Let $n$ be a positive integer and $\{A,B,C\}$ a partition of $\{1,2,\ldots,3n\}$ such that $|A|=|B|=|C|=n$. Prove that there exist $x \in A$, $y \in B$, $z \in C$ such that one of $x,y,z$ is the sum of the other two.

Let $a,b,n$ be positive integers. Prove that $n!$ divides \[b^{n-1}a(a+b)(a+2b)...(a+(n-1)b)\]

Is there a number $s$ in the set $\{\pi,2\pi,3\pi,...,\} $ such that the first three digits after the decimal point of $s$ are $.001$? Fully justify your answer.

Find the number of ordered quadruples of positive integers $(a,b,c, d)$ such that $ab + cd = 10$.

$a)$ Prove that for all positive integers $n \geq 3$ holds: $$\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n-1}=2^n-2$$ where $\binom{n}{k}$ , with integer $k$ such that $n \geq k \geq 0$, is binomial coefficent $b)$ Let $n \geq 3$ be an odd positive integer. Prove that set $A=\left\{ \binom{n}{1},\binom{n}{2},...,\binom{n}{\frac{n-1}{2}} \right\}$ has odd number of odd numbers

Find all integers $n$ for which $n(n + 2010)$ is a perfect square.

Determine all pairs of integers $(a, b)$ that solve the equation $a^3 + b^3 + 3ab = 1$.

Show that if $(a_n)$ is an infinite sequence of distinct positive integers, neither of which contains digit $0$ in the decimal expansion, then $$\sum_{n=1}^{\infty} \frac{1}{a_n}< 29.$$

Determine the number of pairs of positive integers $x,y$ such that $x\le y$, $\gcd (x,y)=5!$ and $\text{lcm}(x,y)=50!$.

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Matt has a twenty dollar bill and buys two items worth $\$7:99$ each. How much change does he receive, in dollars? [b]p2.[/b] The sum of two distinct numbers is equal to the positive difference of the two numbers. What is the product of the two numbers? [b]p3.[/b] Evaluate $$\frac{1 + 2 + 3 + 4 + 5 + 6 + 7}{8 + 9 + 10 + 11 + 12 + 13 + 14}.$$ [b]p4.[/b] A sphere with radius $r$ has volume $2\pi$. Find the volume of a sphere with diameter $r$. [b]p5.[/b] Yannick ran $100$ meters in $14.22$ seconds. Compute his average speed in meters per second, rounded to the nearest integer. [b]p6.[/b] The mean of the numbers $2, 0, 1, 5,$ and $x$ is an integer. Find the smallest possible positive integer value for $x$. [b]p7.[/b] Let $f(x) =\sqrt{2^2 - x^2}$. Find the value of $f(f(f(f(f(-1)))))$. [b]p8.[/b] Find the smallest positive integer $n$ such that $20$ divides $15n$ and $15$ divides $20n$. [b]p9.[/b] A circle is inscribed in equilateral triangle $ABC$. Let $M$ be the point where the circle touches side $AB$ and let $N$ be the second intersection of segment $CM$ and the circle. Compute the ratio $\frac{MN}{CN}$ . [b]p10.[/b] Four boys and four girls line up in a random order. What is the probability that both the first and last person in line is a girl? [b]p11.[/b] Let $k$ be a positive integer. After making $k$ consecutive shots successfully, Andy's overall shooting accuracy increased from $65\%$ to $70\%$. Determine the minimum possible value of $k$. [b]p12.[/b] In square $ABCD$, $M$ is the midpoint of side $CD$. Points $N$ and $P$ are on segments $BC$ and $AB$ respectively such that $ \angle AMN = \angle MNP = 90^o$. Compute the ratio $\frac{AP}{PB}$ . [b]p13.[/b] Meena writes the numbers $1, 2, 3$, and $4$ in some order on a blackboard, such that she cannot swap two numbers and obtain the sequence $1$, $2$, $3$, $4$. How many sequences could she have written? [b]p14.[/b] Find the smallest positive integer $N$ such that $2N$ is a perfect square and $3N$ is a perfect cube. [b]p15.[/b] A polyhedron has $60$ vertices, $150$ edges, and $92$ faces. If all of the faces are either regular pentagons or equilateral triangles, how many of the $92$ faces are pentagons? [b]p16.[/b] All positive integers relatively prime to $2015$ are written in increasing order. Let the twentieth number be $p$. The value of $\frac{2015}{p}-1$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Compute $a + b$. [b]p17.[/b] Five red lines and three blue lines are drawn on a plane. Given that $x$ pairs of lines of the same color intersect and $y$ pairs of lines of different colors intersect, find the maximum possible value of $y - x$. [b]p18.[/b] In triangle $ABC$, where $AC > AB$, $M$ is the midpoint of $BC$ and $D$ is on segment $AC$ such that $DM$ is perpendicular to $BC$. Given that the areas of $MAD$ and $MBD$ are $5$ and $6$, respectively, compute the area of triangle $ABC$. [b]p19.[/b] For how many ordered pairs $(x, y)$ of integers satisfying $0 \le x, y \le 10$ is $(x + y)^2 + (xy - 1)^2$ a prime number? [b]p20.[/b] A solitaire game is played with $8$ red, $9$ green, and $10$ blue cards. Totoro plays each of the cards exactly once in some order, one at a time. When he plays a card of color $c$, he gains a number of points equal to the number of cards that are not of color $c$ in his hand. Find the maximum number of points that he can obtain by the end of the game. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Does there exist positive integers $n_1, n_2, \dots, n_{2022}$ such that the number $$ \left( n_1^{2020} + n_2^{2019} \right)\left( n_2^{2020} + n_3^{2019} \right) \cdots \left( n_{2021}^{2020} + n_{2022}^{2019} \right)\left( n_{2022}^{2020} + n_1^{2019} \right) $$ is a power of $11$?

Let $m$, $n$, and $x$ be positive integers. Prove that \[ \sum_{i = 1}^n \min\left(\left\lfloor \frac{x}{i} \right\rfloor, m \right) = \sum_{i = 1}^m \min\left(\left\lfloor \frac{x}{i} \right\rfloor, n \right). \] [i]Proposed by Yang Liu[/i]

Consider $64$ distinct natural numbers, at most equal to $2012$. Show that it is possible to choose four of them, denoted as $a,b,c,d$ such that $ a+b-c-d$ to be a multiple of $2013$

[b]p1.[/b] Find the smallest positive multiple of $9$ whose digits are all even. [b]p2.[/b] Anika writes down a positive real number $x$ in decimal form. When Nat erases everything to the left of the decimal point, the remaining value is one-fifth of x. Find the sum of all possible values of $x$. [b]p3.[/b] A star-like shape is formed by joining up the midpoints and vertices of a unit square, as shown in the diagram below. Compute the area of this shape. [img]https://cdn.artofproblemsolving.com/attachments/4/8/923b1bf26f6e9872b596e8c81ad1872137f362.png[/img] [b]p4.[/b] Benny and Daria are running a $200$ meter foot race, each at a different constant speed. When Daria finishes the race, she is $14$ meters ahead of Benny. The next time they race, Daria starts 14 meters behind Benny, who starts at the starting line. Both runners run at the same constant speed as in the first race. When Daria reaches the finish line, compute, in centimeters, how far she is ahead of Benny. [b]p5.[/b] In one semester, Ronald takes ten biology quizzes, earning a distinct integer score from $91$ to $100$ on each quiz. He notices that after the first three quizzes, the average of his three most recent scores always increased. Compute the number of ways Ronald could have earned the ten quiz scores. [b]p6.[/b] Ant and Ben are playing a game with stones. They begin with $Z$ stones on the ground. Ant and Ben take turns removing a prime number of stones. Ant moves first. The player who is first unable to make a valid move loses. Find the sum of all positive integers $Z \le 30$ such that Ben can guarantee a win with perfect play. [b]p7.[/b] Let $P$ be a point in a regular octagon as shown in the diagram below. The areas of three triangles are shown. Find the area of the octagon. [img]https://cdn.artofproblemsolving.com/attachments/0/9/6fde77eeafd04614046292175e4b1411158e85.png[/img] [b]p8.[/b] Find the number of ordered triples $(a, b, c)$ of nonnegative integers with $a \le b \le c$ for which $5a + 4b + 6c = 1200$. [b]p9.[/b] Define $$f(x) = \begin{cases} 2x \,\,\,\, ,\,\,\,\, 0 \le x < \frac12 \\ 2 - 2x \,\,\,\, , \,\,\,\, \frac12 \le x \le 1 \end{cases}$$ Michael picks a real number $0 \le x \le 1$. Michael applies $f$ repeatedly to $x$ until he reaches $x$ again. Find the number of real numbers $x$ for which Michael applies $f$ exactly $12$ times. [b]p10.[/b] In $\vartriangle ABC$, let point $H$ be the intersection of its altitudes and let $M$ be the midpoint of side $BC$. Given that $BC = 4$, $MA = 3$, and $\angle HMA = 60^o$, find the circumradius of $\vartriangle ABC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A [i]special number[/i] is a positive integer $n$ for which there exists positive integers $a$, $b$, $c$, and $d$ with \[ n = \frac {a^3 + 2b^3} {c^3 + 2d^3}. \] Prove that i) there are infinitely many special numbers; ii) $2014$ is not a special number. [i]Romania[/i]

A set of $k$ integers is said to be a [i]complete residue system modulo[/i] $k$ if no two of its elements are congruent modulo $k$. Find all positive integers $m$ so that there are infinitely many positive integers $n$ wherein $\{ 1^n,2^n, \dots , m^n \}$ is a complete residue system modulo $m$.

Determine all pairs of natural numbers $(x; n)$ that satisfy the equation \[x^{3}+2x+1 = 2^{n}.\]

Let $k$ be a positive integer. Show that exists positive integer $n$, such that sets $A = \{ 1^2, 2^2, 3^2, ...\}$ and $B = \{1^2 + n, 2^2 + n, 3^2 + n, ... \}$ have exactly $k$ common elements.

[b](1)[/b] Find the number of positive integer solutions $(m,n,r)$ of the indeterminate equation $mn+nr+mr=2(m+n+r)$. [b](2)[/b] Given an integer $k (k>1)$, prove that indeterminate equation $mn+nr+mr=k(m+n+r)$ has at least $3k+1$ positive integer solutions $(m,n,r)$.

Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that \[n \mid a^{f(n)}-1.\] Prove that $S$ has density $0$; that is, prove that $\lim_{n\rightarrow \infty} \frac{|S\cap \{1,...,n\}|}{n}=0$.

Let a, b, c be natural numbers , so that c> b> a> 0. Show that, among any 2c consecutive natural numbers, there are three distinct numbers x, y, z so abc divides xyz.

Find prime numbers $p$, $q$, $r$ and $s$, pairwise distinct, such that their sum is prime number and numbers $p^2+qr$ and $p^2+qs$ are perfect squares

Find all pairs of prime numbers $(p,q)$, such that $\frac{(2p^2-1)^q+1}{p+q}$ and $\frac{(2q^2-1)^p+1}{p+q}$ are both integers.

[u]Round 9[/u] [b]p25.[/b] Let $a$, $b$, and $c$ be positive numbers with $a +b +c = 4$. If $a,b,c \le 2$ and $$M =\frac{a^3 +5a}{4a^2 +2}+\frac{b^3 +5b}{4b^2 +2}+\frac{c^3 +5c}{4c^2 +2},$$ then find the maximum possible value of $\lfloor 100M \rfloor$. [b]p26.[/b] In $\vartriangle ABC$, $AB = 15$, $AC = 16$, and $BC = 17$. Points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, such that $CE = 1$ and $BF = 3$. A point $D$ is chosen on side $BC$, and let the circumcircles of $\vartriangle BFD$ and $\vartriangle CED$ intersect at point $P \ne D$. Given that $\angle PEF = 30^o$, the length of segment $PF$ can be expressed as $\frac{m}{n}$ . Find $m+n$. [b]p27.[/b] Arnold and Barnold are playing a game with a pile of sticks with Arnold starting first. Each turn, a player can either remove $7$ sticks or $13$ sticks. If there are fewer than $7$ sticks at the start of a player’s turn, then they lose. Both players play optimally. Find the largest number of sticks under $200$ where Barnold has a winning strategy [u]Round 10[/u] [b]p28.[/b] Let $a$, $b$, and $c$ be positive real numbers such that $\log_2(a)-2 = \log_3(b) =\log_5(c)$ and $a +b = c$. What is $a +b +c$? [b]p29.[/b] Two points, $P(x, y)$ and $Q(-x, y)$ are selected on parabola $y = x^2$ such that $x > 0$ and the triangle formed by points $P$, $Q$, and the origin has equal area and perimeter. Find $y$. [b]p30.[/b] $5$ families are attending a wedding. $2$ families consist of $4$ people, $2$ families consist of $3$ people, and $1$ family consists of $2$ people. A very long row of $25$ chairs is set up for the families to sit in. Given that all members of the same family sit next to each other, let the number of ways all the people can sit in the chairs such that no two members of different families sit next to each other be $n$. Find the number of factors of $n$. [u]Round 11[/u] [b]p31.[/b] Let polynomial $P(x) = x^3 +ax^2 +bx +c$ have (not neccessarily real) roots $r_1$, $r_2$, and $r_3$. If $2ab = a^3 -20 = 6c -21$, then the value of $|r^3_1+r^3_2+r^3_3|$ can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$. [b]p32.[/b] In acute $\vartriangle ABC$, let $H$, $I$ , $O$, and $G$ be the orthocenter, incenter, circumcenter, and centroid of $\vartriangle ABC$, respectively. Suppose that there exists a circle $\omega$ passing through $B$, $I$ , $H$, and $C$, the circumradius of $\vartriangle ABC$ is $312$, and $OG = 80$. Let $H'$, distinct from $H$, be the point on $\omega$ such that $\overline{HH'}$ is a diameter of $\omega$. Given that lines $H'O$ and $BC$ meet at a point $P$, find the length $OP$. [b]p33.[/b] Find the number of ordered quadruples $(x, y, z,w)$ such that $0 \le x, y, z,w \le 1000$ are integers and $$x!+ y! =2^z \cdot w!$$ holds (Note: $0! = 1$). [u]Round 12[/u] [b]p34.[/b] Let $Z$ be the product of all the answers from the teams for this question. Estimate the number of digits of $Z$. If your estimate is $E$ and the answer is $A$, your score for this problem will be $$\max \left( 0, \lceil 15- |A-E| \rceil \right).$$ Your answer must be a positive integer. [b]p35.[/b] Let $N$ be number of ordered pairs of positive integers $(x, y)$ such that $3x^2 -y^2 = 2$ and $x < 2^{75}$. Estimate $N$. If your estimate is $E$ and the answer is $A$, your score for this problem will be $$\max \left( 0, \lceil 15- 2|A-E| \rceil \right).$$ [b]p36.[/b] $30$ points are located on a circle. How many ways are there to draw any number of line segments between the points such that none of the line segments overlap and none of the points are on more than one line segment? (It is possible to draw no line segments). If your estimate is $E$ and the answer is $A$, your score for this problem will be $$\max \left( 0, \left \lceil 15- \ln \frac{A}{E} \right \rceil \right).$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166472p28814057]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166476p28814111]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose that $m$ and $k$ are non-negative integers, and $p = 2^{2^m}+1$ is a prime number. Prove that [b](a)[/b] $2^{2^{m+1}p^k} \equiv 1$ $(\text{mod } p^{k+1})$; [b](b)[/b] $2^{m+1}p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1$ $(\text{mod } p^{k+1})$.