Compute the number of positive integers $n < 2012$ that share exactly two positive factors with 2012. [i]Proposed by Aaron Lin[/i]

Compute the remainder when $98!$ is divided by $101$.

Among all polynomials $P(x)$ with integer coefficients for which $P(-10) = 145$ and $P(9) = 164$, compute the smallest possible value of $|P(0)|.$

Prove that there is a constant $c>0$ and infinitely many positive integers $n$ with the following property: there are infinitely many positive integers that cannot be expressed as the sum of fewer than $cn\log(n)$ pairwise coprime $n$th powers. [i]Canada[/i]

[u]General Round[/u] [b]p1.[/b] Alice can bake a pie in $5$ minutes. Bob can bake a pie in $6$ minutes. Compute how many more pies Alice can bake than Bob in $60$ minutes. [b]p2.[/b] Ben likes long bike rides. On one ride, he goes biking for six hours. For the first hour, he bikes at a speed of $15$ miles per hour. For the next two hours, he bikes at a speed of $12$ miles per hour. He remembers biking $90$ miles over the six hours. Compute the average speed, in miles per hour, Ben biked during the last three hours of his trip. [b]p3.[/b] Compute the perimeter of a square with area $36$. [b]p4.[/b] Two ants are standing side-by-side. One ant, which is $4$ inches tall, casts a shadow that is $10$ inches long. The other ant is $6$ inches tall. Compute, in inches, the length of the shadow that the taller ant casts. [b]p5.[/b] Compute the number of distinct line segments that can be drawn inside a square such that the endpoints of the segment are on the square and the segment divides the square into two congruent triangles. [b]p6.[/b] Emily has a cylindrical water bottle that can hold $1000\pi$ cubic centimeters of water. Right now, the bottle is holding $100\pi$ cubic centimeters of water, and the height of the water is $1$ centimeter. Compute the radius of the water bottle. [b]p7.[/b] Given that $x$ and $y$ are nonnegative integers, compute the number of pairs $(x, y)$ such that $5x + y = 20$. [b]p8.[/b] A sequence an is recursively defined where $a_n = 3(a_{n-1}-1000)$ for $n > 0$. Compute the smallest integer $x$ such that when $a_0 = x$, $a_n > a_0$ for all $n > 0$. [b]p9.[/b] Compute the probability that two random integers, independently chosen and both taking on an integer value between $1$ and $10$ with equal probability, have a prime product. [b]p10.[/b] If $x$ and $y$ are nonnegative integers, both less than or equal to $2$, then we say that $(x, y)$ is a friendly point. Compute the number of unordered triples of friendly points that form triangles with positive area. [b]p11.[/b] Cindy is thinking of a number which is $4$ less than the square of a positive integer. The number has the property that it has two $2$-digit prime factors. What is the smallest possible value of Cindy's number? [b]p12.[/b] Winona can purchase a pencil and two pens for $250$ cents, or two pencils and three pens for $425$ cents. If the cost of a pencil and the cost of a pen does not change, compute the cost in cents of five pencils and six pens. [b]p13.[/b] Colin has an app on his phone that generates a random integer betwen $1$ and $10$. He generates $10$ random numbers and computes the sum. Compute the number of distinct possible sums Colin can end up with. [b]p14.[/b] A circle is inscribed in a unit square, and a diagonal of the square is drawn. Find the total length of the segments of the diagonal not contained within the circle. [b]p15.[/b] A class of six students has to split into two indistinguishable teams of three people. Compute the number of distinct team arrangements that can result. [b]p16.[/b] A unit square is subdivided into a grid composed of $9$ squares each with sidelength $\frac13$ . A circle is drawn through the centers of the $4$ squares in the outermost corners of the grid. Compute the area of this circle. [b]p17.[/b] There exists exactly one positive value of $k$ such that the line $y = kx$ intersects the parabola $y = x^2 + x + 4$ at exactly one point. Compute the intersection point. [b]p18.[/b] Given an integer $x$, let $f(x)$ be the sum of the digits of $x$. Compute the number of positive integers less than $1000$ where $f(x) = 2$. [b]p19.[/b] Let $ABCD$ be a convex quadrilateral with $BA = BC$ and $DA = DC$. Let $E$ and $F$ be the midpoints of $BC$ and $CD$ respectively, and let $BF$ and $DE$ intersect at $G$. If the area of $CEGF$ is $50$, what is the area of $ABGD$? [b]p20.[/b] Compute all real solutions to $16^x + 4^{x+1} - 96 = 0$. [b]p21.[/b] At an M&M factory, two types of M&Ms are produced, red and blue. The M&Ms are transported individually on a conveyor belt. Anna is watching the conveyor belt, and has determined that four out of every five red M&Ms are followed by a blue one, while one out of every six blue M&Ms is followed by a red one. What proportion of the M&Ms are red? [b]p22.[/b] $ABCDEFGH$ is an equiangular octagon with side lengths $2$, $4\sqrt2$, $1$, $3\sqrt2$, $2$, $3\sqrt2$, $3$, and $2\sqrt2$,in that order. Compute the area of the octagon. [b]p23.[/b] The cubic $f(x) = x^3 +bx^2 +cx+d$ satisfies $f(1) = 3$, $f(2) = 6$, and $f(4) = 12$. Compute $f(3)$. [b]p24.[/b] Given a unit square, two points are chosen uniformly at random within the square. Compute the probability that the line segment connecting those two points touches both diagonals of the square. [b]p25.[/b] Compute the remainder when: $$5\underbrace{666...6666}_{2016 \,\, sixes}5$$ is divided by $17$. [u]General Tiebreaker [/u] [b]Tie 1.[/b] Trapezoid $ABCD$ has $AB$ parallel to $CD$, with $\angle ADC = 90^o$. Given that $AD = 5$, $BC = 13$ and $DC = 18$, compute the area of the trapezoid. [b]Tie 2.[/b] The cubic $f(x) = x^3- 7x - 6$ has three distinct roots, $a$, $b$, and $c$. Compute $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ . [b]Tie 3.[/b] Ben flips a fair coin repeatedly. Given that Ben's first coin flip is heads, compute the probability Ben flips two heads in a row before Ben flips two tails in a row. PS. You should use hide for answers.

Defined as $N_0$ as the set of all non-negative integers. Set $S \subset N_0$ with not so many elements is called beautiful if for every $a, b \in S$ with $a \ge b$ ($a$ and $b$ do not have to be different), exactly one of $a + b$ or $a - b$ is in $S$. Set $T \subset N_0$ with not so many elements is called charming if the largest number $k$ such that up to 3$^k | a$ is the same for each element $a \in T$. Prove that each beautiful set must be charming.

For each positive integer $n$ larger than $1$ with prime factorization $p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$, its [i]signature[/i] is defined as the sum $\alpha_1+\alpha_2+\cdots+\alpha_k$. Does there exist $2020$ consecutive positive integers such that among them, there are exactly $1812$ integers whose signatures are strictly smaller than $11$?

Let $ a$ and $b$ be positive integers. Prove that there exist positive integers $ x $ and $ y $ such that \[ \binom{x+y}{2} = ax + by . \]

Show that if $x, y, z$ are positive integers, then $(xy+1)(yz+1)(zx+1)$ is a perfect square if and only if $xy+1$, $yz+1$, $zx+1$ are all perfect squares.

We say that a positive integer is [i]decomposed [/i] if it is prime and also If a line is drawn separating it into two numbers, those two numbers are never composite. For example 1997 is [i]decomposed [/i] since it is prime, it is divided into: $1$, $997$; $19$, $97$; $199$, $7$ and none of those numbers are compound. How many [i]decomposed [/i] numbers are there between $2000$ and $3000$?

Let $n$ be a nonnegative integer less than $2023$ such that $2n^2 + 3n$ is a perfect square. What is the sum of all possible $n$? [i]Proposed by Giacomo Rizzo[/i]

Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.

Find all natural numbers that can be expressed in a unique way as a sum of five or less perfect squares.

Say that two sets of positive integers $S, T$ are $\emph{k-equivalent}$ if the sum of the $i$th powers of elements of $S$ equals the sum of the $i$th powers of elements of $T$, for each $i= 1, 2, \ldots, k$. Given $k$, prove that there are infinitely many numbers $N$ such that $\{1,2,\ldots,N^{k+1}\}$ can be divided into $N$ subsets, all of which are $k$-equivalent to each other.

Find the largest integer $k$ ($k \ge 2$), for which there exists an integer $n$ ($n \ge k$) such that from any collection of $n$ consecutive positive integers one can always choose $k$ numbers, which verify the following conditions: 1. each chosen number is not divisible by $6$, by $7$, nor by $8$; 2. the positive difference of any two distinct chosen numbers is not divisible by at least one of the numbers $6$, $7$, and $8$.

$\tfrac11+\tfrac13+\tfrac15=\tfrac12+\tfrac14+\tfrac16+\tfrac m n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Determine all positive integers $n$ for which there exists an integer $m$ such that $2^{n}-1$ divides $m^{2}+9$.

Let $x$ be the positive integer satisfying $5^2 +28^2 +39^2 = 24^2 +35^2 + x^2$. Find $x$.

Find all $ x, y, z\in\mathbb{Z}^+ $ such that \[ (x-y)(y-z)(z-x)=x+y+z \]

Let $n$ be a positive integer. Prove that there exist an infinity of multiples of $n$ which do not contain the digit “$9$” in their decimal representation

[url=https://artofproblemsolving.com/community/c675693][b]Macedonia JBMO TST 2017[/b][/url] [url=http://artofproblemsolving.com/community/c6h1663908p10569198][b]Problem 1[/b][/url]. Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$. [url=http://artofproblemsolving.com/community/c6h1663916p10569261][b]Problem 2[/b][/url]. In the triangle $ABC$, the medians $AA_1$, $BB_1$, and $CC_1$ are concurrent at a point $T$ such that $BA_1=TA_1$. The points $C_2$ and $B_2$ are chosen on the extensions of $CC_1$ and $BB_2$, respectively, such that $$C_1C_2 = \frac{CC_1}{3} \quad \text{and} \quad B_1B_2 = \frac{BB_1}{3}.$$ Show that $TB_2AC_2$ is a rectangle. [url=http://artofproblemsolving.com/community/c6h1663918p10569305][b]Problem 3[/b][/url]. Let $x,y,z$ be positive reals such that $xyz=1$. Show that $$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$ When does equality happen? [url=http://artofproblemsolving.com/community/c6h1663920p10569326][b]Problem 4[/b][/url]. In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $$BM + CM > AY.$$ [url=http://artofproblemsolving.com/community/c6h1663922p10569370][b]Problem 5[/b][/url]. Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.

Prove that$$p (n)= 2+ \left (p (1) + \cdots + p\left ( \left [\frac {n}{2} \right ] + \chi_1 (n)\right ) + \left (p'_2(n) + \cdots + p' _{ \left [\frac {n}{2} \right ] - 1}(n)\right )\right )$$for every $n \in \mathbb {N}$ with $n>2$ where $\chi $ denotes the principal character Dirichlet modulo 2, i.e.$$ \chi _1 (n) = \begin{cases} 1 & \text{if } (n,2)=1 \\ 0 &\text{if } (n,2)>1 \end{cases} $$with $p (n) $ we denote number of possible partitions of $n $ and $p' _m(n) $ we denote the number of partitions of $n$ in exactly $m$ sumands.

There are no integers $ a,b,c $ that satisfy $ \left( a+b\sqrt{-3}\right)^{17}=c+\sqrt{-3} . $ [i]Dorin Andrica, Mihai Piticari[/i]

Let $a,b,c\in\mathbb{Z^{+}}$ such that $$(a^2-1, b^2-1, c^2-1)=1$$ Prove that $$(ab+c, bc+a, ca+b)=(a,b,c)$$ (As usual, $(x,y,z)$ means the greatest common divisor of numbers $x,y,z$) [I]Proposed by A. Golovanov[/i]

Let $n$ be a positive integer such that the number \[\frac{1^k + 2^k + \dots + n^k}{n}\] is an integer for any $k \in \{1, 2, \dots, 99\}$. Prove that $n$ has no divisors between 2 and 100, inclusive.