Let $p \neq 5$ be a prime number. Prove that $p^5-1$ has a prime divisor of the form $5x+1$.

Let $BB_1$ and $CC_1$ be altitudes of triangle $ABC$. The tangents to the circumcircle of $AB_1C_1$ at $B_1$ and $C_1$ meet AB and $AC$ at points $M$ and $N$ respectively. Prove that the common point of circles $AMN$ and $AB_1C_1$ distinct from $A$ lies on the Euler line of $ABC$.

Let $m$ be the largest real solution to the equation \[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\] There are positive integers $a,b,c$ such that $m = a + \sqrt{b+\sqrt{c}}$. Find $a+b+c$.

Let $\sigma(n)=\sum_{d|n} d$, the sum of positive divisors of an integer $n>0$. [list] [b](a)[/b] Show that $\sigma(mn)=\sigma(m)\sigma(n)$ for positive integers $m$ and $n$ with $gcd(m,n)=1$ [b](b)[/b] Find all positive integers $n$ such that $\sigma(n)$ is a power of $2$.[/list]

To a manufacturer of three products whose unit prices are $50$, $70$, and $65$ pta, a retailer asks him for $100$ units, remitting him $6850$ pta as payment, on the condition that you send as many of the higher-priced product as possible and the rest of the other two. How many of each product should he send to serve the request?

Two circles are internally tangent at $N$. The chords $BA$ and $BC$ of the larger circle are tangent to the smaller circle at $K$ and $M$ respectively. $Q$ and $P$ are midpoint of arcs $AB$ and $BC$ respectively. Circumcircles of triangles $BQK$ and $BPM$ are intersect at $L$. Show that $BPLQ$ is a parallelogram.

At a recent math contest, Evan was asked to find $2^{2016} \pmod{p}$ for a given prime number $p$ with $100 < p < 500$. Evan has forgotten what the prime $p$ was, but still remembers how he solved it: [list] [*] Evan first tried taking $2016$ modulo $p - 1$, but got a value $e$ larger than $100$. [*] However, Evan noted that $e - \frac{1}{2}(p - 1) = 21$, and then realized the answer was $-2^{21} \pmod{p}$. [/list] What was the prime $p$?

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f'(x) > f(x)>0$ for all real numbers $x$. Show that $f(8) > 2022f(0)$. [i]Proposed by Ethan Tan[/i]

Poldavia is a strange kingdom. Its currency unit is the bourbaki and there exist only two types of coins: gold ones and silver ones. Each gold coin is worth $ n$ bourbakis and each silver coin is worth $ m$ bourbakis ($ n$ and $ m$ are positive integers). Using gold and silver coins, it is possible to obtain sums such as 10000 bourbakis, 1875 bourbakis, 3072 bourbakis, and so on. But Poldavia’s monetary system is not as strange as it seems: [b](a)[/b] Prove that it is possible to buy anything that costs an integral number of bourbakis, as long as one can receive change. [b](b)[/b] Prove that any payment above $ mn\minus{}2$ bourbakis can be made without the need to receive change.

A sequence of polynomials $\{f_n\}_{n=0}^{\infty}$ is defined recursively by $f_0(x)=1$, $f_1(x)=1+x$, and \[(k+1)f_{k+1}(x)-(x+1)f_k(x)+(x-k)f_{k-1}(x)=0, \quad k=1,2,\ldots\] Prove that $f_k(k)=2^k$ for all $k\geq 0$.

Four sequences of real numbers $(a_n), (b_n), (c_n), (d_n)$ satisfy for all $n$, $$a_{n+1} = a_n +b_n, b_{n+1} = b_n +c_n,$$ $$c_{n+1} = c_n +d_n, d_{n+1} = d_n +a_n.$$ Prove that if $a_{k+m} = a_m, b_{k+m} = b_m, c_{k+m} = c_m, d_{k+m} = d_m$ for some $k\ge 1,n \ge 1$, then $a_2 = b_2 = c_2 = d_2 = 0$.

Let $a_1,a_2,...,a_{2020}$ be positive real numbers. Prove that: $$\max{(a^2_1-a_2,a^2_2-a_3,...,a^2_{2020}-a_1)}\ge\max{(a^2_1-a_1,a^2_2-a_2,...,a^2_{2020}-a_{2020})}$$

Let $ABC$ be an acute-angled triangle inscribed in the circle $(O)$. Let $AD$ be the diameter of $(O)$. The points $M,N$ are chosen on $BC$ such that $OM\parallel AB, ON\parallel AC$. The lines $DM,DN$ cut $(O)$ again at $P,Q$. Prove that $BC=DP=DQ$. Tran Quang Hung, Vietnam

Let $x,y$ and $z$ be non-negative real numbers satisfying the equation $x+y+z=xyz$. Prove that $2(x^2+y^2+z^2)\geq3(x+y+z)$.

A right triangle has all its sides rational numbers and the area $S $. Prove that there exists a right triangle, different from the original one, such that all its sides are rational numbers and its area is $S $. Tuymaada 2017 Q4 Juniors

Steve says to Jon, "I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form $P(x)=2x^3-2ax^2+(a^2-81)x-c$ for some positive integers $a$ and $c$. Can you tell me the values of $a$ and $c$?" After some calculations, Jon says, "There is more than one such polynomial." Steve says, "You’re right. Here is the value of $a$." He writes down a positive integer and asks, "Can you tell me the value of $c$?" Jon says, "There are still two possible values of $c$." Find the sum of the two possible values of $c$.

A circle is tangential to sides $AB$ and $AD$ of convex quadrilateral $ABCD$ at $G$ and $H$ respectively, and cuts diagonal $AC$ at $E$ and $F$. What are the necessary and sufficient conditions such that there exists another circle which passes through $E$ and $F$, and is tangential to $DA$ and $DC$ extended?

On a $n \times n$ grid, each edge are written with $=$ or $\neq$. We need to filled every cells with color black or white. Find the largest constant $k$, such that for every $n>777771449$ and any layout of $=$ and $\neq$, we can always find a way to colored every cells, such that at least $k \cdot 2n(n-1)$ neighboring cells, there colors conform to the symbols on the edge. (Namely, two cells are filled with the same color if $=$ was written on their edge; two cells are filled with different colors if $\neq$ was written on their edge) [i]Proposed by chengbilly & sn6dh[/i]

Georg has a board displaying the numbers from $1$ to $50$. Georg may strike out a number if it can be formed by starting with the number $2$ and doing one or more calculations where he either multiplies by $10$ or subtracts $3$. Which of the board’s numbers may Georg strike out?[img]https://cdn.artofproblemsolving.com/attachments/c/e/1bea13b691d3591d782e698bedee3235f8512f.png[/img] Example: Georg may strike out $26$ because it may, for example, be formed by starting with $2$, multiplying by $10$, subtracting $3$ three times, multiplying by $10$ and subtracting $3$ twenty-eight times.

Let $x_0,x_1,\dots, x_{n-1}$ be real numbers such that $0<|x_0|<|x_1|<\dots<|x_{n-1}|$. We will write the sum of the elements of each one of the $2^n$ subsets of $\{x_0,x_1,\dots,x_{n-1}\}$ in a paper. Prove that the $2^n$ written numbers are consecutive elements of a arithmetic progression if and only if the ratios $$|\frac{x_i}{x_j}|, 0\leq j<i\leq n-1$$ are equal(s) to the ratio(s) obtained with the numbers $2^0,2^1,\dots,2^{n-1}$. Note: The sum of the elements of the empty set is $0$.

Starting with a list of three numbers, the “[i]Make-My-Day[/i]” procedure creates a new list by replacing each number by the sum of the other two. For example, from $\{1, 3, 8\}$ “[i]Make-My-Day[/i]” gives $\{11, 9, 4\}$ and a new “[i]MakeMy-Day[/i]” leads to $\{13, 15, 20\}$. If we begin with $\{20, 1, 8\}$, what is the maximum difference between two numbers on the list after $2018$ consecutive “[i]Make-My-Day[/i]”s?

Prove that for all positive integers $m$ and $n$, $$\frac1m\cdot\binom{2n}0-\frac1{m+1}\cdot\binom{2n}1+\frac1{m+2}\cdot\binom{2n}2-\ldots+\frac1{m+2n}\cdot\binom{2n}{n2}>0$$

Some circles of radius 2 are drawn on the plane. Prove that the numerical value of the total area covered by these circles is at least as big as the total length of arcs bounding the area.

Real numbers $a, b, c$ are such that $$a^2+c-bc = b^2+a-ca = c^2+b-ab$$ Does it follow that $a=b=c$? [i]Proposed by Mykhailo Shtandenko[/i]

Chameleons of five colors live on the island. When one chameleon bites another, the color of bitten chameleon changes to one of these five colors according to some rule, and the new color depends only on the color of the bitten and the color of the bitting. It is known that $2023$ red chameleons can agree on a sequence of bites between themselves, after which they will all turn blue. What is the smallest $k$ that can guarantee that $k$ red chameleons, biting only each other, can turn blue? (For example, the rules might be: if a red chameleon bites a green one, the bitten one changes color to blue; if a green one bites a red one, the bitten one remains red, that is, "changes color to red"; if red bites red, the bitten one changes color to yellow, etc. The rules for changing colors may be different.)