The difference of fractions $\frac{2024}{2023} - \frac{2023}{2024}$ was represented as an irreducible fraction $\frac{p}{q}$. Find the value of $p$.

Let $A\in M_4(\mathbb R)$ be an invertible matrix s.t. $\det(A+^tA)=5\det A$ and $\det (A-^tA)=\det A$. Prove that for every complex root $\omega$ of order 5 of unitity (i.e. $\omega^5=1,\omega\not\in\mathbb R$) the following relation holds $\det(\omega A+^tA)=0$. [i]Dan Popescu[/i]

Let $a$, $b$ and $c$ be real numbers with $0 \le a, b, c \le 2$. Prove that $$(a - b)(b - c)(a- c) \le 2.$$ When does equality hold? [i](Karl Czakler)[/i]

Find all polynomials $P(x)$ with real coefficients such that for all real $x$ holds the equality $$(1 + 2x)P(2x) = (1 + 2^{1999}x)P(x) .$$

Let $f : \mathbb{N} \longrightarrow \mathbb{N}$ be a function such that $(x + y)f (x) \le x^2 + f (xy) + 110$, for all $x, y$ in $\mathbb{N}$. Determine the minimum and maximum values of $f (23) + f (2011)$.

Find all functions $f:\mathbb N\to\mathbb N_0$ such that for all $m,n\in\mathbb N$, \begin{align*} f(mn)&=f(m)f(n)\\ f(m+n)&=\min(f(m),f(n))\qquad\text{if }f(m)\ne f(n)\end{align*}

Nine numbers $a, b, c, \dots$ are arranged around a circle. All numbers of the form $a+b^c, \dots$ are prime. What is the largest possible number of different numbers among $a, b, c, \dots$?

Do there exist two bounded sequences $a_1, a_2,\ldots$ and $b_1, b_2,\ldots$ such that for each positive integers $n$ and $m>n$ at least one of the two inequalities $|a_m-a_n|>1/\sqrt{n},$ and $|b_m-b_n|>1/\sqrt{n}$ holds?

An equilateral triangle $ABC$ is given, together with a point $P$ inside it. [asy] draw((0,0)--(4,0)--(2,3.464)--(0,0)); draw((1.3, 1.2)--(0,0)); draw((1.3, 1.2)--(2,3.464)); draw((1.3, 1.2)--(4,0)); label("$A$",(0,0),SW); label("$B$",(4,0),SE); label("$C$",(2,3.464),N); label("$P$",(1.3,1.2),S); [/asy] Given that $PA = 3$ cm, $PB = 5$ cm, and $PC = 4$ cm, find the side of the equilateral triangle.

An [i]arithmetic sequence[/i] is an infinite sequence of the form $a_n=a_0+n\cdot d$ with $d\neq 0$. A [i]geometric sequence[/i] is an infinite sequence of the form $b_n=b_0 \cdot q^n$ where $q\neq 1,0,-1$. [list=a] [*] Does every arithmetic sequence of [b]integers[/b] have an infinite subsequence which is geometric? [*] Does every arithmetic sequence of [b]real numbers[/b] have an infinite subsequence which is geometric? [/list]

Find all functions $f : N \to N$ such that $f(m)+f(n)$ divides $m+n$ for all positive integers $m$ and $n$.

Let $A$ be a finite set of pairs of real numbers such that for any pairs $(a,b)$ in $A$ we have $a>0$. Let $X_0=(x_0, y_0)$ be a pair of real numbers(not necessarily from $A$). We define $X_{j+1}=(x_{j+1}, y_{j+1})$ for all $j\ge 0$ as follows: for all $(a,b)\in A$, if $ax_j+by_j>0$ we let $X_{j+1}=X_j$; otherwise we choose a pair $(a,b)$ in $A$ for which $ax_j+by_j\le 0$ and set $X_{j+1}=(x_j+a, y_j+b)$. Show that there exists an integer $N\ge 0$ such that $X_{N+1}=X_N$.

Calculate the sum of all unsimplified fractions whose numerator and denominator are positive divisors of $1000$.

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.

Let $n$ be a positive integer and let $a_1$, $a_2$,..., $a_n$ be different real numbers, placed one after the other in any order. We say we have a [i]local minimum[/i] in one of the numbers if this is less than both of their neighbors. Which is the average number of local minima over all possible ways of ordering the numbers each other?

Let $S$ be the domain surrounded by the two curves $C_1:y=ax^2,\ C_2:y=-ax^2+2abx$ for constant positive numbers $a,b$. Let $V_x$ be the volume of the solid formed by the revolution of $S$ about the axis of $x$, $V_y$ be the volume of the solid formed by the revolution of $S$ about the axis of $y$. Find the ratio of $\frac{V_x}{V_y}$.

Find all non-zero real numbers $a, b, c$ such that the following polynomial has four (not necessarily distinct) positive real roots. $$P(x) = ax^4 - 8ax^3 + bx^2 - 32cx + 16c$$

Let $(F_n)_{n\geq 1} $ be the Fibonacci sequence $F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),$ and $P(x)$ the polynomial of degree $990$ satisfying \[ P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.\] Prove that $P(1983) = F_{1983} - 1.$

Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$

Prove for positive $a,b,c$ that $$ (a^2+\frac{b^2}{c^2})(b^2+\frac{c^2}{a^2})(c^2+\frac{a^2}{b^2}) \geq abc (a+\frac{1}{a})(b+\frac{1}{b})(c+\frac{1}{c})$$

A positive integer $N$ is divided in $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \ldots$ The smallest part is equal to $\frac{1}{400} N$. Find the value of $n$.

Given a positive integer $n$ and $x_1 \leq x_2 \leq \ldots \leq x_n, y_1 \geq y_2 \geq \ldots \geq y_n$, satisfying \[\displaystyle\sum_{i = 1}^{n} ix_i = \displaystyle\sum_{i = 1}^{n} iy_i\] Show that for any real number $\alpha$, we have \[\displaystyle\sum_{i =1}^{n} x_i[i\alpha] \geq \displaystyle\sum_{i =1}^{n} y_i[i\alpha]\] Here $[\beta]$ denotes the greastest integer not larger than $\beta$.

For some positive numbers $A, B, C$ and $D$, the system of equations $$\begin{cases} x^2 + y^2 = A \\ |x| + |y| = B \end{cases}$$ has $m$ solutions, while the system of equations $$\begin{cases} x^2 + y^2 +z^2= X\\ |x| + |y| +|z| = D \end{cases}$$ has $n$ solutions. If $m > n > 1$, find $m$ and $n$. ( G Galperin)

[b]p1.[/b] Arrange the whole numbers $1$ through $15$ in a row so that the sum of any two adjacent numbers is a perfect square. In how many ways this can be done? [b]p2.[/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]p3.[/b] If a polyleg has even number of legs he always tells truth. If he has an odd number of legs he always lies. Once a green polyleg told a dark-blue polyleg ”- I have $8$ legs. And you have only $6$ legs!” The offended dark-blue polyleg replied ”-It is me who has $8$ legs, and you have only $7$ legs!” A violet polyleg added ”-The dark-blue polyleg indeed has $8$ legs. But I have $9$ legs!” Then a stripped polyleg started ”None of you has $8$ legs. Only I have $8$ legs!” Which polyleg has exactly $8$ legs? [b][b]p4.[/b][/b] There is a small puncture (a point) in the wall (as shown in the figure below to the right). The housekeeper has a small flag of the following form (see the figure left). Show on the figure all the points of the wall where you can hammer in a nail such that if you hang the flag it will close up the puncture. [img]https://cdn.artofproblemsolving.com/attachments/a/f/8bb55a3fdfb0aff8e62bc4cf20a2d3436f5d90.png[/img] [b]p5.[/b] Assume $ a, b, c$ are odd integers. Show that the quadratic equation $ax^2 + bx + c = 0$ has no rational solutions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].