Let $P(x), Q(x), $ and $R(x)$ be three monic quadratic polynomials with only real roots, satisfying $$P(Q(x))=(x-1)(x-3)(x-5)(x-7)$$$$Q(R(x))=(x-2)(x-4)(x-6)(x-8)$$ for all real numbers $x.$ What is $P(0)+Q(0)+R(0)?$ [i]Proposed by Kyle Lee[/i]

For the Georg Mohr game, a playing piece is used, a Georg Mohr cube (i.e. a die whose six sides show the letters G, E, O, R, M and H) as well as a game board: [img]https://cdn.artofproblemsolving.com/attachments/0/9/30ca5cd2579bfcc1d702b40f3ef58916ac768f.png[/img] With each stroke, you advance to the next field with that letter the cube shows; if it is not possible to advance, one remains standing. Peter playing the georg mohr game. Determine the probability that he completes played in two strokes.

Jacob has a piece of bread shaped like a figure $8$, marked into sections and all initially connected as one piece of bread. The central part of the “$8$” is a single section, and each of the two loops of “$8$” is divided into an additional $1010$ pieces. For each section, there is a $50$ percent chance that Jacob will decide to cut it out and give it to a friend, and this is done independently for each section. The remaining sections of bread form some number of connected pieces. If $E$ is the expected number of these pieces, and $k$ is the smallest positive integer so that $2^k(E - \lfloor E \rfloor ) \ge 1$, find $\lfloor E \rfloor +k$. (Here, we say that if Jacob donates all pieces, there are $0$ pieces left).

The real-valued function $f(x)$ is defined on the reals. It satisfies $|f(x)| \le A$, $|f''(x)| \le B$ for some positive $A, B$ (and all $x$). Show that $|f'(x)| \le C$, for some fixed$ C$, which depends only on $A$ and $B$. What is the smallest possible value of $C$?

Can You fill the squares of the infinite cross-lined paper with integers so, that the sum of the numbers in every $4\times 6$ fields rectangle would be a) $10$? b) $1$?

[b]Q1.[/b] Assum that $a-b=-(a-b).$ Then: $(A) \; a=b; \qquad (B) \; a<b; \qquad (C) \; a>b \qquad (D) \; \text{ It is impossible to compare those of a and b.}$

Determine all triples $(x,y,z)$ of real numbers that satisfy all of the following three equations: $$\begin{cases} \lfloor x \rfloor + \{y\} =z \\ \lfloor y \rfloor + \{z\} =x \\ \lfloor z \rfloor + \{x\} =y \end{cases}$$

Let $a$, $b$, $c$, and $d$ be positive real numbers such that $a+b=c+d$ and $a^2+b^2>c^2+d^2$. Prove that $a^3+b^3>c^3+d^3$.

Find the largest positive integer $N$ such that the equation $99x + 100y + 101z = N$ has an unique solution in the positive integers $x, y, z$. [i](7 points)[/i]

Prove that if $n$ is an odd natural number, then $1^n +2^n +\cdots +n^n$ is divisible by $n^2$.

Let $P(x) = x^3 - px^2 + qx - r$ be a cubic polynomial with integer roots $a, b, c$. [b](a)[/b] Show that the greatest common divisor of $p, q, r$ is equal to $1$ if the greatest common divisor of $a, b, c$ is equal to $1$. [b](b)[/b] What are the roots of polynomial $Q(x) = x^3-98x^2+98sx-98t$ with $s, t$ positive integers.

Let $n$ be a natural number and suppose that $ w_1, w_2, \ldots , w_n$ are $n$ weights . We call the set of $\{ w_1, w_2, \ldots , w_n\}$ to be a [i]Perfect Set [/i]if we can achieve all of the $1,2, \ldots, W$ weights with sums of $ w_1, w_2, \ldots , w_n$, where $W=\sum_{i=1}^n w_i $. Prove that if we delete the maximum weight of a Perfect Set, the other weights make again a Perfect Set.

A container is shaped like a right circular cone with base diameter $18$ and height $12$. The vertex of the container is pointing down, and the container is open at the top. Four spheres, each with radius $3$, are placed inside the container as shown. The first sphere sits at the bottom and is tangent to the cone along a circle. The second, third, and fourth spheres are placed so they are each tangent to the cone and tangent to the rst sphere, and the second and fourth spheres are each tangent to the third sphere. The volume of the tetrahedron whose vertices are at the centers of the spheres is $K$. Find $K^2$. [img]https://cdn.artofproblemsolving.com/attachments/9/c/648ec2cf0f0c2f023cd00b1c0595a9396d0ddc.png[/img]

Given an acute triangle $ ABC$. The incircle of triangle $ ABC$ touches $ BC,CA,AB$ respectively at $ D,E,F$. The angle bisector of $ \angle A$ cuts $ DE$ and $ DF$ respectively at $ K$ and $ L$. Suppose $ AA_1$ is one of the altitudes of triangle $ ABC$, and $ M$ be the midpoint of $ BC$. (a) Prove that $ BK$ and $ CL$ are perpendicular with the angle bisector of $ \angle BAC$. (b) Show that $ A_1KML$ is a cyclic quadrilateral.

A rectangle with a diagonal of length $ x$ is twice as long as it is wide. What is the area of the rectangle? $ \textbf{(A)}\ \frac14x^2 \qquad \textbf{(B)}\ \frac25x^2 \qquad \textbf{(C)}\ \frac12x^2 \qquad \textbf{(D)}\ x^2 \qquad \textbf{(E)}\ \frac32x^2$

A figure on a computer screen shows $n$ points on a sphere, no four coplanar. Some pairs of points are joined by segments. Each segment is colored red or blue. For each point there is a key that switches the colors of all segments with that point as endpoint. For every three points there is a sequence of key presses that makes the three segments between them red. Show that it is possible to make all the segments on the screen red. Find the smallest number of key presses that can turn all the segments red, starting from the worst case.

If $P$ is the area of a triangle $ABC$ with sides $a,b,c$, prove that $\frac{ab+bc+ca}{4P} \ge \sqrt3$

Segments $AB$, $BC$ and $CA$ are, respectively, diagonals of squares $K_1$, $K_2$, $K3$. Prove that if triangle $ABC$ is acute, then it completely covered by squares $K_1$, $K_2$ and $K_3$.

Each of the $2N = 2004$ real numbers $x_1, x_2, \ldots , x_{2004}$ equals either $\sqrt 2 -1 $ or $\sqrt 2 +1$. Can the sum $\sum_{k=1}^N x_{2k-1}x_2k$ take the value $2004$? Which integral values can this sum take?

$A$ is a set satisfying the following the condition. Show that $2001+\sqrt{2001}$ is an element of $A$. [b]Condition[/b] (1) $1 \in A$ (2) If $x \in A$, then $x^2 \in A$. (3) If $(x-3)^2 \in A$, then $x \in A$.

The incircle $\omega$ of a triangle $ABC$ is tangent to the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The angle bisector at $A$ meets $PQ$ at point $S$. Prove $\angle ASC = 90^o$ .

A coordinate system was drawn on the board and points $A (1,2)$ and $B (3,1)$ were marked. The coordinate system was erased. Restore it by the two marked points.

suppose that polynomial $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$ does not have a real root. what is the maximum number of coefficients to be $-1$?(14 points)

Given a set $S$ of integers, an allowed operation consists of the following three steps: $\bullet$ Choose a positive integer $n$. $\bullet$ Choose $n+1$ elements $a_0, a_1, \dots, a_n \in S$, not necessarily distinct. $\bullet$ Add to the set $S$ all the integer roots of the polynomial $a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0$. Beto must choose an initial set $S$ and perform several allowed operations, so that at the end of the process $S$ contains among its elements the integers $1, 2, 3, \dots, 2023, 2024$. Determine the smallest $k$ for which there exists an initial set $S$ with $k$ elements that allows Beto to achieve his objective.

Let $F_1$ be an arbitrary convex quadrilateral. For $k\ge2$, $F_k$ is obtained by cutting $F_{k-1}$ into two pieces along one of its diagonals, flipping one piece over, and the glueing them back together along the same diagonal. What is the maximum number of non-congruent quadrilaterals in the sequence $\{F_k\}$?