For positive integer $n$, note $S_n=1^{2024}+2^{2024}+ \cdots +n^{2024}$. Prove that there exists infinitely many positive integers $n$, such that $S_n$ isn’t divisible by $1865$ but $S_{n+1}$ is divisible by $1865$

The numbers on the faces of this cube are consecutive whole numbers. The sums of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is [asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); label("$15$",(1.5,1.2),N); label("$11$",(4,2.3),N); label("$14$",(2.5,3.7),N);[/asy] $ \text{(A)}\ 75\qquad\text{(B)}\ 76\qquad\text{(C)}\ 78\qquad\text{(D)}\ 80\qquad\text{(E)}\ 81 $

Determine the largest integer $n$ such that $7^{2048}-1$ is divisible by $2^n$.

A positive integer $n$ is defined as a $\textit{stepstool number}$ if $n$ has one less positive divisor than $n + 1$. For example, $3$ is a stepstool number, as $3$ has $2$ divisors and $4$ has $2 + 1 = 3$ divisors. Find the sum of all stepstool numbers less than $300$. [i]Proposed by [b]Th3Numb3rThr33[/b][/i]

Suppose that $ABCD$ is a parallelogram such that $\angle DAC = 90^o$. Let $H$ be the foot of perpendicular from $A$ to $DC$, also let $P$ be a point along the line $AC$ such that the line $PD$ is tangent to the circumcircle of the triangle $ABD$. Prove that $\angle PBA = \angle DBH$. Proposed by Iman Maghsoudi

Determine all strictly increasing functions $f: R \to R$ satisfying $f (f(x) + y) = f(x + y) + f (0)$ for all $x,y \in R$.

An auditorium with $20$ rows of seats has $10$ seats in the first row. Each successive row has one more seat than the previous row. If students taking an exam are permitted to sit in any row, but not next to another student in that row, then the maximum number of students that can be seated for an exam is $\text{(A)}\ 150 \qquad \text{(B)}\ 180 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 400 \qquad \text{(E)}\ 460$

Let the point $P$ in the interior of the triangle $ABC$. $(AP, (BP, (CP$ intersect the circumcircle of $ABC$ at $A_{1}, B_{1}, C_{1}$. Prove that the maximal value of the sum of the areas $A_{1}BC$, $B_{1}AC$, $C_{1}AB$ is $p(R-r)$, where $p, r, R$ are the usual notations for the triangle $ABC$.

Four lines in the plane intersect in six points. Each line is thus divided into two segments and two rays. Is it possible for the eight segments to have lengths $1, 2, 3, ... , 8$? Can the lengths of the eight segments be eight distinct integers?

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

Determine the quadratic functions $f(x) = ax^2 + bx + c$ for which there exists an interval $(h, k)$ such that for all $x \in (h, k)$ it holds that $f(x)f(x + 1) < 0$ and $f(x)f(x -1) < 0$.

It is given a cube with sidelength $a$. Find the surface of the intersection of the cube with a plane, perpendicular to one of its diagonals and whose distance from the centre of the cube is equal to $h$.

Find the domain of the pairs of positive real numbers $(a,\ b)$ such that there is a $\theta\ (0<\theta \leq \pi)$ such that $\cos a\theta =\cos b\theta$, then draw the domain on the coordinate plane. 30 points

Compute the number of ordered quintuples of nonnegative integers $(a_1,a_2,a_3,a_4,a_5)$ such that $0\leq a_1,a_2,a_3,a_4,a_5\leq 7$ and $5$ divides $2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$.

Let $x$ and $y$ be positive integers we have a table $x\times y$ where $(x + 1)(y + 1)$ points are red(the points are the vertices of the squares). Initially there is one ant in each red point, in a moment the ants walk by the lines of the table with same speed, each turn that an ant arrive in a red point the ant moves $90º$ to some direction. Determine all values of $x$ and $y$ where is possible that the ants move indefinitely where can't be in any moment two ants in the same red point.

It is given a graph whose vertices are positive integers and an edge between numbers $a$ and $b$ exists if and only if $a + b + 1 | a^2 + b^2 + 1$. Is this graph connected?

Find minimum number $n$ that: 1) $80|n$ 2) we can permute 2 different numbers in $n$ to get $m$ and $80|m$

You have some white one-by-one tiles and some black and white two-bye-one tiles as shown below. There are four different color patterns that can be generated when using these tiles to cover a three-by-one rectangoe by laying these tiles side by side (WWW, BWW, WBW, WWB). How many different color patterns can be generated when using these tiles to cover a ten-by-one rectangle? [asy] import graph; size(5cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((12,0)--(12,1)--(11,1)--(11,0)--cycle); fill((13.49,0)--(13.49,1)--(12.49,1)--(12.49,0)--cycle, black); draw((13.49,0)--(13.49,1)--(14.49,1)--(14.49,0)--cycle); draw((15,0)--(15,1)--(16,1)--(16,0)--cycle); fill((17,0)--(17,1)--(16,1)--(16,0)--cycle, black); [/asy]

Let $n \geq 2$ be an integer. Prove that if $$\frac{n^2+4^n+7^n}{n}$$ is an integer, then it is divisible by 11.

Let $n\geqslant 2$ be a fixed positive integer. Determine the minimum value of the expression $$\frac{a_{a_1}}{a_1}+\frac{a_{a_2}}{a_2}+\dots +\frac{a_{a_n}}{a_n},$$ where $a_1, a_2, \dots, a_n$ are positive integers at most $n$. [i]Proposed by David Anghel[/i]

Three circles $C_1,C_2,C_3$, with radii $1, 2, 3$ respectively, are externally tangent. In the area enclosed by these circles, there is a circle $C_4$ which is externally tangent to all three circles. Find the radius of $C_4$. [asy] unitsize(0.4 cm); pair[] O; real[] r; O[1] = (-12/5,16/5); r[1] = 1; O[2] = (0,5); r[2] = 2; O[3] = (0,0); r[3] = 3; O[4] = (-132/115, 351/115); r[4] = 6/23; draw(Circle(O[1],r[1])); draw(Circle(O[2],r[2])); draw(Circle(O[3],r[3])); draw(Circle(O[4],r[4])); label("$C_1$", O[1]); label("$C_2$", O[2]); label("$C_3$", O[3]); [/asy]

Find all funtions $f : Z_{>0} \to Z_{>0}$ that satisfy $f(f(f(n))) + f(f(n)) + f(n) = 3n$ for all $n \in Z_{>0}$ .

In triangle $ABC$ we have $\angle C = 90^o$ and $AC = BC$. Furthermore $M$ is an interior pont in the triangle so that $MC = 1 , MA = 2$ and $MB =\sqrt2$. Determine $AB$

If $a(x), b(x), c(x)$ and $d(x)$ are polynomials in $ x$, show that $$ \int_{1}^{x} a(x) c(x)\; dx\; \cdot \int_{1}^{x} b(x) d(x) \; dx - \int_{1}^{x} a(x) d(x)\; dx\; \cdot \int_{1}^{x} b(x) c(x)\; dx$$ is divisible by $(x-1)^4.$

Alice and Brian are playing a game on a $1\times (N + 2)$ board. To start the game, Alice places a checker on any of the $N$ interior squares. In each move, Brian chooses a positive integer $n$. Alice must move the checker to the $n$-th square on the left or the right of its current position. If the checker moves off the board, Alice wins. If it lands on either of the end squares, Brian wins. If it lands on another interior square, the game proceeds to the next move. For which values of $N$ does Brian have a strategy which allows him to win the game in a finite number of moves?