We call a collection of weights (each weighing an integer value) basic if their total weight equals $500$ and each object of integer weight not greater than $500$ can be balanced exactly with a uniquely determined set of weights from the collection. (Uniquely means that we are not concerned with order or which weights of equal value are chosen to balance against a particular object, if in fact there is a choice.) (a) Find an example of a basic collection other than the collection of $500$ weights each of value $1$. (b) How many different basic collections are there? (D. Fomin, Leningrad)

Let $\omega_1$ and $\omega_2$ be two circles with centers $C_1$ and $C_2$, respectively, which intersect at two points $P$ and $Q$. Suppose that the circumcircle of triangle $PC_1C_2$ intersects $\omega_1$ at $A \neq P$ and $\omega_2$ at $B \neq P$. Suppose further that $Q$ is inside the triangle $PAB$. Show that $Q$ is the incenter of triangle $PAB$.

Find all functions from $\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}$ such that $f(m^2+mf(n))=mf(m+n)$, for all $m,n\in \mathbb{N}\cup\{0\}$.

The area of the largest triangle that can be inscribed in a semi-circle whose radius is $r$ is: $\textbf{(A)}\ r^2 \qquad \textbf{(B)}\ r^3 \qquad \textbf{(C)}\ 2r^2 \qquad \textbf{(D)}\ 2r^3 \qquad \textbf{(E)}\ \dfrac{1}{2}r^2$

Let $L$ and $N$ be the mid-points of the diagonals $[AC]$ and $[BD]$ of the cyclic quadrilateral $ABCD$, respectively. If $BD$ is the bisector of the angle $ANC$, then prove that $AC$ is the bisector of the angle $BLD$.

In the sequence $00, 01, 02, 03,\ldots , 99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19, 39$, or $28$, but not by $30$ or $20$). What is the maximal number of terms that could remain on their places?

Let $(a_n)_{n\in \mathbb N}$ be the sequence of integers defined recursively by $a_1 = a_2 = 1, a_{n+2} = 7a_{n+1} - a_n - 2$ for $n \geq 1$. Prove that $a_n$ is a perfect square for every $n.$

Find the maximum of \[P=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}\] where $x,y,z$ are positive real numbers.

In every cell of the table $3 \times 3$ a monomial with a positive coefficient is written (cells (1,1); (2,3); (3,2) have the degree of two, cells (1,2);(2,1);(3,3) have a degree of one, cells (3,1);(2,2);(1,3) have a constant). Vuga added up monomials in every row and got three quadratic polynomials. It turned out that exactly $N$ of them have real roots. Leka added up monomials in every column and got three quadratic polynomials. It turned out that exactly $M$ of them have real roots. Find the maximum possible value of $N-M$.

Let \( n \geq 4 \) be an integer. Show that at a party of \( n \) people, it is possible for each person to have greeted exactly three other people if and only if \( n \) is even.

The positive integers $a, b, c$ are such that $$gcd \,\,\, (a, b, c) = 1,$$ $$gcd \,\,\,(a, b + c) > 1,$$ $$gcd \,\,\,(b, c + a) > 1,$$ $$gcd \,\,\,(c, a + b) > 1.$$ Determine the smallest possible value of $a + b + c$. Clarification: gcd stands for greatest common divisor.

Let $x$ be a real number in the interval $\left(0,\dfrac{\pi}{2}\right)$ such that $\dfrac{1}{\sin x \cos x}+2\cot 2x=\dfrac{1}{2}$. Evaluate $\dfrac{1}{\sin x \cos x}-2\cot 2x$.

A sequence ($a_n$) is given by $a_0 = 0, a_1 = 1$ and $a_{k+2} = a_{k+1} +a_k$ for all integers $k \ge 0$. Prove that the inequality $\sum_{k=0}^n \frac{a_k}{2^k}< 2$ holds for all positive integers $n$.

Suppose a $m \times n$ table. We write an integer in each cell of the table. In each move, we chose a column, a row, or a diagonal (diagonal is the set of cells which the difference between their row number and their column number is constant) and add either $+1$ or $-1$ to all of its cells. Prove that if for all arbitrary $3 \times 3$ table we can change all numbers to zero, then we can change all numbers of $m \times n$ table to zero. ([i]Hint[/i]: First of all think about it how we can change number of $ 3 \times 3$ table to zero.)

There are several counters of various colours and sizes. No two of them have, simultaneously, the same colour and the same size. On each counter $F$ two numbers are written. One of them is the number of counters that have the same colour as $F$ but a different size than $F$. The other number is the number of counters that have the same size as $F$ but a different colour. It is known that each of the $101$ numbers $0,1,\ldots,100$ is written at least once. Determine the smallest number of counters for which this is possible.

An isosceles right triangle is inscribed in a circle of radius 5, thereby separating the circle into four regions. Compute the sum of the areas of the two smallest regions.

Prove that, for every tetrahedron $ABCD$, there exists a unique point $P$ in the interior of the tetrahedron such that the tetrahedra $PABC,PABD,PACD,PBCD$ have equal volumes.

For positive integers $x$, $y$ and $z$ holds $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}$. Prove that $xyz\geq 3600$

Region $ABCDEFGHIJ$ consists of $13$ equal squares and is inscribed in rectangle $PQRS$ with $A$ on $\overline{PQ}$, $B$ on $\overline{QR}$, $E$ on $\overline{RS}$, and $H$ on $\overline{SP}$, as shown in the figure on the right. Given that $PQ=28$ and $QR=26$, determine, with proof, the area of region $ABCDEFGHIJ$. [asy] size(200); defaultpen(linewidth(0.7)+fontsize(12)); pair P=(0,0), Q=(0,28), R=(26,28), S=(26,0), B=(3,28); draw(P--Q--R--S--cycle); picture p = new picture; draw(p, (0,0)--(3,0)^^(0,-1)--(3,-1)^^(0,-2)--(5,-2)^^(0,-3)--(5,-3)^^(2,-4)--(3,-4)^^(2,-5)--(3,-5)); draw(p, (0,0)--(0,-3)^^(1,0)--(1,-3)^^(2,0)--(2,-5)^^(3,0)--(3,-5)^^(4,-2)--(4,-3)^^(5,-2)--(5,-3)); transform t = shift(B) * rotate(-aSin(1/26^.5)) * scale(26^.5); add(t*p); label("$P$",P,SW); label("$Q$",Q,NW); label("$R$",R,NE); label("$S$",S,SE); label("$A$",t*(0,-3),W); label("$B$",B,N); label("$C$",t*(3,0),plain.ENE); label("$D$",t*(3,-2),NE); label("$E$",t*(5,-2),plain.E); label("$F$",t*(5,-3),plain.SW); label("$G$",t*(3,-3),(0.81,-1.3)); label("$H$",t*(3,-5),plain.S); label("$I$",t*(2,-5),NW); label("$J$",t*(2,-3),SW);[/asy]

Let $ ABCD $ be a cyclic quadrilateral,$ M $ midpoint of $ AC $ and $ N $ midpoint of $ BD. $ If $ \angle AMB =\angle AMD, $ prove that $ \angle ANB =\angle BNC. $

Let $\omega_1$, $\omega_2$, $\omega_3$ are three externally tangent circles, with $\omega_1$ and $\omega_2$ tangent at $A$. Choose points $B$ and $C$ on $\omega_1$ so that lines $AB$ and $AC$ are tangent to $\omega_3$. Suppose the line $BC$ intersect $\omega_3$ at two distinct points, and $X$ is the intersection further away to $B$ and $C$ than the other one. Prove that one of the tangent lines of $\omega_2$ passing through $X$, is also tangent to an excircle of triangle $ABC$. [i]Proposed by Ivan Chan Kai Chin[/i]

Prove that if real numbers $a,b,c$ satisfy the equality $$\frac{a}{m+2}+\frac{b}{m+1}+\frac{c}{m}= 0$$ for some positive number $m$, then the equation $ax^2 + bx + c = 0$ has a root between $0$ and $1$.

Find the angle between two common sections of the page $2x+y-z=0$ and the cone $4x^2-y^2+3z^2=0.$

Let $N$ be the set of non-negative integers. The function $f:N\to N$ satisfies $f(a+b) = f(f(a)+b)$ for all $a, b$ and $f(a+b) = f(a)+f(b)$ for $a+b < 10$. Also $f(10) = 1$. How many three digit numbers $n$ satisfy $f(n) = f(N)$, where $N$ is the "tower" $2, 3, 4, 5$, in other words, it is $2^a$, where $a = 3^b$, where $b = 4^5$?

Let $P$ and $Q$ be the points on the diagonal $BD$ of a quadrilateral $ABCD$ such that $BP = PQ = QD$. Let $AP$ and $BC$ meet at $E$, and let $AQ$ meet $DC$ at $F$. (a) Prove that if $ABCD$ is a parallelogram, then $E$ and $F$ are the midpoints of the corresponding sides. (b) Prove the converse of (a).