Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A,B,C,D)\in\{1,2,\ldots,N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3+Bn^2+2Cn+D$ is divisible by $N$.

Let $P_n(x)=a_0 + a_1x + \cdots + a_nx^n$, with $n \geq 2$, be a real-coefficient polynomial. Prove that if there exists $a > 0$ such that \begin{align*} P_n(x) = (x + a)^2 \left( \sum_{i=0}^{n-2} b_i x^i \right), \end{align*} where $b_i$ are positive real numbers, then there exists some $i$, with $1 \leq i \leq n-1$, such that \[a_i^2 - 4a_{i-1}a_{i+1} \leq 0.\]

Let $Q_0(x)=1$, $Q_1(x)=x,$ and \[Q_n(x)=\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}\] for all $n\ge 2.$ Show that, whenever $n$ is a positive integer, $Q_n(x)$ is equal to a polynomial with integer coefficients.

Find the smallest $n$ such that $n^2 -n+11$ is the product of four primes (not necessarily distinct).

How many paths from $(0, 0)$ to $(2020, 2020)$, consisting of unit steps up and to the right, pass through at most one point with both coordinates even, other than $(0,0)$ and $(2020,2020)$?

In an acute scalene triangle $ABC$ the bisector of the acute angle between the altitudes $AA_1$ and $CC_1$ meets the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The bisector of the angle $B$ intersects the segment joining the orthocenter of $ABC$ and the midpoint of $AC$ at point $R$. Prove that $P$, $B$, $Q$, $R$ lie on a circle.

Let $a,b$ be two constants within the open interval $(0,\frac{1}{2})$. Find all continous functions $f(x)$ such that \[f(f(x))=af(x)+bx\] holds for all real $x$. This is much harder than the problems we had in the 1st TST...

Fin the functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ such that: $$ \frac{f(x+y)+f(x)}{2x+f(y)} =\frac{2y+f(x)}{f(x+y)+f(y)} ,\quad\forall x,y\in\mathbb{N} . $$

How many $7$-digit numbers with distinct digits can be made that are divisible by $3$?

An equilateral triangle with side $n$ is built with $n^{2}$ [i]plates[/i] - equilateral triangles with side $1$. Each plate has one side black, and the other side white. We name [i]the move[/i] the following operation: we choose a plate $P$, which has common sides with at least two plates, whose visible side is the same color as the visible side of $P$. Then, we turn over plate $P$. For any $n\geq 2$ decide whether there exists an innitial configuration of plates permitting for an infinite sequence of moves.

A table consists of $17 \times 17$ squares. In each square one positive integer from $1$ to $17$ is written, every such number is written in exactly $17$ squares. Prove that there is a row or a column of the table that contains at least $5$ different numbers.

a) A student takes a subway to an Olympiad, pays one ruble and gets his change. Prove that if he takes a tram (street car) on his way home, he will have enough coins to pay the fare without change. b) A student is going to a club. (S)he takes a tram, pays one ruble and gets the change. Prove that on the way back by a tram (s)he will be able to pay the fare without any need to change. Note: In $1957$, the price of a subway ticket was $50$ kopeks, that of a tram ticket $30$ kopeks, the denominations of the coins were $1, 2, 3, 5, 10, 15$, and $20$ kopeks. ($1$ rouble = $100$ kopeks.)

Given points $ A $, $ B $, $ C $ and $ D $ that do not lie in the same plane. Draw a plane through the point $ A $ such that the orthogonal projection of the quadrilateral $ ABCD $ on this plane is a parallelogram.

Let $E(x,y)=\frac{(1+x)(1+y)(1+xy)}{(1+x^2)(1+y^2)}$. Find the minimum and maximum value of $E$ on $\mathbb{R}^2$. [i]Dorel Miheț[/i]

Let be given integers k and n such that 1<=k<=n. Find the number of ordered k-tuples (a_1,a_2,...,a_n), where a_1, a_2, ..., a_k are different and in the set {1,2,...,n}, satisfying 1) There exist s, t such that 1<=s<t<=k and a_s>a_t. 2) There exists s such that 1<=s<=k and a_s is not congruent to s mod 2. P.S. This is the only problem from VMO 1996 I cannot find a solution or I cannot solve. But I'm not good at all in combinatoric...

Find all prime numbers $p$ and nonnegative integers $x\neq y$ such that $x^4- y^4=p(x^3-y^3)$. [i]Proposed by Bulgaria[/i]

Assume that a triangle $ABC$ satisfies the following property: For any point from the triangle, the sum of distances from $D$ to the lines $AB,BC$ and $CA$ is less than $1$. Prove that the area of the triangle is less than or equal to $\frac{1}{\sqrt3}$

Let $\Omega_1$ and $\Omega_2$ be two circles intersecting at distinct points $P$ and $Q$. The line tangent to $\Omega_1$ at $P$ passes through $\Omega_2$ at a second point $A$, and the line tangent to $\Omega_2$ at $P$ passes through $\Omega_1$ at a second point $B$. Ray $AQ$ intersects $\Omega_1$ at a second point $C$, and ray $BQ$ intersects $\Omega_2$ at a second point $D$. Suppose that $\angle CPD > \angle APB$ (measuring both angles as the non-reflex angle) and that \[ \frac{\text{Area}(CPD)}{PA \cdot PB} = \frac{1}{4}. \] Find the sum of all possible measures of $\angle APB$ in degrees.

Let $n$ and $k$ be positive integers and let $$T = \{ (x,y,z) \in \mathbb{N}^3 \mid 1 \leq x,y,z \leq n \}$$ be the length $n$ lattice cube. Suppose that $3n^2 - 3n + 1 + k$ points of $T$ are colored red such that if $P$ and $Q$ are red points and $PQ$ is parallel to one of the coordinate axes, then the whole line segment $PQ$ consists of only red points. Prove that there exists at least $k$ unit cubes of length $1$, all of whose vertices are colored red.

Find the sum \[\frac{1}{1(3)}+\frac{1}{3(5)}+\dots+\frac{1}{(2n-1)(2n+1)}+\dots+\frac{1}{255(257)}.\] $\textbf{(A) }\frac{127}{255}\qquad\textbf{(B) }\frac{128}{255}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{128}{257}\qquad \textbf{(E) }\frac{129}{257}$

Circles $A$, $B$, and $C$ are externally tangent to each other and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius 1 and passes through the center of $D$. What is the radius of circle $B$? [asy] size(200); defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(Circle(origin, 2)); draw(Circle((-1,0), 1)); draw(Circle((6/9, 8/9), 8/9)); draw(Circle((6/9, -8/9), 8/9)); label("$A$", (-1.2, -0.2), NE); label("$B$", (6/9, 7/9), N); label("$C$", (6/9, -7/9), S); label("$D$", 2*dir(110), dir(110));[/asy] $ \textbf{(A)}\; \frac23\qquad \textbf{(B)}\; \frac{\sqrt{3}}2\qquad \textbf{(C)}\; \frac78\qquad \textbf{(D)}\; \frac89\qquad \textbf{(E)}\; \frac{1+\sqrt3}3 $

[b]6.[/b] Prove or disprove the following two propositions: [b](i)[/b] If $a$ and $b$ are positive integers such that $a<b$, then in any set of $b$ consecutive integers there are two whose product is divisible by $ab$ [b](ii)[/b] If $a,b$ and $c$ are positive integers such that $a<b<c$, then in any set of $c$ consecutive integers there are three whose product is divisible by $abc$. [b](N.8)[/b]

Let $d\geq 1$ and $n\geq 0$ be integers. Find the number of ways to write down a nonnegative integer in each square of a $d\times d$ grid such that the numbers in any set of $d$ squares, no two in the same row or column, sum to $n$.

A point $T$ is given on the altitude through point $C$ in the acute triangle $ABC$ with circumcenter $O$, such that $\measuredangle TBA=\measuredangle ACB$. If the line $CO$ intersects side $AB$ at point $K$, prove that the perpendicular bisector of $AB$, the altitude through $A$ and the segment $KT$ are concurrent.

On the side $AC$ of an acute triangle $\triangle ABC$, a point $D$ is taken such that $2AD=CD=2, BD\perp AC$. A circle of radius $2$ passes through $A$ and $D$ and is tangent to the circumcircle of $\triangle BDC$. Find $[\text{Area}(\triangle ABC)]$ where $[.]$ is the greatest integer function.