible by $g(n)$ for all positive integers $n{}$ such that $g(n)\neq 0$. [i]From the folklore[/i]

4. Consider spherical surface $S$ which radius is $1$, central point $(0,0,1)$ in $xyz$ space. If point $Q$ move to points on S expect $(0,0,2)$. Let $R$ be an intersection of plane $z=0$ and line $l$ pass point $Q$ and point $P (1,0,2)$. Find the range of moving $R$, then illustrate it.

$n$ players ($n \ge 4$) took part in the tournament. Each player played exactly one match with every other player, there were no draws. There was no four players $(A, B, C, D)$, such that $A$ won with $B$, $B$ won with $C$, $C$ won with $D$ and $D$ won with $A$. Determine, depending on $n$, maximum number of trios of players $(A, B, C)$, such that $A$ won with $B$, $B$ won with $C$ and $C$ won with $A$. (Attention: Trios $(A, B, C)$, $(B, C, A)$ and $(C, A, B)$ are the same trio.)

Find the polynomial $f$ with the following properties: $\bullet$ its leading coefficient is $1$, $\bullet$ its coefficients are nonnegative integers, $\bullet$ $72|f(x)$ if $x$ is an integer, $\bullet$ if $g$ is another polynomial with the same properties, then $g - f$ has a nonnegative leading coecient.

Let $(x_1, y_1, z_1), (x_2, y_2, z_2), \cdots, (x_{19}, y_{19}, z_{19})$ be integers. Prove that there exist pairwise distinct subscripts $i, j, k$ such that $x_i+x_j+x_k$, $y_i+y_j+y_k$, $z_i+z_j+z_k$ are all multiples of $3$.

On the side $BC$ of the triangle $ABC$ there are two points $D$ and $E$ such that $BD < BE$. Denote by $p_1$ and $p_2$ the perimeters of triangles $ABC$ and $ADE$ respectively. Prove that \[p_1 > p_2 + 2\cdot \min\{BD, EC\}.\]

Angle $C$ of an isosceles triangle $ABC$ equals $120^o$. Each of two rays emitting from vertex $C$ (inwards the triangle) meets $AB$ at some point ($P_i$) reflects according to the rule the angle of incidence equals the angle of reflection" and meets lateral side of triangle $ABC$ at point $Q_i$ ($i = 1,2$). Given that angle between the rays equals $60^o$, prove that area of triangle $P_1CP_2$ equals the sum of areas of triangles $AQ_1P_1$ and $BQ_2P_2$ ($AP_1 < AP_2$).

Given $1000$ linear functions $f_k(x)=p_k x + q_k$ where $k = 1 , 2 ,... , 1000$, it is necessary to evaluate their composite $f(x) =f_1 (f_2(f_3 ... f_{1000}(x)...))$ at the point $x_0$ . Prove that this can be done in no more than $30$ steps, where at each step one may execute simultaneously any number of arithmetic operations on pairs of numbers obtained from the previous step (at the first step one may use the numbers $p_1 , p_2 ,... ,p_{1000}, q_l , q_2 ,... ,q_{1000} , x_o$). {S. Fomin, Leningrad)

Let $ABCD$ be a rhombus and $P$ be a point on its side $BC$. The circle passing through $A, B$, and $P$ intersects $BD$ once more at the point $Q$ and the circle passing through $C,P$ and $Q$ intersects $BD$ once more at the point $R$. Prove that $A, R$ and $P$ lie on the one straight line. (D. Fomin, Leningrad)

Let $n$ be the smallest nonprime integer greater than 1 with no prime factor less than 10. Then A. $100 < n \leq 110$ B. $110 < n \leq 120$ C. $120 < n \leq 130$ D. $130 < n \leq 140$ E. $140 < n \leq 150$

The digits $0, 1 , 2, ..., 9$ are written in a $10 x 10$ table , each number appearing $10$ times . (a) Is it possible to write them in such a way that in any row or column there would be not more than $4$ different digits? (b) Prove that there must be a row or column containing more than $3$ different digits . { L . D . Kurlyandchik , Leningrad)

Let $ a\plus{}ar_1\plus{}ar_1^2\plus{}ar_1^3\plus{}\cdots$ and $ a\plus{}ar_2\plus{}ar_2^2\plus{}ar_2^3\plus{}\cdots$ be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is $ r_1$, and the sum of the second series is $ r_2$. What is $ r_1\plus{}r_2$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ \frac{1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \frac{1\plus{}\sqrt{5}}{2}\qquad \textbf{(E)}\ 2$

Let $a,\ b$ be positive real numbers with $a<b$. Define the definite integrals $I_1,\ I_2,\ I_3$ by $I_1=\int_a^b \sin\ (x^2)\ dx,\ I_2=\int_a^b \frac{\cos\ (x^2)}{x^2}\ dx,\ I_3=\int_a^b \frac{\sin\ (x^2)}{x^4}\ dx$. (1) Find the value of $I_1+\frac 12I_2$ in terms of $a,\ b$. (2) Find the value of $I_2-\frac 32I_3$ in terms of $a,\ b$. (3) For a positive integer $n$, define $K_n=\int_{\sqrt{2n\pi}}^{\sqrt{2(n+1)\pi}} \sin\ (x^2)\ dx+\frac 34\int_{\sqrt{2n\pi}}^{\sqrt{2(n+1)\pi}}\frac{\sin\ (x^2)}{x^4}\ dx$. Find the value of $\lim_{n\to\infty} 2n\pi \sqrt{2n\pi} K_n$. [i]2011 Tokyo University of Science entrance exam/Information Sciences, Applied Chemistry, Mechanical Enginerring, Civil Enginerring[/i]

Let $a, b, c, d$ be natural numbers such that $a + b + c + d = 2018$. Find the minimum value of the expression: $$E = (a-b)^2 + 2(a-c)^2 + 3(a-d)^2+4(b-c)^2 + 5(b-d)^2 + 6(c-d)^2.$$

Solve the following system of equations in positive integers \[\left\{\begin{array}{cc}a^3-b^3-c^3=3abc\\ \\ a^2=2(b+c)\end{array}\right.\]

Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was $\$12.48$, but in January her bill was $\$17.54$ because she used twice as much connect time as in December. What is the fixed monthly fee? $\mathrm{(A)} \$2.53 \qquad\mathrm{(B)} \$5.06 \qquad\mathrm{(C)} \$6.24 \qquad\mathrm{(D)} \$7.42 \qquad\mathrm{(E)} \$8.77$

How many different sentences with two words can be written using all letters of the word $\text{YARI\c{S}MA}$? (The Turkish word $\text{YARI\c{S}MA}$ means $\text{CONTEST}$. It will produce same result.) $ \textbf{(A)}\ 2520 \qquad\textbf{(B)}\ 5040 \qquad\textbf{(C)}\ 15120 \qquad\textbf{(D)}\ 20160 \qquad\textbf{(E)}\ \text{None of the above} $

Given a board $3 \times 2021$, all cells of which are white. Two players in turns colour two white cells, which are either in the same row or column, in black. A player, which can not make a move, loses. Which of the player can guarantee his win regardless of the moves of his opponent?

Is it possible to fill the cells of a table of size $2019\times2019$ with pairwise distinct positive integers in such a way that in each rectangle of size $1\times2$ or $2\times1$ the larger number is divisible by the smaller one, and the ratio of the largest number in the table to the smallest one is at most $2019^4?$

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

Prove that there exists exactly one function $ƒ$ which is defined for all $x \in R$, and for which holds: $\bullet$ $x \le y \Rightarrow f(x) \le f(y)$, for all $x, y \in R$, and $\bullet$ $f(f(x)) = x$, for all $x \in R$.