Given several points, not all lying on one straight line. Some number is assigned to every point. It is known, that if a straight line contains two or more points, than the sum of the assigned to those points equals zero. Prove that all the numbers equal to zero.

Suppose that $ ABC$ is an acute triangle such that $ \tan A$, $ \tan B$, $ \tan C$ are the three roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} p \equal{} 0$, where $ q\neq 1$. Show that $ p \le \minus{} 3\sqrt 3$ and $ q > 1$.

Solve the system $\begin{cases} x+ y +z = a \\ x^2 + y^2 + z^2 = a^2 \\ x^3 + y^3 +z^3 = a^3 \end{cases}$

Let $k$ be a positive real number such that $$\frac{1}{k+a} + \frac{1}{k+b} + \frac{1}{k+c} \leq 1$$ for any positive positive real numbers $a$, $b$ and $c$ with $abc = 1$. Find the minimum value of $k$.

Let $a,b$ and $c$ be real numbers such that $|a+b|+|b+c|+|c+a|=8.$ Find the maximum and minimum value of the expression $P=a^2+b^2+c^2.$

Let $n$ be an integer and $a, b$ real numbers such that $n > 1$ and $a > b > 0$. Prove that $$(a^n - b^n) \left ( \frac{1}{b^{n- 1}} - \frac{1}{a^{n -1}}\right) > 4n(n -1)(\sqrt{a} - \sqrt{b})^2$$

Let $a, b, c, d, e$ be real numbers such that $a<b<c<d<e.$ The least possible value of the function $f: \mathbb R \to \mathbb R$ with $f(x) = |x-a| + |x - b|+ |x - c| + |x - d|+ |x - e|$ is $$ \mathrm a. ~ e+d+c+b+a\qquad \mathrm b.~e+d+c-b-a\qquad \mathrm c. ~e+d+|c|-b-a \qquad \mathrm d. ~e+d+b-a \qquad \mathrm e. ~e+d-b-a $$

Let $m,n$ be positive integers such that the equation (in respect of $x$) \[(x+m)(x+n)=x+m+n\] has at least one integer root. Prove that $\frac{1}{2}n<m<2n$.

Let $a_0$ be an arbitrary positive integer. Consider the infinite sequence $(a_n)_{n\geq 1}$, defined inductively as follows: given $a_0, a_1, ..., a_{n-1}$ define the term $a_n$ as the smallest positive integer such that $a_0+a_1+...+a_n$ is divisible by $n$. Prove that there exist a positive integer a positive integer $M$ such that $a_{n+1}=a_n$ for all $n\geq M$.

Let $S_n=\sum_{k=0}^{n}\frac{1}{\sqrt{k+1}+\sqrt{k}}$. What is the value of $\sum_{n=1}^{99}\frac{1}{S_n+S_{n-1}}$ ?

Let $f : \mathbb{Z} \rightarrow \mathbb{Z}^+$ be a function, and define $h : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}^+$ by $h(x, y) = \gcd (f(x), f(y))$. If $h(x, y)$ is a two-variable polynomial in $x$ and $y$, prove that it must be constant.

A sequence $ (S_n), n \geq 1$ of sets of natural numbers with $ S_1 = \{1\}, S_2 = \{2\}$ and \[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}. \] Determine $ S_{1024}.$

Prove that for all real numbers $x$ holds: $$\sin 5x = 16 \sin x \cdot \sin \left(x -\frac{\pi}{5} \right) \cdot \sin\left(x -\frac{2\pi}{5} \right) \sin \left(x +\frac{2\pi}{5} \right) $$

Determine all pairs $(m,n)$ of integers with $n \ge m$ satisfying the equation \[n^3+m^3-nm(n+m)=2023.\]

Let be four functions $ f,g,s,i:\mathbb{N}\longrightarrow\mathbb{N} $ such that $ s(x)=\max (f(x),g(x)) $ and $ i(x)=\min (f(x),g(x)) , $ for any natural number $ x. $ Prove that $ f=g $ if $ s $ is surjective and $ i $ injective.

Let $a_0,a_1,\dots,a_{2024}$ be real numbers such that $\left|a_{i+1}-a_i\right| \le 1$ for $i=0,1,\dots,2023$. a) Find the minimum possible value of $$a_0a_1+a_1a_2+\dots+a_{2023}a_{2024}$$ b) Does there exist a real number $C$ such that $$a_0a_1-a_1a_2+a_2a_3-a_3a_4+\dots+a_{2022}a_{2023}-a_{2023}a_{2024} \ge C$$ for all real numbers $a_0,a_1,\dots,a_2024$ such that $\left|a_{i+1}-a_i\right| \le 1$ for $i=0,1,\dots,2023$.

Determine the number of times and the positions in which it appears $\frac12$ in the following sequence of fractions: $$ \frac11, \frac21, \frac12 , \frac31 , \frac22 , \frac13 , \frac41,\frac32,\frac23,\frac14,..., \frac{1}{1992}$$

Let $r,s,t$ be the roots of $x^3+6x^2+7x+8$. Find $$(r^2+s+t)(s^2+t+r)(t^2+r+s).$$ [i]Proposed by Evan Chang (squareman), USA[/i]

Let $m$ and $n$ be two positive integers for which $m<n$. $n$ distinct points $X_1,\ldots , X_n$ are in the interior of the unit disc and at least one of them is on its border. Prove that we can find $m$ distinct points $X_{i_1},\ldots , X_{i_m}$ so that the distance between their center of gravity and the center of the circle is at least $\frac{1}{1+2m(1- 1/n)}$.

[b]p1.[/b] If $n$ is a positive integer such that $2n+1 = 144169^2$, find two consecutive numbers whose squares add up to $n + 1$. [b]p2.[/b] Katniss has an $n$-sided fair die which she rolls. If $n > 2$, she can either choose to let the value rolled be her score, or she can choose to roll a $n - 1$ sided fair die, continuing the process. What is the expected value of her score assuming Katniss starts with a $6$ sided die and plays to maximize this expected value? [b]p3.[/b] Suppose that $f(x) = x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + f$, and that $f(1) = f(2) = f(3) = f(4) = f(5) = f(6) = 7$. What is $a$? [b]p4.[/b] $a$ and $b$ are positive integers so that $20a+12b$ and $20b-12a$ are both powers of $2$, but $a+b$ is not. Find the minimum possible value of $a + b$. [b]p5.[/b] Square $ABCD$ and rhombus $CDEF$ share a side. If $m\angle DCF = 36^o$, find the measure of $\angle AEC$. [b]p6.[/b] Tom challenges Harry to a game. Tom first blindfolds Harry and begins to set up the game. Tom places $4$ quarters on an index card, one on each corner of the card. It is Harry’s job to flip all the coins either face-up or face-down using the following rules: (a) Harry is allowed to flip as many coins as he wants during his turn. (b) A turn consists of Harry flipping as many coins as he wants (while blindfolded). When he is happy with what he has flipped, Harry will ask Tom whether or not he was successful in flipping all the coins face-up or face-down. If yes, then Harry wins. If no, then Tom will take the index card back, rotate the card however he wants, and return it back to Harry, thus starting Harry’s next turn. Note that Tom cannot touch the coins after he initially places them before starting the game. Assuming that Tom’s initial configuration of the coins weren’t all face-up or face-down, and assuming that Harry uses the most efficient algorithm, how many moves maximum will Harry need in order to win? Or will he never win? PS. You had better use hide for answers.

Let $K$ be a finite field, $n \ge 2$ an integer, $f \in K[X]$ an irreducible polynomial of degree $n,$ and $g$ the product of all the nonconstant polynomials in $K[X]$ of degree at most $n-1.$ Prove that $f$ divides $g-1.$

Let $ \alpha ,\beta $ be real numbers. Find the greatest value of the expression $$ |\alpha x +\beta y| +|\alpha x-\beta y| $$ in each of the following cases: [b]a)[/b] $ x,y\in \mathbb{R} $ and $ |x|,|y|\le 1 $ [b]b)[/b] $ x,y\in \mathbb{C} $ and $ |x|,|y|\le 1 $

Find all values of the positive integer $ m$ such that there exists polynomials $ P(x),Q(x),R(x,y)$ with real coefficient satisfying the condition: For every real numbers $ a,b$ which satisfying $ a^m-b^2=0$, we always have that $ P(R(a,b))=a$ and $ Q(R(a,b))=b$.

Find all positive integer solutions $x,y,z$ such that $1/x +2/y - 3/z=1$

When planning a trip from Temuco to the extreme north of the country, a truck driver notices that to cross the Atacama desert you must cross a distance of $800$ km between two stations consecutive service. Your truck can only store $50$ liters of benzene, and has a yield of $10$ km per liter. The trucker can leave gasoline stored in cans on the side of the road in different points along the way. For example, with an initial total charge of $50$ liters you can travel $100$ km, leave $30$ liters stored at the point you reached, and return to the starting point (with zero load) to refuel. The trucker decides to start the trip and arrives at the first service station with a zero load of fuel. a) Can the trucker cross the desert if at this service station the total supply is $140$ liters? b) Can the trucker cross the desert if the total supply of gasoline at the station is $180$ liters?