Define $p(x) = 4x^{3}-2x^{2}-15x+9, q(x) = 12x^{3}+6x^{2}-7x+1$. Show that each polynomial has just three distinct real roots. Let $A$ be the largest root of $p(x)$ and $B$ the largest root of $q(x)$. Show that $A^{2}+3 B^{2}= 4$.

Let $\Lambda$ denote the set of points $(x,y)$ in 2D space with integer coordinates such that $0\leq x\leq 4$ and $0\leq y\leq 2$. That is, \[ \Lambda=\{ (x,y) \in \mathbb{Z}^2: 0\leq x\leq 4, \ 0\leq y\leq 2 \}. \] Find the number of ways to connect points of $\Lambda$ with segments of length $\sqrt{2}$ or $\sqrt{5}$ such that the interior of any unit square with vertices in $\Lambda$ contains part of exactly one segment; an example is shown below (connections that differ by reflections are distinct). [asy] unitsize(1cm); dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((4,0)); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((4,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); dot((4,2)); draw((0,0)--(1,1)); draw((0,2)--(2,1)); draw((1,1)--(2,0)); draw((2,0)--(3,2)); draw((3,1)--(4,2)); draw((3,0)--(4,1)); [/asy]

In the plane, six circles are given so that none of the circles contain one the center of the other. Show that there is no point that lies in all the circles.

Let $a, b$ and $c$ be integers such that $a + b + c = 0$. (a) Show that $a^4 + b^4 + c^4$ is divisible by $a^2 + b^2 + c^2$. (b) Show that $a^{100} + b^{100} + c^{100}$ is divisible by $a^2 + b^2 + c^2$. .

If $a+b=b-c=c-a=3$, find $a+b+c$. $\text{(A) }3\qquad\text{(B) }4\frac{1}{2}\qquad\text{(C) }6\qquad\text{(D) }7\frac{1}{2}\qquad\text{(E) }9$

Let $n\ge 3$ be an integer and $a_1,a_2,\dots ,a_n$ be a finite sequence of positive integers, such that, for $k=2,3,\dots ,n$ $$n(a_k+1)-(n-1)a_{k-1}=1.$$ Prove that $a_n$ is not divisible by $(n-1)^2$.

Let $ A$ be a family of proper closed subspaces of the Hilbert space $ H\equal{}l^2$ totally ordered with respect to inclusion (that is , if $ L_1,L_2 \in A$, then either $ L_1\subset L_2$ or $ L_2\subset L_1$). Prove that there exists a vector $ x \in H$ not contaied in any of the subspaces $ L$ belonging to $ A$. [i]B. Szokefalvi Nagy[/i]

An equilateral triangle is dissected into white and black triangles. It is known that all white triangles are right-angled and mutually congruent, and all black triangles are isosceles and also mutually congruent. Is it necessarily true that a) all angles of white triangles are multiples of $30^{\circ}$; (4 marks) b) all angles of black triangles are multiples of $30^{\circ}$ ? (5 marks)

Let $p$ be a prime number and $n$ a positive integer. Prove that the product \[{N=\frac{1}{p^{n^2}}} \prod_{i=1;2 \nmid i}^{2n-1} \biggl[ \left( (p-1)! \right) \binom{p^2 i}{pi}\biggr]\] Is a positive integer that is not divisible by $p.$

Find all pairs $(x,y)$ of positive integers such that $x^{x+y} =y^{y-x}$.

Fins all ordered triples $ \left(a,b,c\right)$ of positive integers such that $ abc \plus{} ab \plus{} c \equal{} a^3$.

A sequence of positive integers $a_1, a_2,\ldots, a_n$ is called [i]ladder[/i] of length $n$ if it consists of $n$ consecutive integers in ascending order. (a) Prove that for every positive integer $n$ there exist two ladders of length $n$, with no elements in common, $a_1, a_2,\ldots, a_n$ and $b_1, b_2,\ldots, b_n$, such that for all $i$ between $1$ and $n$, the greatest common divisor of $a_i$ and $b_i$ is equal to $1$. (b) Prove that for every positive integer $n$ there exist two ladders of length $n$, with no elements in common, $a_1, a_2,\ldots, a_n$ and $b_1, b_2,\ldots, b_n$, such that for all $i$ between $1$ and $n$, the greatest common divisor of $a_i$ and $b_i$ is greater than $1$.

Let $ p\ge 2 $ be a natural number that divides $ \binom{p}{k} , $ for any natural number $ k $ smaller than $ p. $ Prove that: [b]a)[/b] $ p $ is prime. [b]b)[/b] $ p^2 $ divides $ -2+\binom{2p}{p} . $

Five persons of different heights stand next to the another on numbered booths to take a picture. From how many ways can be arranged so that people in positions $ 1$ and $3$ are both taller than the person in the position $2$?

Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are $60$, $20$, and $15$ respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area.

Let $x, y$, and $z$ be positive integers satisfying the following system of equations: $$x^2 +\frac{2023}{x}= 2y^2$$ $$y +\frac{2028}{y^2} = z^2$$ $$2z +\frac{2025}{z^2} = xy$$ Find $x + y + z$.

The extension of the bisector of angle $A$ of triangle $ABC$ intersects with the circumscribed circle of this triangle at point $W$. A straight line is drawn through $W$, which is parallel to side $AB$ and intersects sides $BC$ and $AC$ , at points $N$ and $K$, respectively. Prove that the line $AW$ is tangent to the circumscribed circle of $\vartriangle CNW$. (Sergey Yakovlev)

Which bond is strongest? ${ \textbf{(A)}\ \text{C=C}\qquad\textbf{(B)}\ \text{C=N}\qquad\textbf{(C)}\ \text{C=O}\qquad\textbf{(D)}}\ \text{C=S}\qquad $

Find the largest positive integer $n$ such that no two adjacent digits are the same, and for any two distinct digits $0 \leq a,b \leq 9 $, you can't get the string $abab$ just by removing digits from $n$.

Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.

Is there an injective function $f : Z^+ \to Q$ satisfying the equation $f(xy) = f(x) + f(y)$ for all positive integers $x$ and $y$?

For a positive integer $n$, let $p(n)$ denote the product of the digits of $n$, and let $s(n)$ denote the sum of the digits of $n$. Find the sum of all positive integers $n$ satisfying $p(n)s(n)=8$.

Let be a natural number $ n\ge 2, $ and a matrix $ A\in\mathcal{M}_n\left( \mathbb{C} \right) $ whose determinant vanishes. Show that $$ \left( A^* \right)^2 =A^*\cdot\text{tr} A^*, $$ where $ A^* $ is the adjugate of $ A. $

It is possible to partition the set $\{100, 101,\cdots , 1000\}$ into two subsets so that for any two distinct elements $x$ and $y$ belonging to the same subset $ \sqrt[3]{x + y}$ is irrational?

Prove that any triangle can be divided into $22$ triangles, each of which has an angle of $22^o$, and another $23$ triangles, each of which has an angle of $23^o$.