Let $0 \leq b \leq c \leq d \leq a$ and $a>14$ are integers. Prove, that there is such natural $n$ that can not be represented as $$n=x(ax+b)+y(ay+c)+z(az+d)$$ where $x,y,z$ are some integers. [i]K. Kohas[/i]

Let $S=\{1,2,\ldots ,98\}$. Find the least natural number $n$ such that we can pick out $10$ numbers in any $n$-element subset of $S$ satisfying the following condition: no matter how we equally divide the $10$ numbers into two groups, there exists a number in one group such that it is coprime to the other numbers in that group, meanwhile there also exists a number in the other group such that it is not coprime to any of the other numbers in the same group.

Let $\left\{ a_n \right\}$ and $\left\{ b_n \right\}$ be sequences defined recursively by $a_0 =2$; $b_0 = 2$, and $a_{n+1} = a_n \sqrt{1+a_n^2+b_n^2}-b_n$; $b_{n+1} = b_n\sqrt{1+a_n^2+b_n^2} + a_n$. Find the ternary (base 3) representation of $a_4$ and $b_4$.

Suppose $n$ is a positive integer, $4n$ teams participate in a football tournament. In each round of the game, we will divide the $4n$ teams into $2n$ pairs, and each pairs play the game at the same time. After the tournament, it is known that every two teams have played at most one game. Find the smallest positive integer $a$, so that we can arrange a schedule satisfying the above conditions, and if we take one more round, there is always a pair of teams who have played in the game.

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

Let $x_1, x_2, \ldots, x_n$ be positive reals; for any positive integer $k$, let $S_k=x_1^k+x_2^k+\ldots+x_n^k$. (a) Given that $S_1<S_2$, show that $S_1, S_2, S_3, \ldots$ is strictly increasing. (b) Prove that there exists a positive integer $n$ and positive reals $x_1, x_2, \ldots, x_n$, such that $S_1>S_2$ and $S_1, S_2, S_3, \ldots$ is not strictly decreasing.

The satellites $A$ and $B$ circle the Earth in the equatorial plane at altitude $h$. They are separated by distance $2r$, where $r$ is the radius of the Earth. For which $h$ can they be seen in mutually perpendicular directions from some point on the equator?

Compute \[\sum_{k=1}^{420} \gcd(k,420).\] [i]2016 CCA Math Bonanza Individual #14[/i]

[b]p1.[/b] $2011$ distinct points are arranged along the perimeter of a circle. We choose without replacement four points $P$, $Q$, $R$, $S$. What is the probability that no two of the segments $P Q$, $QR$, $RS$, $SP$ intersect (disregarding the endpoints)? [b]p2.[/b] In Soviet Russia, all phone numbers are between three and six digits and contain only the digits $1$, $2$, and $3$. No phone number may be the prefix of another phone number, so, for example, we cannot have the phone numbers $123$ and $12332$. If the Soviet bureaucracy has preassigned $10$ phone numbers of length $3$, $20$ numbers of length $4$, and $77$ phone numbers of length $6$, what is the maximum number of phone numbers of length $5$ that the authorities can allocate? [b]p3.[/b] The sequence $\{a_n\}_{n\ge 1}$ is defined as follows: we have $a_1 = 1$, $a_2 = 0$, and for $n \ge 3$ we have $$a_n = \frac12 \sum\limits_{\substack{1\le i,j\\ i+j+k=n}} a_ia_ja_k.$$ Find $$\sum^{\infty}_{n=1} \frac{a_n}{2^n}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let all positive integers $n$ satisfy the following condition: for each non-negative integers $k, m$ with $k + m \le n$, the numbers $\binom{n-k}{m}$ and $\binom{n-m}{k}$ leave the same remainder when divided by $2$. (Poland) PS. The translation was done using Google translate and in case it is not right, there is the original text in Slovak

Find the number of triples of real numbers $(a,b,c)$ such that $a^{20} + b^{20} + c^{20} = a^{24} + b^{24} + c^{24} =1$.

How many $6$-tuples $(a, b, c, d, e, f)$ of natural numbers are there for which $a>b>c>d>e>f$ and $a+f=b+e=c+d=30$ are simultaneously true?

The real numbers $x$, $y$, $z$, satisfy $0\leq x \leq y \leq z \leq 4$. If their squares form an arithmetic progression with common difference $2$, determine the minimum possible value of $|x-y|+|y-z|$.

a) Let $a,n,m$ be natural numbers, $a > 1$. Prove that if $(a^m + 1)$ is divisible by $(a^n + 1)$ than $m$ is divisible by $n$. b) Let $a,b,n,m$ be natural numbers, $a>1, a$ and $b$ are relatively prime. Prove that if $(a^m+b^m)$ is divisible by $(a^n+b^n)$ than $m$ is divisible by $n$.

Which of the following claims are true, and which of them are false? If a fact is true you should prove it, if it isn't, find a counterexample. a) Let $a,b,c$ be real numbers such that $ a^{2013} + b^{2013} + c^{2013} = 0 $. Then $ a^{2014} + b^{2014} + c^{2014} = 0 $. b) Let $a,b,c$ be real numbers such that $ a^{2014} + b^{2014} + c^{2014} = 0 $. Then $ a^{2015} + b^{2015} + c^{2015} = 0 $. c) Let $a,b,c$ be real numbers such that $ a^{2013} + b^{2013} + c^{2013} = 0 $ and $ a^{2015} + b^{2015} + c^{2015} = 0 $. Then $ a^{2014} + b^{2014} + c^{2014} = 0 $. [i]Proposed by Matko Ljulj[/i]

A social club has $2k+1$ members, each of whom is fluent in the same $k$ languages. Any pair of members always talk to each other in only one language. Suppose that there were no three members such that they use only one language among them. Let $A$ be the number of three-member subsets such that the three distinct pairs among them use different languages. Find the maximum possible value of $A$.

On the sides of triangle $ABC$ take the points $M_1, N_1, P_1$ such that each line $MM_1, NN_1, PP_1$ divides the perimeter of $ABC$ in two equal parts ($M, N, P$ are respectively the midpoints of the sides $BC, CA, AB$). [b]I.[/b] Prove that the lines $MM_1, NN_1, PP_1$ are concurrent at a point $K$. [b]II.[/b] Prove that among the ratios $\frac{KA}{BC}, \frac{KB}{CA}, \frac{KC}{AB}$ there exist at least a ratio which is not less than $\frac{1}{\sqrt{3}}$.

The triangle $ABC$ acute with gravity center $M$ is such that $\angle AMB = 2 \angle ACB$. Prove that: [b](a)[/b] $AB^4=AC^4+BC^4-AC^2 \cdot BC^2,$ [b](b)[/b] $\angle ACB \geq 60^o$.

The incircle $\omega$ of triangle $ABC$, right angled at $C$, touches the circumcircle of its medial triangle at point $F$. Let $OE$ be the tangent to $\omega$ from the midpoint $O$ of the hypotenuse $AB$, distinct from $AB$. Prove that $CE = CF$.

Let $R$ be a set of exactly $6$ elements. A set $F$ of subsets of $R$ is called an $S$-family over $R$ if and only if it satisfies the following three conditions: (i) For no two sets $X, Y$ in $F$ is $X \subseteq Y$ ; (ii) For any three sets $X, Y,Z$ in $F$, $X \cup Y \cup Z \neq R,$ (iii) $\bigcup_{X \in F} X = R$

Let $\alpha,\beta,\gamma$ be angles of a triangle. Two of them can be expressed using an auxiliary angle $\varphi$ in a way that $$\alpha=\varphi+\frac\pi4,\quad\beta=\pi-3\varphi.$$ Show that $\alpha>\gamma.$

Show that if $n \ge 4, n \in N$ and $\big [ \frac{2^n}{n} ]$ is a power of $2$, then $n$ is a power of $2$.

What is the last digit in the number $2007^{2007}$?

Call integer number $x$ as far from squares and cubes, if for every integer $k$ it is true : $|x-k^2|>10^6,|x-k^3|>10^6$. Prove, that there are infinitely many far from squares and cubes degrees of $2$

A circle with center $O$ passes through the ends of the hypotenuse of a right-angled triangle and intersects its legs at points $M$ and $K$. Prove that the distance from point $O$ to line $MK$ is half the hypotenuse.