$(SWE 6)$ Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.

Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational.

Find all groups of positive integers $ (a,x,y,n,m)$ that satisfy $ a(x^n \minus{} x^m) \equal{} (ax^m \minus{} 4) y^2$ and $ m \equiv n \pmod{2}$ and $ ax$ is odd.

Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying (i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$. (ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.

Let $ABC$ be a scalene triangle. Points $A_1,B_1$ and $C_1$ are chosen on segments $BC,CA$ and $AB$, respectively, such that $\triangle A_1B_1C_1$ and $\triangle ABC$ are similar. Let $A_2$ be the unique point on line $B_1C_1$ such that $AA_2=A_1A_2$. Points $B_2$ and $C_2$ are defined similarly. Prove that $\triangle A_2B_2C_2$ and $\triangle ABC$ are similar. [i]Fedir Yudin [/i]

Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is [i]superabundant[/i] (P.Erdos, $1944$) if $\forall k \in \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$ Prove that there exists an infinity of [i]superabundant[/i] numbers.

The value of $ \log_5 \frac {(125)(625)}{25}$ is equal to: $\textbf{(A)}\ 725 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 3125 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \text{None of these}$

Let $\mathbb C$ denote the set of all complex numbers. Define $$A=\{(z,w)|z,w\in\mathbb C\text{ and }|z|=|w|\}$$$$B=\{(z,w)|z,w\in\mathbb C\text{ and }z^2=w^2\}$$$\textbf{(A)}~A=B$ $\textbf{(B)}~A\subset B\text{ and }A\ne B$ $\textbf{(C)}~B\subset A\text{ and }B\ne A$ $\textbf{(D)}~\text{None of the above}$

Points $A(0,0)$ and $B(1,1)$ are located on the parabola $y=x^2$. A third point $C$ is positioned on this parabola between $A$ and $B$ such that $AC=CB=r$. What is $r^2$?

Let $N$ be a given positive integer. Consider a permutation of $1,2,3,\cdots,N$, denoted as $p_1,p_2,\cdots,p_N$. For a section $p_l, p_{l+1},\cdots, p_r$, we call it "extreme" if $p_l$ and $p_r$ are the maximum and minimum value of that section. We say a permutation $p_1,p_2,\cdots,p_N$ is "super balanced" if there isn't an "extreme" section with a length at least $3$. For example, $1,4,2,3$ is "super balanced", but $3,1,2,4$ isn't. Please answer the following questions: 1. How many "super balanced" permutations are there? 2. For each integer $M\leq N$. How many "super balanced" permutations are there such that $p_1=M$? [i]Proposed by ltf0501[/i]

There are $2n$ points on the plane. No three of them lie on the same straight line and no four lie on the same circle. Prove that it is possible to split these points into $n$ pairs and cover each pair of points with a circle containing no other points.

Is it true that in every convex polygon $3$ adjacent vertices can be selected such that their circumcirscribed circle can cover the entire polygon?

The complex number $z$ is equal to $9+bi$, where $b$ is a positive real number and $i^{2}=-1$. Given that the imaginary parts of $z^{2}$ and $z^{3}$ are equal, find $b$.

Let's call integer square-free if it's not divisible by $p^2$ for any prime $p$. You are given a square-free integer $n>1$, which has exactly $d$ positive divisors. Find the largest number of its divisors that you can choose, such that $a^2 + ab - n$ isn't a square of an integer for any $a, b$ among chosen divisors. [i](Proposed by Oleksii Masalitin)[/i]

Find all functions $f : R \to R$ that satisfy $f(x + y^2 - f(y)) = f(x)$ for all $x,y \in R$. (Vo Quoc Ba Can)

Inside the triangle $ABC $, the point $P $ is selected so that $BC = AP $ and $\angle APC = 180 {} ^ \circ - \angle ABC $. On the side $AB $ there is a point $K $, for which $AK = KB + PC $. Prove that $\angle AKC = 90 {} ^ \circ $. (Danilo Hilko)

How many rectangles are there in the diagram below such that the sum of the numbers within the rectangle is a multiple of 7? [asy] int n; n=0; for (int i=0; i<=7;++i) { draw((i,0)--(i,7)); draw((0,i)--(7,i)); for (int a=0; a<=7;++a) { if ((a != 7)&&(i != 7)) { n=n+1; label((string) n,(a,i),(1.5,2)); } } } [/asy]

Two circles with radii 1 meet in points $X, Y$, and the distance between these points also is equal to $1$. Point $C$ lies on the first circle, and lines $CA, CB$ are tangents to the second one. These tangents meet the first circle for the second time in points $B', A'$. Lines $AA'$ and $BB'$ meet in point $Z$. Find angle $XZY$.

Let be three nonzero complex numbers $ a,b,c $ satisfying $$ |a|=|b|=|c|=\left| \frac{a+b+c-abc}{ab+bc+ca-1} \right| . $$ Prove that these three numbers have all modulus $ 1 $ or there are two distinct numbers among them whose sum is $ 0. $ [i]Costel Anghel[/i]

Determine the largest positive integer $n$ which cannot be written as the sum of three numbers bigger than $1$ which are pairwise coprime.

Find all integers $x$ and $y$ such that \[x^3-117y^3=5.\]

Let $ n$ be a positive integer. Show that \[ \left(\sqrt{2} \plus{} 1 \right)^n \equal{} \sqrt{m} \plus{} \sqrt{m\minus{}1}\] for some positive integer $ m.$

Let $X$ denotes the set of integers from $1$ to $239$. A magician with an assistant perform a trick. The magician leaves the hall and the spectator writes a sequence of $10$ elements on the board from the set $X$. The magician’s assistant looks at them and adds $k$ more elements from $X$ to the existing sequence. After that the spectator replaces three of these $k+10$ numbers by random elements of $X$ (it is permitted to change them by themselves, that is to not change anything at all, for example). The magician enters and looks at the resulting row of $k+10$ numbers and without error names the original $10$ numbers written by the spectator. Find the minimal possible $k$ for which the trick is possible.

Determine the real numbers $ a,b, $ such that $$ [ax+by]+[bx+ay]=(a+b)\cdot [x+y],\quad\forall x,y\in\mathbb{R} , $$ where $ [t] $ is the greatest integer smaller than $ t. $

Let $ABC$ be a triangle with $AB=5$, $BC=4$, and $CA=3$. On each side of $ABC$, externally erect a semicircle whose diameter is the corresponding side. Let $X$ be on the semicircular arc erected on side $\overline{BC}$ such that $\angle CBX$ has measure $15^\circ$. Let $Y$ be on the semicircular arc erected on side $\overline{CA}$ such that $\angle ACY$ has measure $15^\circ$. Similarly, let $Z$ be on the semicircular arc erected on side $\overline{AB}$ such that $\angle BAZ$ has measure $15^\circ$. What is the area of triangle $XYZ$?