What $n$ is the smallest such that “there is a $n$-gon that can be cut into a triangle, a quadrilateral, ..., a $2006$-gon''?

You are traveling in a foreign country whose currency consists of five different-looking kinds of coins. You have several of each coin in your pocket. You remember that the coins are worth $1, 2, 5, 10$, and $20$ florins, but you have no idea which coin is which and you don’t speak the local language. You find a vending machine where a single candy can be bought for $1$ florin: you insert any kind of coin, and receive $1$ candy plus any change owed. You can only buy one candy at a time, but you can buy as many as you want, one after the other. What is the least number of candies that you must buy to ensure that you can determine the values of all the coins? Prove that your answer is correct.

Let $ABC$ a right isosceles triangle with hypotenuse $AB=\sqrt2$ . Determine the positions of the points $X,Y,Z$ on the sides $BC,CA,AB$ respectively so that the triangle $XYZ$ is isosceles, right, and with minimum area.

Consider sequences $a_1, a_2, a_3,...$ of positive integers. Determine the smallest possible value of $a_{2010}$ if (i) $a_n < a_{n+1}$ for all $n\ge 1$, (ii) $a_i + a_l > a_j + a_k$ for all quadruples $ (i, j, k, l)$ which satisfy $1 \le i < j \le k < l$.

Find the smallest value of the expression $|3 \cdot 5^m - 11 \cdot 13^n|$ for all $m,n \in N$. (Folklore)

Two circles centers $O$ and $O'$, radii $R$ and $R'$, meet at two points. A variable line $L$ meets the circles at $A, C, B, D$ in that order and $\frac{AC}{AD} = \frac{CB}{BD}$. The perpendiculars from $O$ and $O'$ to $L$ have feet $H$ and $H'$. Find the locus of $H$ and $H'$. If $OO'^2 < R^2 + R'^2$, find a point $P$ on $L$ such that $PO + PO'$ has the smallest possible value. Show that this value does not depend on the position of $L$. Comment on the case $OO'^2 > R^2 + R'^2$.

On the legs of $\angle AOB$, the segments $OA$ and $OB$ lie, $OA > OB$. Points $M$ and $N$ on lines $OA$ and $OB$, respectively, are such that $AM = BN = x$. Find $x$ for which the length of $MN$ is minimal.

Find minimum of $x+y+z$ where $x$, $y$ and $z$ are real numbers such that $x \geq 4$, $y \geq 5$, $z \geq 6$ and $x^2+y^2+z^2 \geq 90$

The positive numbers $a, b,c, d, p, q$ are such that $(x+a)(x+b)(x+c)(x+d) = x^4+4px^3+6x^2+4qx+1$ holds for all real numbers $x$. Find the smallest value of $p$ or the largest value of $q$.

Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]

Consider all sets $A$ of one hundred different natural numbers with the property that any three elements $a,b,c \in A$ (not necessarily different) are the sides of a non-obtuse triangle. Denote by $S(A)$ the sum of the perimeters of all such triangles. Compute the smallest possible value of $S(A)$.

Sofia and Viktor are playing the following game on a $2022 \times 2022$ board: - Firstly, Sofia covers the table completely by dominoes, no two are overlapping and all are inside the table; - Then Viktor without seeing the table, chooses a positive integer $n$; - After that Viktor looks at the table covered with dominoes, chooses and fixes $n$ of them; - Finally, Sofia removes the remaining dominoes that aren't fixed and tries to recover the table with dominoes differently from before. If she achieves that, she wins, otherwise Viktor wins. What is the minimum number $n$ for which Viktor can always win, no matter the starting covering of dominoes. [i]Proposed by Viktor Simjanoski[/i]

Find minimal value of $A=\frac{\left(x+\frac{1}{x}\right)^6-\left(x^6+\frac{1}{x^6}\right)-2}{\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}$

At a time $t = 0$, a navy ship is at a point $O$, while an enemy ship is at a point $A$ cruising with speed $v$ perpendicular to $OA = a$. The speed and direction of the enemy ship do not change. The strategy of the navy ship is to travel with constant speed $u$ at a angle $0 < \phi < \pi /2$ to the line $OA$. 1) Let $\phi$ be chosen. What is the minimum distance between the two ships? Under what conditions will the distance vanish? 2) If the distance does not vanish, what is the choice of $\phi$ to minimize the distance? What are directions of the two ships when their distance is minimum?

Find the smallest real number $x$ for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is $x$.

Let Akbar and Birbal together have $n$ marbles, where $n > 0$. Akbar says to Birbal, “ If I give you some marbles then you will have twice as many marbles as I will have.” Birbal says to Akbar, “ If I give you some marbles then you will have thrice as many marbles as I will have.” What is the minimum possible value of $n$ for which the above statements are true?

A honeybug crawls along the honeycombs with the unite length of their hexagons. He has moved from the node $A$ to the node $B$ along the shortest possible trajectory. Prove that the half of his way he moved in one direction.

A triangle $ABC$ is inscribed in a circle $G$. For any point $M$ inside the triangle, $A_1$ denotes the intersection of the ray $AM$ with $G$. Find the locus of point $M$ for which $\frac{BM\cdot CM}{MA_1}$ is minimal, and find this minimum value.

Given a triangle $ABC$ with the are $1$. Let $A',B'$ and $C' $ are the midpoints of the sides $[BC], [CA]$ and $[AB]$ respectively. What is the minimal possible area of the common part of two triangles $A'B'C'$ and $KLM$, if the points $K,L$ and $M$ are lying on the segments $[AB'], [CA']$ and $[BC']$ respectively?

a) Each of the numbers $x_1,x_2,...,x_n$ can be $1, 0$, or $-1$. What is the minimal possible value of the sum of all products of couples of those numbers. b) Each absolute value of the numbers $x_1,x_2,...,x_n$ doesn't exceed $1$. What is the minimal possible value of the sum of all products of couples of those numbers.

Through a point $M$ inside an angle $a$ line is drawn. It cuts off this angle a triangle of the least possible area. Prove that $M$ is the midpoint of the segment on this line that the angle intercepts.

What is the least positive integer $k$ such that, in every convex $101$-gon, the sum of any $k$ diagonals is greater than or equal to the sum of the remaining diagonals?

a) Find the minimal value of the polynomial $$P(x,y) = 4 + x^2y^4 + x^4y^2 - 3x^2y^2$$ b) Prove that it cannot be represented as a sum of the squares of some polynomials of $x,y$.

The positive real numbers $a, b, c$ satisfy the equality $a + b + c = 1$. For every natural number $n$ find the minimal possible value of the expression $$E=\frac{a^{-n}+b}{1-a}+\frac{b^{-n}+c}{1-b}+\frac{c^{-n}+a}{1-c}$$

At a quiz show there are three doors. Behind exactly one of the doors, a prize is hidden. You may ask the quizmaster whether the prize is behind the left-hand door. You may also ask whether the prize is behind the right-hand door. You may ask each of these two questions multiple times, in any order that you like. Each time, the quizmaster will answer ‘yes’ or ‘no’. The quizmaster is allowed to lie at most $10$ times. You have to announce in advance how many questions you will be asking (but which questions you will ask may depend on the answers of the quizmaster). What is the smallest number you can announce, such that you can still determine with absolute certainty the door behind which the prize is hidden?