Prove that if $n$ is a natural number such that $1 + 2^n + 4^n$ is prime then $n = 3^k$ for some $k \in N_0$.

Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$

Prove that the smallest dihedral angle between faces of an arbitrary tetrahedron is not greater than the dihedral angle between faces of a regular tetrahedron. (S. Shosman, O. Ogievetsky)

Let $S$ be the statement "If the sum of the digits of the whole number $n$ is divisible by 6, then $n$ is divisible by 6." A value of $n$ which shows $S$ to be false is $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 33 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ \text{None of these} $

Let $n \geq 4$ be an integer. Consider a regular $2n-$gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this $2n-$gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum. Determine for which (not necessarily positive) integers $m$ the integers $m + 1, m + 2, . . . , m + 2n$ can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)

What is the least possible ratio of two isosceles triangles areas, if three vertices of the first one belong to three different sides of the second one?

On a large piece of paper, Dana creates a “rectangular spiral” by drawing line segments of lengths, in cm, of 1, 1, 2, 2, 3, 3, 4, 4, ... as shown. Dana’s pen runs out of ink after the total of all the lengths he has drawn is 3000 cm. What is the length of the longest line segment that Dana draws? $\textbf{(A)}\ 38 \qquad \textbf{(B)}\ 39 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 30$

[b]a)[/b] Find the group $ H $ that is isomorphic with the multiplicative group of positive real numbers, having an isomorphism $$ \iota :(0,\infty )\longrightarrow H,\quad\iota (x)=\frac{x-1}{x+1} . $$ [b]b)[/b] Calculate the $ 2012\text{-th} $ power of an arbitrary element of $ H. $

A computer program that works only with integer numbers reads the numbers on the screen, identifies the selected numbers and performs one of the following actions: • If button $A$ is pressed, the user selects $5$ numbers and then each selected number is changed to its successor; • If button $B$ is pressed, the user selects $5$ numbers and then each selected number is changed to its triple. Bento has this program on his computer with the numbers $1, 3, 3^2, · · ·, 3^{19}$ on the screen, each one appearing just once. a) By simply pressing button $A$ several times, is Bento able to make the sum of the numbers on the screen be $2024^{2025}$? b) What is the minimum number of times that Bento must press button $B$ to make all the numbers on the screen turn equal, without pressing button $A$?

Let $S_n$ denote the sum of the first $n$ prime numbers. Prove that for any $n$ there exists the square of an integer between $S_n$ and $S_{n+1}$.

Let five points on a circle be labelled $A, B, C, D$, and $E$ in clockwise order. Assume $AE = DE$ and let $P$ be the intersection of $AC$ and $BD$. Let $Q$ be the point on the line through $A$ and $B$ such that $A$ is between $B$ and $Q$ and $AQ = DP$ Similarly, let $R$ be the point on the line through $C$ and $D$ such that $D$ is between $C$ and $R$ and $DR = AP$. Prove that $PE$ is perpendicular to $QR$.

Given a random positive integer $N$. Prove that there exist infinitely many positive integers $M$ whose none of its digits is $0$ and such that the sum of the digits of $N \cdot M$ is same as sum of digits $M$.

Let $n$ be a positive integer. $2n+1$ tokens are in a row, each being black or white. A token is said to be [i]balanced[/i] if the number of white tokens on its left plus the number of black tokens on its right is $n$. Determine whether the number of [i]balanced[/i] tokens is even or odd.

The the parallel lines through an inner point $P$ of triangle $\triangle ABC$ split the triangle into three parallelograms and three triangles adjacent to the sides of $\triangle ABC$. (a) Show that if $P$ is the incenter, the perimeter of each of the three small triangles equals the length of the adjacent side. (b) For a given triangle $\triangle ABC$, determine all inner points $P$ such that the perimeter of each of the three small triangles equals the length of the adjacent side. (c) For which inner point does the sum of the areas of the three small triangles attain a minimum? [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 4)[/i]

Determine if there exists an infinite sequence $a_0, a_1, a_2, a_3,...$ of nonegative integers that satisfies the following conditions: (i) All nonegative integers appear in the sequence exactly once. (ii) The succession $b_n=a_{n}+n,$, $n\geq0$, is formed by all prime numbers and each one appears exactly once.

One point of the plane is called $rational$ if both coordinates are rational and $irrational$ if both coordinates are irrational. Check whether the following statements are true or false: [b]a)[/b] Every point of the plane is in a line that can be defined by $2$ rational points. [b]b)[/b] Every point of the plane is in a line that can be defined by $2$ irrational points. This maybe is not algebra so sorry if I putted it in the wrong category!

Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties: (i) $f(1) = 2$; (ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.

Let $U$ and $C$ be two circles, and kite $BERK$ have vertices that lie on $U$ and sides that are tangent to $C.$ Given that the diagonals of the kite measure $5$ and $6,$ find the ratio of the area of $U$ to the area of $C.$

Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that \[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}. \]

Let $C$ be a right circular cone with apex $A$. Let $P_1, P_2, P_3, P_4$ and $P_5$ be points placed evenly along the circular base in that order, so that $P_1P_2P_3P_4P_5$ is a regular pentagon. Suppose that the shortest path from $P_1$ to $P_3$ along the curved surface of the cone passes through the midpoint of $AP_2$. Let $h$ be the height of $C$, and $r$ be the radius of the circular base of $C$. If $\left(\frac{h}{r}\right)^2$ can be written in simplest form as $\frac{a}{b}$ , find $a + b$.

Which of the following numbers is [b]not[/b] a perfect square? $\textbf{(A) }1^{2016}\qquad\textbf{(B) }2^{2017}\qquad\textbf{(C) }3^{2018}\qquad\textbf{(D) }4^{2019}\qquad \textbf{(E) }5^{2020}$

For a positive integer $n$ and a subset $S$ of $\{1, 2, \dots, n\}$, let $S$ be "$n$-good" if and only if for any $x$, $y\in S$ (allowed to be same), if $x+y\leq n$, then $x+y\in S$. Let $r_n$ be the smallest real number such that for any positive integer $m\leq n$, there is always a $m$-element "$n$-good" set, so that the sum of its elements is not more than $m\cdot r_n$. Prove that there exists a real number $\alpha$ such that for any positive integer $n$, $|r_n-\alpha n|\leq 2024.$

Inscribed circle of triangle $ABC$ touches sides $BC$, $CA$ and $AB$ at the points $X$, $Y$ and $Z$, respectively. Let $AA_{1}$, $BB_{1}$ and $CC_{1}$ be the altitudes of the triangle $ABC$ and $M$, $N$ and $P$ be the incenters of triangles $AB_{1}C_{1}$, $BC_{1}A_{1}$ and $CA_{1}B_{1}$, respectively. a) Prove that $M$, $N$ and $P$ are orthocentres of triangles $AYZ$, $BZX$ and $CXY$, respectively. b) Prove that common external tangents of these incircles, different from triangle sides, are concurent at orthocentre of triangle $XYZ$.

For how many nonnegative integer values of $k$ does the equation $7x^2 +kx +11 = 0$ have no real solutions?

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.