Haydn picks two different integers between $1$ and $100$, inclusive, uniformly at random. The probability that their product is divisible by $4$ can be expressed in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

[b]p1.[/b] Given two irreducible fractions. The denominator of the first fraction is $4$, the denominator of the second fraction is $6$. What can the denominator of the product of these fractions be equal to if the product is represented as an irreducible fraction? [b]p2.[/b] Three horses compete in the race. The player can bet a certain amount of money on each horse. Bets on the first horse are accepted in the ratio $1: 4$. This means that if the first horse wins, then the player gets back the money bet on this horse, and four more times the same amount. Bets on the second horse are accepted in the ratio $1:3$, on the third -$ 1:1$. Money bet on a losing horse is not returned. Is it possible to bet in such a way as to win whatever the outcome of the race? [b]p3.[/b] A quadrilateral is inscribed in a circle, such that the center of the circle, point $O$, is lies inside it. Let $K$, $L$, $M$, $N$ be the midpoints of the sides of the quadrilateral, following in this order. Prove that the bisectors of angles $\angle KOM$ and $\angle LOC$ are perpendicular (Fig.). [img]https://cdn.artofproblemsolving.com/attachments/b/8/ea4380698eba7f4cc2639ce20e3057e0294a7c.png[/img] [b]p4.[/b] Prove that the number$$\underbrace{33...33}_{1999 \,\,\,3s}1$$ is not divisible by $7$. [b]p5.[/b] In triangle $ABC$, the median drawn from vertex $A$ to side $BC$ is four times smaller than side $AB$ and forms an angle of $60^o$ with it. Find the greatest angle of this triangle. [b]p6.[/b] Given a $7\times 8$ rectangle made up of 1x1 cells. Cut it into figures consisting of $1\times 1$ cells, so that each figure consists of no more than $5$ cells and the total length of the cuts is minimal (give an example and prove that this cannot be done with a smaller total length of the cuts). You can only cut along the boundaries of the cells. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Ana and Bety play a game alternating turns. Initially, Ana chooses an odd possitive integer and composite $n$ such that $2^j<n<2^{j+1}$ with $2<j$. In her first turn Bety chooses an odd composite integer $n_1$ such that \[n_1\leq \frac{1^n+2^n+\dots+(n-1)^n}{2(n-1)^{n-1}}.\] Then, on her other turn, Ana chooses a prime number $p_1$ that divides $n_1$. If the prime that Ana chooses is $3$, $5$ or $7$, the Ana wins; otherwise Bety chooses an odd composite positive integer $n_2$ such that \[n_2\leq \frac{1^{p_1}+2^{p_1}+\dots+(p_1-1)^{p_1}}{2(p_1-1)^{p_1-1}}.\] After that, on her turn, Ana chooses a prime $p_2$ that divides $n_2,$, if $p_2$ is $3$, $5$, or $7$, Ana wins, otherwise the process repeats. Also, Ana wins if at any time Bety cannot choose an odd composite positive integer in the corresponding range. Bety wins if she manages to play at least $j-1$ turns. Find which of the two players has a winning strategy.

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

Prove that the product of numbers $1, 2, ..., n$ ($n \ge 2$) is divisible by their sum if and only if the number $n + 1$ is not prime.

For all integral values of parameter $t$, find all integral solutions $(x,y)$ of the equation $$ y^2 = x^4-22x^3+43x^2+858x+t^2+10452(t+39)$$ .

Let $f_n = 2^{2^n}+ 1$, $n = 1,2,3,...$, be the Fermat’s numbers. Find the least real number $C$ such that $$\frac{1}{f_1}+\frac{2}{f_2}+\frac{2^2}{f_3}+...+\frac{2^{n-1}}{f_n} <C$$ for all positive integers $n$

Are there integers $m$ and $n$ such that $m^2 + 1954 = n^2$?

For an integer $n\geq 1$ let $a(n)$ denote the total number of carries which arise when adding $2017$ and $n\cdot 2017$. The first few values are given by $a(1)=1$, $a(2)=1$, $a(3)=0$, which can be seen from the following: \begin{align*} 001 &&001 && 000 \\ 2017 &&4034 &&6051 \\ +2017 &&+2017 &&+2017\\ =4034 &&=6051 &&=8068\\ \end{align*} Prove that $$a(1)+a(2)+...+a(10^{2017}-1)=10\cdot\frac{10^{2017}-1}{9}.$$

[b]p1.[/b] Compute the sum of sharp angles at all five nodes of the star below. ( [url=http://www.math.msu.edu/~mshapiro/NewOlympiad/Olymp2009/10_12_2009.pdf]figure missing[/url] ) [b]p2.[/b] Arrange the integers from $1$ to $15$ in a row so that the sum of any two consecutive numbers is a perfect square. In how many ways this can be done? [b]p3.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 -q^2$ is divisible by $ 24$. [b]p4.[/b] A city in a country is called Large Northern if comparing to any other city of the country it is either larger or farther to the North (or both). Similarly, a city is called Small Southern. We know that in the country all cities are Large Northern city. Show that all the cities in this country are simultaneously Small Southern. [b]p5.[/b] You have four tall and thin glasses of cylindrical form. Place on the flat table these four glasses in such a way that all distances between any pair of centers of the glasses' bottoms are equal. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all the positive perfect cubes that are not divisible by $10$ such that the number obtained by erasing the last three digits is also a perfect cube.

Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number $$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$ is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$. [i]Proposed by Bayarmagnai Gombodorj, Mongolia[/i]

[b]p1.[/b] There are $4$ mirrors facing the inside of a $5\times 7$ rectangle as shown in the figure. A ray of light comes into the inside of a rectangle through $A$ with an angle of $45^o$. When it hits the sides of the rectangle, it bounces off at the same angle, as shown in the diagram. How many times will the ray of light bounce before it reaches any one of the corners $A$, $B$, $C$, $D$? A bounce is a time when the ray hit a mirror and reflects off it. [img]https://cdn.artofproblemsolving.com/attachments/1/e/d6ea83941cdb4b2dab187d09a0c45782af1691.png[/img] [b]p2.[/b] Jerry cuts $4$ unit squares out from the corners of a $45\times 45$ square and folds it into a $43\times 43\times 1$ tray. He then divides the bottom of the tray into a $43\times 43$ grid and drops a unit cube, which lands in precisely one of the squares on the grid with uniform probability. Suppose that the average number of sides of the cube that are in contact with the tray is given by $\frac{m}{n}$ where $m, n$ are positive integers that are relatively prime. Find $m + n$. [b]p3.[/b] Compute $2021^4 - 4 \cdot 2023^4 + 6 \cdot 2025^4 - 4 \cdot 2027^4 + 2029^4$. [b]p4.[/b] Find the number of distinct subsets $S \subseteq \{1, 2,..., 20\}$, such that the sum of elements in $S$ leaves a remainder of $10$ when divided by $32$. [b]p5.[/b] Some $k$ consecutive integers have the sum $45$. What is the maximum value of $k$? [b]p6.[/b] Jerry picks $4$ distinct diagonals from a regular nonagon (a regular polygon with $9$-sides). A diagonal is a segment connecting two vertices of the nonagon that is not a side. Let the probability that no two of these diagonals are parallel be $\frac{m}{n}$ where $m, n$ are positive integers that are relatively prime. Find $m + n$. [b]p7.[/b] The Olympic logo is made of $5$ circles of radius $1$, as shown in the figure [img]https://cdn.artofproblemsolving.com/attachments/1/7/9dafe6b72aa8471234afbaf4c51e3e97c49ee5.png[/img] Suppose that the total area covered by these $5$ circles is $a+b\pi$ where $a, b$ are rational numbers. Find $10a + 20b$. [b]p8.[/b] Let $P(x)$ be an integer polynomial (polynomial with integer coefficients) with $P(-5) = 3$ and $P(5) = 23$. Find the minimum possible value of $|P(-2) + P(2)|$. [b]p9. [/b]There exists a unique tuple of rational numbers $(a, b, c)$ such that the equation $$a \log 10 + b \log 12 + c \log 90 = \log 2025.$$ What is the value of $a + b + c$? [b]p10.[/b] Each grid of a board $7\times 7$ is filled with a natural number smaller than $7$ such that the number in the grid at the $i$th row and $j$th column is congruent to $i + j$ modulo $7$. Now, we can choose any two different columns or two different rows, and swap them. How many different boards can we obtain from a finite number of swaps? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

Let $x$ be an integer of $n$ digits, all equal to $ 1$. Show that if $x$ is prime, then $n$ is also prime.

Let $n,k$ be positive integers such that n is not divisible by 3 and $k \geq n$. Prove that there exists a positive integer $m$ which is divisible by $n$ and the sum of its digits in decimal representation is $k$.

Prove that there do not exist polynomials $ P$ and $ Q$ such that \[ \pi(x)\equal{}\frac{P(x)}{Q(x)}\] for all $ x\in\mathbb{N}$.

For each natural number $x$, let $S(x)$ be the sum of its digits. Find the smallest natural number $n$ such that $9S(n) = 16S(2n)$.

Let $p_1, p_2, \ldots, p_{25}$ are primes which don’t exceed 2004. Find the largest integer $T$ such that every positive integer $\leq T$ can be expressed as sums of distinct divisors of $(p_1\cdot p_2 \cdot \ldots \cdot p_{25})^{2004}.$

Consider the sequence $a_1 = 3$ and $a_{n + 1} =\frac{3a_n^2+1}{2}-a_n$ for $n = 1 ,2 ,...$. Prove that if $n$ is a power of $3$ then $n$ divides $a_n$ .

Find the number of even integers n such that $0 \le n \le 100$ and $5 | n^2 \cdot 2^{{2n}^2}+ 1$.

Find all integer solutions to the equation $y^k = x^2 + x$, where $k$ is a natural number greater than $1$.

[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats, and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were there in each bus? [b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $50$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a 6 cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $150$ coins? [b]p3.[/b] Pinocchio multiplied two $2$ digit numbers. But witch Masha erased some of the digits. The erased digits are the ones marked with a $*$. Could you help Pinocchio to restore all the erased digits? $\begin{tabular}{ccccc} & & & 9 & 5 \\ x & & & * & * \\ \hline & & & * & * \\ + & 1 & * & * & \\ \hline & * & * & * & * \\ \end{tabular}$ Find all solutions. [b]p4.[/b] There are $50$ senators and $435$ members of House of Representatives. On Friday all of them voted a very important issue. Each senator and each representative was required to vote either "yes" or "no". The announced results showed that the number of "yes" votes was greater than the number of "no" votes by $24$. Prove that there was an error in counting the votes. [b]p5.[/b] Was there a year in the last millennium (from $1000$ to $2000$) such that the sum of the digits of that year is equal to the product of the digits? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f \in Z[X]$, $f = X^2 + aX + b$, be a quadratic polynomial. Prove that $f$ has integer zeros if and only if for each positive integer $n$ there is an integer $u_n$ such that $n | f(u_n)$.

Bernado and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernado. Whenever Bernado receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernado. The winner is the last person who produces a number less than 1000. Let $N$ be the smallest initial number that results in a win for Bernado. What is the sum of the digits of $N$? $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$