In a mathematics test number of participants is $N < 40$. The passmark is fixed at $65$. The test results are the following: The average of all participants is $66$, that of the promoted $71$ and that of the repeaters $56$. However, due to an error in the wording of a question, all scores are increased by $5$. At this point the average of the promoted participants becomes $75$ and that of the non-promoted $59$. (a) Find all possible values ​​of $N$. (b) Find all possible values ​​of $N$ in the case where, after the increase, the average of the promoted had become $79$ and that of non-promoted $47$.

Let $k,x,y$ be postive integers. The quotients of $k$ divided by $x^2, y^2$ are $n,n+148$ respectively.($k$ is divisible by $x^2$ and $y^2$) (a) If $\gcd(x,y)=1$, then find $k$. (b) If $\gcd(x,y)=4$, then find $k$.

Show that if $15$ numbers lie between $2$ and $1992$ and each pair is coprime, then at least one is prime.

Determine all positive integers $n$ such that the number $n(n+2)(n+4)$ has at most $15$ positive divisors.

Isabel wants to partition the set $\mathbb{N}$ of the positive integers into $n$ disjoint sets $A_{1}, A_{2}, \ldots, A_{n}$. Suppose that for each $i$ with $1\leq i\leq n$, given any positive integers $r, s\in A_{i}$ with $r\neq s$, we have $r+s\in A_{i}$. If $|A_{j}|=1$ for some $j$, find the greatest positive integer that may belong to $A_{j}$.

Let $M$ be a subset of $\{1,2,..., 1998\}$ with $1000$ elements. Prove that it is always possible to find two elements $a$ and $b$ in $M$, not necessarily distinct, such that $a + b$ is a power of $2$.

Pedro writes all the numbers with four different digits that can be made with digits $a, b, c, d$, that meet the following conditions: $$ a\ne 0 \, , \, b=a+2 \, , \, c=b+2 \, , \, d=c+2$$ Find the sum of all the numbers Pedro wrote.

Prove that there exists a sequence of positive integers $a_1,a_2,a_3, ...$ such that $a_1^2+a_2^2+...+a_n^2$ is a perfect square for all positive integers $n$.

[u]Round 5[/u] [b]p13.[/b] Find the number of six-digit positive integers that satisfy all of the following conditions: (i) Each digit does not exceed $3$. (ii) The number $1$ cannot appear in two consecutive digits. (iii) The number $2$ cannot appear in two consecutive digits. [b]p14.[/b] Find the sum of all distinct prime factors of $103040301$. [b]p15.[/b] Let $ABCA'B'C'$ be a triangular prism with height $3$ where bases $ABC$ and $A'B'C'$ are equilateral triangles with side length $\sqrt6$. Points $P$ and $Q$ lie inside the prism so that $ABCP$ and $A'B'C'Q$ are regular tetrahedra. The volume of the intersection of these two tetrahedra can be expressed in the form $\frac{\sqrt{m}}{n}$ , where $m$ and $n$ are positive integers and $m$ is not divisible by the square of any prime. Find $m + n$. [u]Round 6[/u] [b]p16.[/b] Let $a_0, a_1, ...$ be an infinite sequence such that $a^2_n -a_{n-1}a_{n+1} = a_n -a_{n-1}$ for all positive integers $n$. Given that $a_0 = 1$ and $a_1 = 4$, compute the smallest positive integer $k$ such that $a_k$ is an integer multiple of $220$. [b]p17.[/b] Vincent the Bug is on an infinitely long number line. Every minute, he jumps either $2$ units to the right with probability $\frac23$ or $3$ units to the right with probability $\frac13$ . The probability that Vincent never lands exactly $15$ units from where he started can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$? [b]p18.[/b] Battler and Beatrice are playing the “Octopus Game.” There are $2022$ boxes lined up in a row, and inside one of the boxes is an octopus. Beatrice knows the location of the octopus, but Battler does not. Each turn, Battler guesses one of the boxes, and Beatrice reveals whether or not the octopus is contained in that box at that time. Between turns, the octopus teleports to an adjacent box and secretly communicates to Beatrice where it teleported to. Find the least positive integer $B$ such that Battler has a strategy to guarantee that he chooses the box containing the octopus in at most $B$ guesses. [u]Round 7[/u] [b]p19.[/b] Given that $f(x) = x^2-2$ the number $f(f(f(f(f(f(f(2.5)))))))$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Find the greatest positive integer $n$ such that $2^n$ divides $ab+a+b-1$. [b]p20.[/b] In triangle $ABC$, the shortest distance between a point on the $A$-excircle $\omega$ and a point on the $B$-excircle $\Omega$ is $2$. Given that $AB = 5$, the sum of the circumferences of $\omega$ and $\Omega$ can be written in the form $\frac{m}{n}\pi$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? (Note: The $A$-excircle is defined to be the circle outside triangle $ABC$ that is tangent to the rays $\overrightarrow{AB}$ and $\overrightarrow{AC}$ and to the side $ BC$. The $B$-excircle is defined similarly for vertex $B$.) [b]p21.[/b] Let $a_0, a_1, ...$ be an infinite sequence such that $a_0 = 1$, $a_1 = 1$, and there exists two fixed integer constants $x$ and $y$ for which $a_{n+2}$ is the remainder when $xa_{n+1}+ya_n$ is divided by $15$ for all nonnegative integers $n$. Let $t$ be the least positive integer such that $a_t = 1$ and $a_{t+1} = 1$ if such an integer exists, and let $t = 0$ if such an integer does not exist. Find the maximal value of t over all possible ordered pairs $(x, y)$. [u]Round 8[/u] [b]p22.[/b] A mystic square is a $3$ by $3$ grid of distinct positive integers such that the least common multiples of the numbers in each row and column are the same. Let M be the least possible maximal element in a mystic square and let $N$ be the number of mystic squares with $M$ as their maximal element. Find $M + N$. [b]p23.[/b] In triangle $ABC$, $AB = 27$, $BC = 23$, and $CA = 34$. Let $X$ and $Y$ be points on sides $ AB$ and $AC$, respectively, such that $BX = 16$ and $CY = 7$. Given that $O$ is the circumcenter of $BXY$ , the value of $CO^2$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p24.[/b] Alan rolls ten standard fair six-sided dice, and multiplies together the ten numbers he obtains. Given that the probability that Alan’s result is a perfect square is $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers, compute $a$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949416p26408251]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f:\mathbb{Q}^+\to\mathbb{Q}^+$ such that \[xy(f(x)-f(y))|x-f(f(y))\] holds for all positive rationals $x$, $y$ (we define that $a|b$ if and only if exist $n \in \mathbb{Z}$ such that $b=an$) [i]Proposed by supercarry & windleaf1A[/i]

Consider the set $$M = \left\{\frac{a}{\overline{ba}}+\frac{b}{\overline{ab}} \, | a,b\in\{1,2,3,4,5,6,7,8,9\} \right\}.$$ a) Show that the set $M$ contains no integer. b) Find the smallest and the largest element of $M$

Prove that if a,b,c are integers with $a+b+c=0$, then $2a^4+2b^4+2c^4$ is a perfect square.

For a natural number $n$, $f(n)$ is defined as the number of positive integers less than $n$ which are neither coprime to $n$ nor a divisor of it. Prove that for each positive integer $k$ there exist only finitely many $n$ satisfying $f(n) = k$.

Are there any (not necessarily positive) integers $m$ and $n$ such that a) $\frac{1}{m}-\frac{1}{n}=\frac{1}{m-n}$ ? b) $\frac{1}{m}-\frac{1}{n}=\frac{1}{n-m}$

5. Let $S$ denote the set of all positive integers whose prime factors are elements of $\{2,3,5,7,11\}$. (We include 1 in the set $S$.) If $$ \sum_{q \in S} \frac{\varphi(q)}{q^{2}} $$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, find $a+b$. (Here $\varphi$ denotes Euler's totient function.)

Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] $(i)$ $f(n) \neq 0$ for at least one $n$; [*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; [*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. [/list]

A sequence $a_1, a_2, a_3, \ldots$ of positive integers satisfies $a_1 > 5$ and $a_{n+1} = 5 + 6 + \cdots + a_n$ for all positive integers $n$. Determine all prime numbers $p$ such that, regardless of the value of $a_1$, this sequence must contain a multiple of $p$.

Given a square grid where the distance between two adjacent grid points is $1$. Can the distance between two grid points be $\sqrt5, \sqrt6, \sqrt7$ or $\sqrt{2007}$ ?

How many positive integers $n \le \text{lcm}(1,2, \ldots, 100)$ have the property that $n$ gives different remainders when divided by each of $2,3, \ldots, 100$?

Determine all positive integers $n$ such that $5^n - 1$ can be written as a product of an even number of consecutive integers.

Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?

Whichever $3$ odd numbers is given. Prove that we can find a $4.$ odd number such that, sum of squares of the these numbers is a perfect square.

How many sequences of integers $1 \le a_1 \le a_2\le ... \le a_{11 }\le 2015$ that satisfy $a_i \equiv i^2$ (mod $12$) for all $1 \le i \le 11$ are there? (Le Anh Vinh)

Determine the smallest positive integer \( n \) with the following property: for every triple of positive integers \( x, y, z \), with \( x \) dividing \( y^3 \), \( y \) dividing \( z^3 \), and \( z \) dividing \( x^3 \), it also holds that \( (xyz) \) divides \( (x + y + z)^n \).

Let $a<b<c$ be three positive integers. Prove that among any $2c$ consecutive positive integers there exist three different numbers $x,y,z$ such that $abc$ divides $xyz$.