Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$

Let $n \ge 3$ be a natural number and let $P$ be a polygon with $n$ sides. Let $a_1,a_2,\cdots, a_n$ be the lengths of sides of $P$ and let $p$ be its perimeter. Prove that \[\frac{a_1}{p-a_1}+\frac{a_2}{p-a_2}+\cdots + \frac{a_n}{p-a_n} < 2 \]

Given real numbers $a, b, c$ such that $a + b - c = ab- bc - ca = abc = 8$. Find all possible values of $a$.

The five numbers $a, b, c, d,$ and $e$ satisfy the inequalities $$a+b < c+d < e+a < b+c < d+e.$$ Among the five numbers, which one is the smallest, and which one is the largest?

Jack and Jill run $10$ kilometers. They start at the same point, run $5$ kilometers up a hill, and return to the starting point by the same route. Jack has a $10$ minute head start and runs at the rate of $15$ km/hr uphill and $20$ km/hr downhill. Jill runs $16$ km/hr uphill and $22$ km/hr downhill. How far from the top of the hill are they when they pass going in opposite directions? $ \textbf{(A)}\ \frac{5}{4}\ km\qquad\textbf{(B)}\ \frac{35}{27}\ km\qquad\textbf{(C)}\ \frac{27}{20}\ km\qquad\textbf{(D)}\ \frac{7}{3}\ km\qquad\textbf{(E)}\ \frac{28}{9}\ km $

Let $k$ be a positive integer. Determine the least positive integer $n$, with $n\geq k+1$, for which the game below can be played indefinitely: Consider $n$ boxes, labelled $b_1,b_2,...,b_n$. For each index $i$, box $b_i$ contains exactly $i$ coins. At each step, the following three substeps are performed in order: [b](1)[/b] Choose $k+1$ boxes; [b](2)[/b] Of these $k+1$ boxes, choose $k$ and remove at least half of the coins from each, and add to the remaining box, if labelled $b_i$, a number of $i$ coins. [b](3)[/b] If one of the boxes is left empty, the game ends; otherwise, go to the next step. [i]Proposed by Demetres Christofides, Cyprus[/i]

The currency exchange trades dinars (D), guilders (G), reals (R) and thalers (T). Exchange players have the right to make a purchase and sale transaction with each pair of currencies no more than once a day. The exchange rates are as follows: $D = 6G$,; $D=25R$, $D=120T$,$G = 4R$; $G=21T$, $R = 5T$. For example, the entry $D = 6G$ means that $1$ dinar can be bought for $6$ guilders (or $6$ guilders can be sold for $1$ dinar). In the morning the player had $32$ dinars. What is the maximum number that he can receive by evening a) in dinars? b) in thalers ?

Prove that for arbitrary positive numbers the following inequality holds \[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.\]

Let $a_1,a_2,a_3,\ldots,a_n$ be a sequence of non-negative integers, where $n$ is a positive integer. Let \[ A_n={a_1+a_2+\cdots+a_n\over n}\ . \] Prove that \[ a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n \] where $\lfloor A_n\rfloor$ is the greatest integer less than or equal to $A_n$, and $a!=1\times 2\times\cdots\times a$ for $a\ge 1$(and $0!=1$). When does equality hold?

Denote $\mathbb{Z}_{>0}=\{1,2,3,...\}$ the set of all positive integers. Determine all functions $f:\mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0}$ such that, for each positive integer $n$, $\hspace{1cm}i) \sum_{k=1}^{n}f(k)$ is a perfect square, and $\vspace{0.1cm}$ $\hspace{1cm}ii) f(n)$ divides $n^3$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Prove that if $x, y$ and $n$ are positive integers such that $$x^{2024} + y^{2024} = 2^n,$$ then $x=y$.

Let $ABC$ be a triangle and let $M,N,P$ be points on the sides $BC$, $CA$ and $AB$ respectively such that \[ \frac{AP}{PB} = \frac{BM}{MC} = \frac{CN}{AN}. \] Prove that triangle if $MNP$ is equilateral then triangle $ABC$ is equilateral.

Let $ABCD$ be a square. Let $E, F, G$ and $H$ be the centers, respectively, of equilateral triangles with bases $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA},$ each exterior to the square. What is the ratio of the area of square $EFGH$ to the area of square $ABCD$? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}$

As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE?$ [asy] size(6cm); pair A = (0,10); label("$A$", A, N); pair B = (0,0); label("$B$", B, S); pair C = (10,0); label("$C$", C, S); pair D = (10,10); label("$D$", D, SW); pair EE = (15,11.8); label("$E$", EE, N); pair F = (3,10); label("$F$", F, N); filldraw(D--arc(D,2.5,270,380)--cycle,lightgray); dot(A^^B^^C^^D^^EE^^F); draw(A--B--C--D--cycle); draw(D--EE--F--cycle); label("$110^\circ$", (15,9), SW); [/asy] $\textbf{(A) }160\qquad\textbf{(B) }164\qquad\textbf{(C) }166\qquad\textbf{(D) }170\qquad\textbf{(E) }174$

Let $n$ be a positive integer such that the sum of all its positive divisors (inclusive of $n$) equals to $2n + 1$. Prove that $n$ is an odd perfect square. related: https://artofproblemsolving.com/community/c6h515011 https://artofproblemsolving.com/community/c6h108341 (Putnam 1976) https://artofproblemsolving.com/community/c6h368488 https://artofproblemsolving.com/community/c6h445330 https://artofproblemsolving.com/community/c6h378928

For a rational point (x,y), if xy is an integer that divided by 2 but not 3, color (x,y) red, if xy is an integer that divided by 3 but not 2, color (x,y) blue. Determine whether there is a line segment in the plane such that it contains exactly 2017 blue points and 58 red points.

Rectangle $ABCD$ has side lengths $AB = 3$ and $BC = 7$. Let $E$ be a point on $BC$, and let $F$ be the intersection of $DE$ and $AC$. Given that $[CDF] = 4$, find $\frac{DF}{FE}$ .

On the side $AB$ of the non-isosceles triangle $ABC$, let the points $P$ and $Q$ be so that $AC = AP$ and $BC = BQ$. The perpendicular bisector of the segment $PQ$ intersects the angle bisector of the $\angle C$ at the point $R$ (inside the triangle). Prove that $\angle ACB + \angle PRQ = 180^o$.

We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?

The city of Neverreturn has $N$ bus stops numbered $1, 2, \cdots , N.$ Each bus route is one-way and has only two stops, the beginning and the end. The route network is such that departing from any stop one cannot return to it using city buses. When the mayor notices a route going from a stop with a greater number to a stop with a lesser number, he orders to exchange the number plates of its beginning and its end. Can the plate changing go on infinitely? [i](K. Ivanov )[/i]

Find all natural numbers $n (n \geq 2)$ such that there exists reals $a_1, a_2, \dots, a_n$ which satisfy \[ \{ |a_i - a_j| \mid 1\leq i<j \leq n\} = \left\{1,2,\dots,\frac{n(n-1)}{2}\right\}. \] Let $A=\{1,2,3,4,5,6\}, B=\{7,8,9,\dots,n\}$. $A_i(i=1,2,\dots,20)$ contains eight numbers, three of which are chosen from $A$ and the other five numbers from $B$. $|A_i \cap A_j|\leq 2, 1\leq i<j\leq 20$. Find the minimum possible value of $n$.

Let $AB$ be a chord of the circle $(O)$. Denote M as the midpoint of the minor arc $AB$. A circle $(O')$ tangent to segment $AB$ and internally tangent to $(O)$. A line passes through $M$, perpendicular to $O'A$, $O'B$ and cuts $AB$ respectively at $C, D$. Prove that $AB = 2CD$.

Three travel companies provide transportation between $n$ cities, such that each connection between a pair of cities is covered by one company only. Prove that, for $n \geq 11$, there must exist a round-trip through some four cities, using the services of a same company, while for $n < 11$ this is not anymore necessarily true. [i]Dan Schwarz[/i]

Point $D$ lies on side $BC$ of $\triangle ABC$ so that $\overline{AD}$ bisects $\angle BAC$. The perpendicular bisector of $\overline{AD}$ intersects the bisectors of $\angle ABC$ and $\angle ACB$ in points $E$ and $F$, respectively. Given that $AB=4$, $BC=5$, $CA=6$, the area of $\triangle AEF$ can be written as $\tfrac{m\sqrt n}p$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.

$5.$ Let $P(x)$ be a non - zero polynomial with integer coefficients. If $P(n)$ is divisible by $n$ for each integer polynomial $n.$ What is the value of $P(0) ?$