Let a binary operation $*$ on ordered pairs of integers be defined by $(a,b)*(c,d)=(a-c,b+d)$. Then, if $(3,2)*(0,0)$ and $(x,y)*(3,2)$ represent idential pairs, $x$ equals: $\textbf{(A) }-3\qquad \textbf{(B) }0\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }6$

On a circle, $n \geq 3$ points are marked. Each marked point is coloured red, green or blue. In one step, one can erase two neighbouring marked points of different colours and mark a new point between the locations of the erased points with the third colour. In a final state, all marked points have the same colour which is called the colour of the final state. Find all $n$ for which there exists an initial state of $n$ marked points with one missing colour, from which one can reach a final state of any of the three colours by applying a suitable sequence of steps.

Let $ABCD$ be an inscribed quadrilateral. Let the circles with diameters $AB$ and $CD$ intersect at two points $X_1$ and $Y_1$, the circles with diameters $BC$ and $AD$ intersect at two points $X_2$ and $Y_2$, the circles with diameters $AC$ and $BD$ intersect at two points $X_3$ and $Y_3$. Prove that the lines $X_1Y_1, X_2Y_2$ and $X_3Y_3$ are concurrent. Maxim Didin

Prove that if the numbers $ p $, $ q $, $ r $ satisfy the equality $$ p+q + r=1$$ $$ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 0$$ then for any numbers $ a $, $ b $, $ c $ equality holds $$a^2 + b^2 + c^2 = (pa + qb + rc)^2 + (qa + rb + pc)^2 + (ra + pb + qc)^2.$$

Let $f:[0,1]\to\mathbb R$ be a twice continuously differentiable increasing function. Define the sequences given by $L_n=\frac1n\sum_{k=0}^{n-1}f\left(\frac kn\right)$ and $U_n=\frac1n\sum_{k=0}^nf\left(\frac kn\right)$, $n\ge1$. 1. The interval $[L_n,U_n]$ is divided into three equal segments. Prove that, for large enough $n$, the number $I=\int^1_0f(x)\text dx$ belongs to the middle one of these three segments.

Let $a,b,c,d,e,f$ be positive real numbers. Given that $def+de+ef+fd=4$, show that \[ ((a+b)de+(b+c)ef+(c+a)fd)^2 \geq\ 12(abde+bcef+cafd). \][i]Proposed by Allen Liu[/i]

Csaba stands in the middle of a $15$ m $\times 15$ m room at a workplace where everyone strictly adheres to $1,5$ m social distancing. At least how many people are there other than Csaba in the room if Csaba cannot reach any wall without the others moving? [i]The people are viewed as points.[/i]

Find all positive integers $x,y,z$ and $t$ such that $2^x3^y+5^z=7^t$.

Given a circle with center $O$ and radius equal to $1$. $AB$ and $AC$ are the tangents to this circle from point $A$. Point $M$ on the circle is such that the areas of quadrilaterals $OBMC$ and $ABMC$ are equal. Find $MA$.

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]

A convex quadrilateral is determined by the points of intersection of the curves $x^4+y^4=100$ and $xy=4$; determine its area.

Different non-zero natural numbers a$_1, a_2,. . . , a_{12}$ satisfy the condition: all positive differences other than two numbers $a_i$ and $a_j$ form many $20$ consecutive natural numbers. a) Show that $\max \{a_1, a_2,. . . , a_{12}\} - \min \{a_1, a_2,. . . , a_{12}\} = 20$. b)Determine $12$ natural numbers with the property from the statement.

Determine whether there exist real numbers $x$, $y$, $z$, such that \[x+\frac{1}{y}=z,\quad y+\frac{1}{z}=x,\quad z+\frac{1}{x}=y.\]

A rectangular $M \times N$ board is divided into $1 \times $ cells. There are also many domino pieces of size $1 \times 2$. These pieces are placed on a board so that each piece occupies two cells. The board is not entirely covered, but it is impossible to move the domino pieces (the board has a frame, so that the pieces cannot stick out of it). Prove that the number of uncovered cells is (a) less than $\frac14 MN$, (b) less than $\frac15 MN$.

The village chatterboxes are exchanging their gossip by phone every day so that any two of them talk to each other exactly once. A certain day, every chatterbox called up at least one of the other chatterboxes. Show that there exist three chatterboxes such that the first called up the second, the second called up the third, and the third called up the first.

Eddie assigns each of Jason, Jerry, and Jonathan a different positive integer. The three are each perfectly logical and currently know that their numbers are distinct but don't know each other's numbers. Additionally, if one of them knows the answer to the question they will say so immediately. They have the following conversation listed below in chronological order: [list] [*] Eddie: Does anyone know who has the smallest number? [*] Jason, Jerry, Jonathan (at the same time): I'm not sure. [*] Jonathan: Now I know who has the smallest number. [*] Eddie: Does anyone know who has the largest number? [*] Jason, Jonathan, Jerry (at the same time): I'm not sure. [*] Jerry: Now I know who has the largest number. [*] Jason: Wow, our numbers are in an geometric sequence! [/list] Find the sum of their numbers.

Let $N>1$ and let $a_1,a_2,\dots,a_N$ be nonnegative reals with sum at most $500$. Prove that there exist integers $k\ge 1$ and $1=n_0<n_1<\dots<n_k=N$ such that \[\sum_{i=1}^k n_ia_{n_{i-1}}<2005.\]

A particle can travel at speeds up to $ \frac{2m}{s}$ along the $ x$-axis, and up to $ \frac{1m}{s}$ elsewhere in the plane. Provide a labelled sketch of the region which can be reached within one second by the particle starting at the origin.

sorry if this has been posted before . given a fixed circle $(O)$ and two fixed point $B,C$ on it.point A varies on circle $(O)$. let $I$ be the midpoint of $BC$ and $H$ be the orthocenter of $\triangle ABC$. ray $IH$ meet $(O)$ at $K$ ,$AH$ meet $BC$ at $D$ ,$KD$ meet $(O)$ at $M$ .a line pass $M$ and perpendicular to $BC$ meet $AI$ at $N$. a) prove that $N$ varies on a fixed circle. b) a circle pass $N$ and tangent to $AK$ at $A$ cut $AB,AC$ at $P,Q$. let $J$ be the midpoint of $PQ$ .prove that $AJ$ pass through a fixed point.

Develop necessary and sufficient conditions that the equation \[ \begin{vmatrix} 0 & a_1 - x & a_2 - x \\ -a_1 - x & 0 & a_3 - x \\ -a_2 - x & -a_3 - x & 0\end{vmatrix} = 0 \qquad (a_i \neq 0) \] shall have a multiple root.

Let $ABCD$ be a rectangle and let $E$ and $F$ be points on $CD$ and $BC$ respectively such that area $(ADE) = 16$, area $(CEF) = 9$ and area $(ABF) = 25$. What is the area of triangle $AEF$ ?

Find all functions $ f\in\mathcal{C}^1 [0,1] $ that satisfy $ f(1)=-1/6 $ and $$ \int_0^1 \left( f'(x) \right)^2 dx\le 2\int_0^1 f(x)dx. $$

Let $ n$ be a positive integer. Find all $ x_1,x_2,\ldots,x_n$ that satisfy the relation: \[ \sqrt{x_1\minus{}1}\plus{}2\cdot \sqrt{x_2\minus{}4}\plus{}3\cdot \sqrt{x_3\minus{}9}\plus{}\cdots\plus{}n\cdot\sqrt{x_n\minus{}n^2}\equal{}\frac{1}{2}(x_1\plus{}x_2\plus{}x_3\plus{}\cdots\plus{}x_n).\]

Let $S$ be a convex region in the euclidean plane containing the origin. Assume that every ray from the origin has at least one point outside $S$. Prove that $S$ is bounded.

The square $DEFG$ is contained in equilateral triangle $ABC$, with $E$ on $\overline{AC}$, $G$ on $\overline{AD}$, and $F$ as the midpoint of $\overline{BC}$. Find $AD$ if $DE = 6$.