Find all 4-digit numbers $n$, such that $n=pqr$, where $p<q<r$ are distinct primes, such that $p+q=r-q$ and $p+q+r=s^2$, where $s$ is a prime number.

Let $p>5$ be a prime number and $A=\{b_1,b_2,\dots,b_{\frac{p-1}{2}}\}$ be the set of all quadratic residues modulo $p$, excluding zero. Prove that there doesn't exist any natural $a,c$ satisfying $(ac,p)=1$ such that set $B=\{ab_1+c,ab_2+c,\dots,ab_{\frac{p-1}{2}}+c\}$ and set $A$ are disjoint modulo $p$. [i]This problem was proposed by Amir Hossein Pooya.[/i]

Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + q^2$ is the square of an integer.

Prove that the equation $a^3-b^3=2004$ does not have any solutions in positive integers.

$n\in \mathbb{N}$ is called [i]“good”[/i], if $n$ can be presented as a sum of the fourth powers of five of its divisors (different). a) Prove that each [i]good[/i] number is divisible by 5; b) Find a [i]good[/i] number; c) Does there exist infinitely many [i]good[/i] numbers?

Let $a,b,c,d$ be positive integers satisfying \[\frac{ab}{a+b}+\frac{cd}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}.\] Determine all possible values of $a+b+c+d$.

For a prime $ p$ an a positive integer $ n$, denote by $ \nu_p(n)$ the exponent of $ p$ in the prime factorization of $ n!$. Given a positive integer $ d$ and a finite set $ \{p_1,p_2,\ldots, p_k\}$ of primes, show that there are infinitely many positive integers $ n$ such that $ \nu_{p_i}(n) \equiv 0 \pmod d$, for all $ 1\leq i \leq k$.

Let $a_1$ , $\ldots$ , $a_n$ be positive integers and $a$ a positive integer that is greater than $1$ and is divisible by the product $a_1a_2\ldots a_n$. Prove that $a^{n+1}+a-1$ is not divisible by the product $(a+a_1-1)(a+a_2-1)\ldots(a+a_n-1)$.

Determine the smallest positive integer $n\geq 2$ for which there exists a positive integer $m$ such that $mn$ divides $m^{2023} + n^{2023} + n$.

Find all positive integers $(a, b)$, such that $\frac{a^2}{2ab^2-b^3+1}$ is also a positive integer.

By dividing $2023$ by a natural number $m$, the remainder is $23$. How many numbers $m$ are there with this property?

Let us define a function $f:\mathbb N\to\mathbb N_0$ by $f(1)=0$ and, for all $n\in\mathbb N$, $$f(2n)=2f(n)+1,\qquad f(2n+1)=2f(n).$$Given a positive integer $p$, define a sequence $(u_n)$ by $u_0=p$ and $u_{k+1}=f(u_k)$ whenever $u_k\ne0$. (a) Prove that, for each $p\in\mathbb N$, there is a unique integer $v(p)$ such that $u_{v(p)}=0$. (b) Compute $v(1994)$. What is the smallest integer $p>0$ for which $v(p)=v(1994)$. (c) Given an integer $N$, determine the smallest integer $p$ such that $v(p)=N$.

A pair of positive integer numbers \((a, b)\) is given. It turns out that for every positive integer number \(n\), for which the numbers \((n - a)(n + b)\) and \(n^2 - ab\) are positive, they have the same number of divisors. Is it necessarily true that \(a = b\)? [i]Proposed by Oleksii Masalitin[/i]

2017 prime numbers $p_1,...,p_{2017}$ are given. Prove that $\prod_{i<j} (p_i^{p_j}-p_j^{p_i})$ is divisible by 5777.

Let $T$ the set of the infinite sequences of integers. For two given elements in $T$: $(a_{1},a_{2},a_{3},...)$ and $(b_{1},b_{2},b_{3},...)$, define the sum $(a_{1},a_{2},a_{3},...)+(b_{1},b_{2},b_{3},...)=(a_{1}+b_{1},a_{2}+b_{2},a_{3}+b_{3},...)$. Let $f: T\rightarrow$ $\mathbb{Z}$ a function such that: i) If $x\in T$ has exactly one of your terms equal $1$ and all the others equal $0$, then $f(x)=0$. ii)$f(x+y)=f(x)+f(y)$, for all $x,y\in T$. Prove that $f(x)=0$ for all $x\in T$

For $2n$ positive integers a matching (i.e. dividing them into $n$ pairs) is called {\it non-square} if the product of two numbers in each pair is not a perfect square. Prove that if there is a non-square matching, then there are at least $n!$ non-square matchings. (By $n!$ denote the product $1\cdot 2\cdot 3\cdot \ldots \cdot n$.)

Let $ m$ be a positive integer and define $ f(m)$ to be the number of factors of $ 2$ in $ m!$ (that is, the greatest positive integer $ k$ such that $ 2^k|m!$). Prove that there are infinitely many positive integers $ m$ such that $ m \minus{} f(m) \equal{} 1989.$

A $tetromino$ is a figure made up of four unit squares connected by common edges. [List=i] [*] If we do not distinguish between the possible rotations of a tetromino within its plane, prove that there are seven distinct tetrominos. [*]Prove or disprove the statement: It is possible to pack all seven distinct tetrominos into $4\times 7$ rectangle without overlapping. [/list]

Find all positive integers $n$, such that there exists a positive integer $m$ and primes $1<p<q$ such that $q-p \mid m$ and $p, q \mid n^m+1$.

Morteza wishes to take two real numbers $S$ and $P$, and then to arrange six pairwise distinct real numbers on a circle so that for each three consecutive numbers at least one of the two following conditions holds: 1) their sum equals $S$ 2) their product equals $P$. Determine if Morteza’s wish could be fulfilled.

Prove that the equation $(3x+4y)(4x+5y)=7^z$ doesn't have solution in natural numbers.

Find all primes $1 < p < 100$ such that the equation $x^2-6y^2=p$ has an integer solution $(x, y)$.

A prime number $p> 3$ is given. Prove that integers less than $p$, it is possible to divide them into two non-empty sets such that the sum of the numbers in the first set will be congruent modulo p to the product of the numbers in the second set.

Consider the system \begin{align*}x + y &= z + u,\\2xy & = zu.\end{align*} Find the greatest value of the real constant $m$ such that $m \leq x/y$ for any positive integer solution $(x,y,z,u)$ of the system, with $x \geq y$.

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Compute $\frac{2024}{2 + 0 \times 2 - 4}.$ [b]p2.[/b] Find the smallest integer that can be written as the product of three distinct positive odd integers. [b]p3.[/b] Bryan’s physics test score is a two-digit number. When Bryan reverses its digits and adds the tens digit of his test score, he once again obtains his test score. Determine Bryan’s physics test score. [b]p4.[/b] Grant took four classes today. He spent $70$ minutes in math class. Had his math class been $40$ minutes instead, he would have spent $15\%$ less total time in class today. Find how many minutes he spent in his other classes combined. [b]p5.[/b] Albert’s favorite number is a nonnegative integer. The square of Albert’s favorite number has $9$ digits. Find the number of digits in Albert’s favorite number. [b]p6.[/b] Two semicircular arcs are drawn in a rectangle, splitting it into four regions as shown below. Given the areas of two of the regions, find the area of the entire rectangle. [img]https://cdn.artofproblemsolving.com/attachments/1/a/22109b346c7bdadeaf901d62155de4c506b33c.png[/img] [b]p7.[/b] Daria is buying a tomato and a banana. She has a $20\%$-off coupon which she may use on one of the two items. If she uses it on the tomato, she will spend $\$1.21$ total, and if she uses it on the banana, she will spend $\$1.31$ total. In cents, find the absolute difference between the price of a tomato and the price of a banana. [b]p8.[/b] Celine takes an $8\times 8$ checkerboard of alternating black and white unit squares and cuts it along a line, creating two rectangles with integer side lengths, each of which contains at least $9$ black squares. Find the number of ways Celine can do this. (Rotations and reflections of the cut are considered distinct.) [b]p9.[/b] Each of the nine panes of glass in the circular window shown below has an area of $\pi$, eight of which are congruent. Find the perimeter of one of the non-circular panes. [img]https://cdn.artofproblemsolving.com/attachments/b/c/0d3644dde33b68f186ba1ff0602e08ce7996f5.png[/img] [b]p10.[/b] In Alan’s favorite book, pages are numbered with consecutive integers starting with $1$. The average of the page numbers in Chapter Five is $95$ and the average of the page numbers in Chapter Six is $114$. Find the number of pages in Chapters Five and Six combined. [b]p11.[/b] Find the number of ordered pairs $(a, b)$ of positive integers such that $a + b = 2024$ and $$\frac{a}{b}>\frac{1000}{1025}.$$ [b]p12.[/b] A square is split into three smaller rectangles $A$, $B$, and $C$. The area of $A$ is $80$, $B$ is a square, and the area of $C$ is $30$. Compute the area of $B$. [img]https://cdn.artofproblemsolving.com/attachments/d/5/43109b964eacaddefd410ddb8bf4e4354a068b.png[/img] [b]p13.[/b] A knight on a chessboard moves two spaces horizontally and one space vertically, or two spaces vertically and one space horizontally. Two knights attack each other if each knight can move onto the other knight’s square. Find the number of ways to place a white knight and a black knight on an $8 \times 8$ chessboard so that the two knights attack each other. One such possible configuration is shown below. [img]https://cdn.artofproblemsolving.com/attachments/2/2/b4a83fbbab7e54dda81ac5805728d268b6db9f.png[/img] [b]p14.[/b] Find the sum of all positive integers $N$ for which the median of the positive divisors of $N$ is $9$. [b]p15.[/b] Let $x$, $y$, and $z$ be nonzero real numbers such that $$\begin{cases} 20x + 24y = yz \\ 20y + 24x = xz \end{cases}$$ Find the sum of all possible values of $z$. [b]p16.[/b] Ava glues together $9$ standard six-sided dice in a $3 \times 3$ grid so that any two touching faces have the same number of dots. Find the number of dots visible on the surface of the resulting shape. (On a standard six-sided die, opposite faces sum to $7$.) [img]https://cdn.artofproblemsolving.com/attachments/5/5/bc71dac9b8ae52a4456154000afde2c89fd83a.png[/img] [b]p17.[/b] Harini has a regular octahedron of volume $1$. She cuts off its $6$ vertices, turning the triangular faces into regular hexagons. Find the volume of the resulting solid. [b]p18.[/b] Each second, Oron types either $O$ or $P$ with equal probability, forming a growing sequence of letters. Find the probability he types out $POP$ before $OOP$. [b]p19.[/b] For an integer $n \ge 10$, define $f(n)$ to be the number formed after removing the first digit from $n$ (and removing any leading zeros) and define $g(n)$ to be the number formed after removing the last digit from $n$. Find the sum of the solutions to the equation $f(n) + g(n) = 2024$. [b]p20.[/b] In convex trapezoid $ABCD$ with $\overline{AB} \parallel \overline{CD}$ and $AD = BC$, let $M$ be the midpoint of $\overline{BC}$. If $\angle AMB = 24^o$ and $\angle CMD = 66^o$, find $\angle ABC$, in degrees. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].