Let $ f(x)$ be a nonnegative, integrable function on $ (0,2\pi)$ whose Fourier series is $ f(x)\equal{}a_0\plus{}\sum_{k\equal{}1}^{\infty} a_k \cos (n_k x)$, where none of the positive integers $ n_k$ divides another. Prove that $ |a_k| \leq a_0$. [i]G. Halasz[/i]

Consider two odd natural numbers $a$ and $b$ where $a$ is a divisor of $b^2+2$ and $b$ is a divisor of $a^2+2.$ Prove that $a$ and $b$ are the terms of the series of natural numbers $\langle v_n\rangle$ defined by \[v_1 = v_2 = 1; v_n = 4v_ {n-1}-v_{n-2} \ \ \text{for} \ n\geq 3.\]

In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? ${ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 27\qquad\textbf{(E)}\ 30 $

For any positive integer $m\ge2$ define $A_m=\{m+1, 3m+2, 5m+3, 7m+4, \ldots, (2k-1)m + k, \ldots\}$. (1) For every $m\ge2$, prove that there exists a positive integer $a$ that satisfies $1\le a<m$ and $2^a\in A_m$ or $2^a+1\in A_m$. (2) For a certain $m\ge2$, let $a, b$ be positive integers that satisfy $2^a\in A_m$, $2^b+1\in A_m$. Let $a_0, b_0$ be the least such pair $a, b$. Find the relation between $a_0$ and $b_0$.

Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

We know that $R^3 = \{(x_1, x_2, x_3) | x_i \in R, i = 1, 2, 3\}$ is a vector space regarding the laws of composition $(x_1, x_2, x_3) + (y_1, y_2, y_3) = (x_1 + y_1, x_2 + y_2, x_3 + y_3)$, $\lambda (x_1, x_2, x_3) = (\lambda x_1, \lambda x_2, \lambda x_3)$, $\lambda \in R$. We consider the following subset of $R^3$ : $L =\{(x_1, x2, x_3) \in R^3 | x_1 + x_2 + x_3 = 0\}$. a) Prove that $L$ is a vector subspace of $R^3$ . b) In $R^3$ the following relation is defined $\overline{x} R \overline{y} \Leftrightarrow \overline{x} -\overline{y} \in L, \overline{x} , \overline{y} \in R^3$. Prove that it is an equivalence relation. c) Find two vectors of $R^3$ that belong to the same class as the vector $(-1, 3, 2)$.

$A, B, C$ are the angles of a triangle. Show that $2\frac{\sin A}{A} + 2\frac{\sin B}{B} + 2\frac{\sin C}{C} \le \left(\frac{1}{B} + \frac{1}{C}\right) \sin A + \left(\frac{1}{C} + \frac{1}{A}\right) \sin B + \left(\frac{1}{A} + \frac{1}{B}\right) \sin C$

For integers $n>1$, define $f(n)$ to be the sum of all postive divisors of $n$ that are less than $n$. Prove that for any positive integer $k$, there exists a positive integer $n>1$ such that $n<f(n)<f^2(n)<\cdots<f^k(n)$, where $f^i(n)=f(f^{i-1}(n))$ for $i>1$ and $f^1(n)=f(n)$.

When Phil and Shelley stand on a scale together, the scale reads $151$ pounds. When Shelley and Ryan stand on the same scale together, the scale reads $132$ pounds. When Phil and Ryan stand on the same scale together, the scale reads $115$ pounds. Find the number of pounds Shelley weighs.

For which positive integers $n$ is the following statement true: if $a_1, a_2, ... , a_n$ are positive integers, $a_k \le n$ for all $k$ and $\sum\limits_{k=1}^{{n}}{a_k}=2n$ then it is always possible to choose $a_{i1} , a_{i2} , ..., a_{ij}$ in such a way that the indices $i_1, i_2,... , i_j$ are different numbers, and $\sum\limits_{k=1}^{{{j}}}{a_{ik}}=n$?

Consider the three squares indicated in the figure. Show that if the lengths of the sides of the smaller square and the square greater are integers, then adding to the area of the smallest square the area of the inclined square, a perfect square is obtained. [img]https://1.bp.blogspot.com/-B0QdvZIjOLw/X4URvs3C0ZI/AAAAAAAAMmw/S5zMpPBXBn8Jj39d-OZVtMRUDn3tXbyWgCLcBGAsYHQ/s0/2002%2Bportugal%2Bp5.png[/img]

Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0. \] Determine when equality occurs.

In how many ways can one choose distinct numbers a and b from {1, 2, 3, ..., 2005} such that a + b is a multiple of 5?

$(a_{n})$ is sequence with positive integer. $a_{1}>10$ $ a_{n}=a_{n-1}+GCD(n,a_{n-1})$, n>1 For some i $a_{i}=2i$. Prove that these numbers are infinite in this sequence.

For a positive integer $n$, let $d_n$ be the units digit of $1 + 2 + \dots + n$. Find the remainder when \[\sum_{n=1}^{2017} d_n\] is divided by $1000$.

Is it possible to divide a $7\times7$ table into a few $\text{connected}$ parts of cells with the same perimeter? ( A group of cells is called $\text{connected}$ if any cell in the group, can reach other cells by passing through the sides of cells.)

Let $A^\prime$ be the point of tangency of the excircle of a triangle $ABC$ (corrsponding to $A$) with the side $BC$. The line $a$ through $A^\prime$ is parallel to the bisector of $\angle BAC$. Lines $b$ and $c$ are analogously defined. Prove that $a, b, c$ have a common point.

Let $n$ be a positive integer, and let $f(n) =1^n + 2^{n-1} + 3^{n-2}+ 4^{n-3}+... + (n-1)^2 + n^1$ Find the smallest possible value of $\frac{f(n+2)}{f(n)}$ .Justify your answer.

In the game of Ric-Rac-Roe, two players take turns coloring squares of a $3 \times 3$ grid in their color; a player wins if they complete a row or column of their color on their turn. If Alice and Bob play this game, picking an uncolored square uniformly at random on their turn, what is the probability that they tie?

Does there exist a triangle, whose side is equal to some of its altitudes, another side is equal to some of its bisectors, and the third is equal to some of its medians?

For all natural numbers $n$, define $f(n) = \tau (n!) - \tau ((n-1)!)$, where $\tau(a)$ denotes the number of positive divisors of $a$. Prove that there exist infinitely many composite $n$, such that for all naturals $m < n$, we have $f(m) < f(n)$.

Consider the set $ S$ of all numbers of the form $ a(n)\equal{}n^2\plus{}n\plus{}1, n \in \mathbb{N}.$ Show that the product $ a(n)a(n\plus{}1)$ is in $ S$ for all $ n \in \mathbb{N}$ and give an example of two elements $ s,t$ of $ S$ such that $ s,t \notin S$.

A circle, its chord $AB$ and the midpoint $W$ of the minor arc $AB$ are given. Take an arbitrary point $C$ on the major arc $AB$. The tangent to the circle at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$ respectively. Lines $WX$ and $WY$ meet $AB$ at points $N$ and $M$. Prove that the length of segment $NM$ doesn’t depend on point $C$.

Player $A$ and player $B$ play the next game on an $8$ by $8$ square chessboard. They in turn color a field that is not yet colored. One player uses red and the other blue. Player $A$ starts. The winner is the first person to color the four squares of a square of $2$ by $2$ squares with his color somewhere on the board. Prove that player $B$ can always prevent player $A$ from winning.

Given four weights in geometric progression and an equal arm balance, show how to find the heaviest weight using the balance only twice.