a) A positive integer is written at each vertex of a triangle. Then on each side of the triangle the greatest common divisor of its ends is written. It is possible that the numbers written on the sides be three consecutive integers, in some order? b) A positive integer is written at each vertex of a tetrahedron. Then, on each edge of the tetrahedron is written the greatest common divisor of its ends . It is possible that the numbers written in the edges are six consecutive integers, in some order?

Let $ a_1, \ldots, a_n$ be distinct positive integers that do not contain a $ 9$ in their decimal representations. Prove that the following inequality holds \[ \sum^n_{i\equal{}1} \frac{1}{a_i} \leq 30.\]

Prove that $1^{1987} + 2^{1987} + ... + n^{1987}$ is divisible by $n+2$.

Let $r_2, r_3,\ldots, r_{1000}$ denote the remainders when a positive odd integer is divided by $2,3,\ldots,1000$, respectively. It is known that the remainders are pairwise distinct and one of them is $0$. Find all values of $k$ for which it is possible that $r_k = 0$.

Let $a>1$ be an integer which is not divisible by four. Prove that there are infinitely many primes $p$ of the form $4k-1$ such that $p | a^d-1$ for some $d<\frac{p-1}{2}$

Find the least positive integer whose digits add to a multiple of 27 yet the number itself is not a multiple of 27. For example, 87999921 is one such number.

Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square.

[i]25 problems for 30 minutes[/i] [b]p1.[/b] You and nine friends spend $4000$ dollars on tickets to attend the new Harry Styles concert. Unfortunately, six friends cancel last minute due to the u. You and your remaining friends still attend the concert and split the original cost of $4000$ dollars equally. What percent of the total cost does each remaining individual have to pay? [b]p2.[/b] Find the number distinct $4$ digit numbers that can be formed by arranging the digits of $2021$. [b]p3.[/b] On a plane, Darnay draws a triangle and a rectangle such that each side of the triangle intersects each side of the rectangle at no more than one point. What is the largest possible number of points of intersection of the two shapes? [b]p4.[/b] Joy is thinking of a two-digit number. Her hint is that her number is the sum of two $2$-digit perfect squares $x_1$ and $x_2$ such that exactly one of $x_i - 1$ and $x_i + 1$ is prime for each $i = 1, 2$. What is Joy's number? [b]p5.[/b] At the North Pole, ice tends to grow in parallelogram structures of area $60$. On the other hand, at the South Pole, ice grows in right triangular structures, in which each triangular and parallelogram structure have the same area. If every ice triangle $ABC$ has legs $\overline{AB}$ and $\overline{AC}$ that are integer lengths, how many distinct possible lengths are there for the hypotenuse $\overline{BC}$? [b]p6.[/b] Carlsen has some squares and equilateral triangles, all of side length $1$. When he adds up the interior angles of all shapes, he gets $1800^o$. When he adds up the perimeters of all shapes, he gets $24$. How many squares does he have? [b]p7.[/b] Vijay wants to hide his gold bars by melting and mixing them into a water bottle. He adds $100$ grams of liquid gold to $100$ grams of water. His liquefied gold bars have a density of $20$ g/ml and water has a density of $1$ g/ml. Given that the density of the mixture in g/mL can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute the sum $m + n$. (Note: density is mass divided by volume, gram (g) is unit of mass and ml is unit of volume. Further, assume the volume of the mixture is the sum of the volumes of the components.) [b]p8.[/b] Julius Caesar has epilepsy. Specifically, if he sees $3$ or more flashes of light within a $0.1$ second time frame, he will have a seizure. His enemy Brutus has imprisoned him in a room with $4$ screens, which flash exactly every $4$, $5$, $6$, and $7$ seconds, respectively. The screens all flash at once, and $105$ seconds later, Caesar opens his eyes. How many seconds after he opened his eyes will Caesar first get a seizure? [b]p9.[/b] Angela has a large collection of glass statues. One day, she was bored and decided to use some of her statues to create an entirely new one. She melted a sphere with radius $12$ and a cone with height of 18 and base radius of $2$. If Angela wishes to create a new cone with a base radius $2$, what would the the height of the newly created cone be? [b]p10.[/b] Find the smallest positive integer $N$ satisfying these properties: (a) No perfect square besides $1$ divides $N$. (b) $N$ has exactly $16$ positive integer factors. [b]p11.[/b] The probability of a basketball player making a free throw is $\frac15$. The probability that she gets exactly $2$ out of $4$ free throws in her next game can be expressed as $\frac{m}{n}$ for relatively prime positive integers m and n. Find $m + n$. [b]p12.[/b] A new donut shop has $1000$ boxes of donuts and $1000$ customers arriving. The boxes are numbered $1$ to $1000$. Initially, all boxes are lined up by increasing numbering and closed. On the first day of opening, the first customer enters the shop and opens all the boxes for taste testing. On the second day of opening, the second customer enters and closes every box with an even number. The third customer then "reverses" (if closed, they open it and if open, they close it) every box numbered with a multiple of three, and so on, until all $1000$ customers get kicked out for having entered the shop and reversing their set of boxes. What is the number on the sixth box that is left open? [b]p13.[/b] For an assignment in his math class, Michael must stare at an analog clock for a period of $7$ hours. He must record the times at which the minute hand and hour hand form an angle of exactly $90^o$, and he will receive $1$ point for every time he records correctly. What is the maximum number of points Michael can earn on his assignment? [b]p14.[/b] The graphs of $y = x^3 +5x^2 +4x-3$ and $y = -\frac15 x+1$ intersect at three points in the Cartesian plane. Find the sum of the $y$-coordinates of these three points. [b]p15.[/b] In the quarterfinals of a single elimination countdown competition, the $8$ competitors are all of equal skill. When any $2$ of them compete, there is exactly a $50\%$ chance of either one winning. If the initial bracket is randomized, the probability that two of the competitors, Daniel and Anish, face off in one of the rounds can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p$, $q$. Find $p + q$. [b]p16.[/b] How many positive integers less than or equal to $1000$ are not divisible by any of the numbers $2$, $3$, $5$ and $11$? [b]p17.[/b] A strictly increasing geometric sequence of positive integers $a_1, a_2, a_3,...$ satisfies the following properties: (a) Each term leaves a common remainder when divided by $7$ (b) The first term is an integer from $1$ to $6$ (c) The common ratio is an perfect square Let $N$ be the smallest possible value of $\frac{a_{2021}}{a_1}$. Find the remainder when $N$ is divided by $100$. [b]p18.[/b] Suppose $p(x) = x^3 - 11x^2 + 36x - 36$ has roots $r, s,t$. Find %\frac{r^2 + s^2}{t}+\frac{s^2 + t^2}{r}+\frac{t^2 + r^2}{s}%. [b]p19.[/b] Let $a, b \le 2021$ be positive integers. Given that $ab^2$ and $a^2b$ are both perfect squares, let $G = gcd(a, b)$. Find the sum of all possible values of $G$. [b]p20.[/b] Jessica rolls six fair standard six-sided dice at the same time. Given that she rolled at least four $2$'s and exactly one $3$, the probability that all six dice display prime numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. What is $m + n$? [b]p21.[/b] Let $a, b, c$ be numbers such $a + b + c$ is real and the following equations hold: $$a^3 + b^3 + c^3 = 25$$ $$\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}= 1$$ $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{25}{9}$$ The value of $a + b + c$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. Find $m + n$. [b]p22.[/b] Let $\omega$ be a circle and $P$ be a point outside $\omega$. Let line $\ell$ pass through $P$ and intersect $\omega$ at points $A,B$ and with $PA < PB$ and let $m$ be another line passing through $P$ intersecting $\omega$ at points $C,D$ with $PC < PD$. Let X be the intersection of $AD$ and $BC$. Given that $\frac{PC}{CD}=\frac23$, $\frac{PC}{PA}=\frac45$, and $\frac{[ABC]}{[ACD]}=\frac79$,the value of $\frac{[BXD]}{[BXA]}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$: Find $m + n$. [b]p23.[/b] Define the operation $a \circ b =\frac{a^2 + 2ab + a - 12}{b}$. Given that $1 \circ (2 \circ (3 \circ (... 2019 \circ (2020 \circ 2021)))...)$ can be expressed as $-\frac{a}{b}$ for some relatively prime positive integers $a,b$, compute $a + b$. [b]p24.[/b] Find the largest integer $n \le 2021$ for which $5^{n-3} | (n!)^4$ [b]p25.[/b] On the Cartesian plane, a line $\ell$ intersects a parabola with a vertical axis of symmetry at $(0, 5)$ and $(4, 4)$. The focus $F$ of the parabola lies below $\ell$, and the distance from $F$ to $\ell$ is $\frac{16}{\sqrt{17}}$. Let the vertex of the parabola be $(x, y)$. The sum of all possible values of $y$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f : \mathbb N \to \mathbb N$ be a function satisfying \[f(f(m)+f(n))=m+n, \quad \forall m,n \in \mathbb N.\] Prove that $f(x)=x$ for all $x \in \mathbb N$.

[b]p1.[/b] What is the value of the sum $2 + 20 + 202 + 2022$? [b]p2.[/b] Find the smallest integer greater than $10000$ that is divisible by $12$. [b]p3.[/b] Valencia chooses a positive integer factor of $6^{10}$ at random. The probability that it is odd can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m + n$. [b]p4.[/b] How many three digit positive integers are multiples of $4$ but not $8$? [b]p5.[/b] At the Jane Street store, Andy accidentally buys $5$ dollars more worth of shirts than he had planned. Originally, including the tip to the cashier, he planned to spend all of the remaining $90$ dollars on his giftcard. To compensate for his gluttony, Andy instead gives the cashier a smaller, $12.5\%$ tip so that he still spends $90$ dollars total. How much percent tip was Andy originally planning on giving? [b]p6.[/b] Let $A,B,C,D$ be four coplanar points satisfying the conditions $AB = 16$, $AC = BC =10$, and $AD = BD = 17$. What is the minimum possible area of quadrilateral $ADBC$? [b]p7.[/b] How many ways are there to select a set of three distinct points from the vertices of a regular hexagon so that the triangle they form has its smallest angle(s) equal to $30^o$? [b]p8.[/b] Jaeyong rolls five fair $6$-sided die. The probability that the sum of some three rolls is exactly $8$ times the sum of the other two rolls can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p9.[/b] Find the least positive integer n for there exists some positive integer $k > 1$ for which $k$ and $k + 2$ both divide $\underbrace{11...1}_{n\,\,\,1's}$. [b]p10.[/b] For some real constant $k$, line $y = k$ intersects the curve $y = |x^4-1|$ four times: points $A$,$B$,$C$ and $D$, labeled from left to right. If $BC = 2AB = 2CD$, then the value of $k$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p11.[/b] Let a be a positive real number and $P(x) = x^2 -8x+a$ and $Q(x) = x^2 -8x+a+1$ be quadratics with real roots such that the positive difference of the roots of $P(x)$ is exactly one more than the positive difference of the roots of $Q(x)$. The value of a can be written as a common fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [b]p12.[/b] Let $ABCD$ be a trapezoid satisfying $AB \parallel CD$, $AB = 3$, $CD = 4$, with area $35$. Given $AC$ and $BD$ intersect at $E$, and $M$, $N$, $P$, $Q$ are the midpoints of segments $AE$,$BE$,$CE$,$DE$, respectively, the area of the intersection of quadrilaterals $ABPQ$ and $CDMN$ can be expressed as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$. [b]p13.[/b] There are $8$ distinct points $P_1, P_2, ... , P_8$ on a circle. How many ways are there to choose a set of three distinct chords such that every chord has to touch at least one other chord, and if any two chosen chords touch, they must touch at a shared endpoint? [b]p14.[/b] For every positive integer $k$, let $f(k) > 1$ be defined as the smallest positive integer for which $f(k)$ and $f(k)^2$ leave the same remainder when divided by $k$. The minimum possible value of $\frac{1}{x}f(x)$ across all positive integers $x \le 1000$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p15.[/b] In triangle $ABC$, let $I$ be the incenter and $O$ be the circumcenter. If $AO$ bisects $\angle IAC$, $AB + AC = 21$, and $BC = 7$, then the length of segment $AI$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer. Prove that for each factor $m$ of the number $1+2+\cdots+n$ such that $m\ge n$, the set $\{1,2,\ldots,n\}$ can be partitioned into disjoint subsets, the sum of the elements of each being equal to $m$. [b]Edit[/b]:Typographical error fixed.

Find the number of integers $ c$ such that $ \minus{}2007 \leq c \leq 2007$ and there exists an integer $ x$ such that $ x^2 \plus{} c$ is a multiple of $ 2^{2007}$.

Let $A$ and $B$ points in the plane and $C$ a point in the perpendiclar bisector of $AB$. It is constructed a sequence of points $C_1,C_2,\dots, C_n,\dots$ in the following way: $C_1=C$ and for $n\geq1$, if $C_n$ does not belongs to $AB$, then $C_{n+1}$ is the circumcentre of the triangle $\triangle{ABC_n}$. Find all the points $C$ such that the sequence $C_1,C_2,\dots$ is defined for all $n$ and turns eventually periodic. Note: A sequence $C_1,C_2, \dots$ is called eventually periodic if there exist positive integers $k$ and $p$ such that $C_{n+p}=c_n$ for all $n\geq{k}$.

Let $n$ be the smallest positive integer such that there exist integers, $a$, $b$, and $c$, satisfying: $$\frac{n}{2}= a^2 \,\,\, \,\,\, \frac{n}{3}= b^3 \,\,\ , \,\,\ \frac{n}{5}= c^5.$$ Find the number of positive integer factors of $n$.

In base $10$ there exist two-digit natural numbers that can be factorized into two natural factors such that the two digits and the two factors form a sequence of four consecutive integers (for example, $12 = 3 \cdot 4$). Determine all such numbers in all bases.

For positive integer $n$ we define $f(n)$ as sum of all of its positive integer divisors (including $1$ and $n$). Find all positive integers $c$ such that there exists strictly increasing infinite sequence of positive integers $n_1, n_2,n_3,...$ such that for all $i \in \mathbb{N}$ holds $f(n_i)-n_i=c$

Find all natural numbers $n$ such that $1+[\sqrt{2n}]~$ divides $2n$. ( For any real number $x$ , $[x]$ denotes the largest integer not exceeding $x$. )

A number n is called charming when $ 4k^2 + n $ is a prime number for every $ 0 \leq k <n $ integer, find all charming numbers.

Prove that every positive integer can be represented in the form $$3^{m_1}\cdot 2^{n_1}+3^{m_2}\cdot 2^{n_2} + \dots + 3^{m_k}\cdot 2^{n_k}$$ where $m_1 > m_2 > \dots > m_k \geq 0$ and $0 \leq n_1 < n_2 < \dots < n_k$ are integers.

Solve the equation $$x^2+y^2+z^2=686$$ where $x$, $y$ and $z$ are positive integers

Show that there are no integers $x, y, z$, and $t$ such that $$\sqrt[3]{x^5 + y^5 + z^5 + t^5} = 2016.$$

$b,c$ are naturals. $b|c+1$ Prove that exists such natural $x,y,z$ that $x+y=bz,xy=cz$

There is some natural number $n>1$ on the board. Operation is adding to number on the board it maximal non-trivial divisor. Prove, that after some some operations we get number, that is divisible by $3^{2000}$ [i]A. Golovanov[/i]

Clara computed the product of the first $n$ positive integers, and Valerie computed the product of the first $m$ even positive integers, where $m\ge2$. They got the same answer. Prove that one of them had made a mistake.

Find all triplets of natural numbers $(a,b,c)$ that satisfy the equation $abc=a+b+c+1$.