Determine all functions $f : \mathbb R \to \mathbb R$ such that $$f(f(x)- f(y)) = f(f(x)) - 2x^2f(y) + f\left(y^2\right),$$ for all reals $x, y$.

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A total of $731$ objects are put into $n$ nonempty bags where $n$ is a positive integer. These bags can be distributed into $17$ red boxes and also into $43$ blue boxes so that each red and each blue box contain $43$ and $17$ objects, respectively. Find the minimum value of $n$.

If $x,y$ and $y - \frac{1}{x}$ are not 0, then \[\frac{x - \frac{1}{y}}{y - \frac{1}{x}}\] equals $\textbf{(A) }1\qquad\textbf{(B) } \frac{x}{y}\qquad\textbf{(C) }\frac{y}{x}\qquad\textbf{(D) }\frac{x}{y} - \frac{y}{x}\qquad\textbf{(E) } xy - \frac{1}{xy}$

Let $S$ be the set of bijections \[ T\colon\{1,\,2,\,3\}\times\{1,\,2,\,\ldots,\,2024\}\to\{1,\,2,\,\ldots,\,6072\} \] such that $T(1,\,j)<T(2,\,j)<T(3,\,j)$ for all $j\in\{1,\,2,\,\ldots,\,2024\}$ and $T(i,\,j)<T(i,\,j+1)$ for all $i\in\{1,\,2,\,3\}$ and $j\in\{1,\,2,\,\ldots,\,2023\}$. Do there exist $a$ and $c$ in $\{1,\,2,\,3\}$ and $b$ and $d$ in $\{1,\,2,\,\ldots,\,2024\}$ such that the fraction of elements $T$ in $S$ for which $T(a,\,b)<T(c,\,d)$ is at least $1/3$ and at most $2/3$.

In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

prove that if $r_n$ is a rational function whose numerator and denominator have at most degrees $n$, then $$||r_n||_{1/2}+\left\|\frac{1}{r_n}\right\|_2\geq\frac{1}{2^{n-1}}$$ where $||\cdot||_a$ denotes the supremum over a circle of radius $a$ around the origin.

A parallelogram $ABCD$ is given. The incircle of triangle $ABC$ with center $I$ touches $AB,BC,CA$ at $R,P,Q$. Ray $DI$ intersects segment $AB$ at $S$. It turned out that $\angle DPR=90$ Prove that the circle with diameter $AS$ is tangent to the circumcircle of triangle $DPQ$ [i]M. Zorka[/i]

What is the sum of all positive real solutions $x$ to the equation \[2\cos(2x)\left(\cos(2x)-\cos\left(\frac{2014\pi^2}{x}\right)\right)=\cos(4x)-1?\] $\textbf{(A) }\pi\qquad \textbf{(B) }810\pi\qquad \textbf{(C) }1008\pi\qquad \textbf{(D) }1080\pi\qquad \textbf{(E) }1800\pi\qquad$

[b](a)[/b] What is the maximal number of acute angles in a convex polygon? [b](b)[/b] Consider $m$ points in the interior of a convex $n$-gon. The $n$-gon is partitioned into triangles whose vertices are among the $n + m$ given points (the vertices of the $n$-gon and the given points). Each of the $m$ points in the interior is a vertex of at least one triangle. Find the number of triangles obtained.

The numbers $1$ to $25$ will be written in a table $5 \times 5$. First, Ana chooses $k$ of these numbers($1$ to $25$), and write in some cells. Then, Enrique writes the remaining numbers with the following goal: The product of the numbers in some column/row is a perfect square. [b]a)[/b] Prove that if $k=5$, Ana can [b]avoid[/b] Enrique to reach his goal. [b]b)[/b] Prove that if $k=4$, Enrique can reach his goal.

What minimal number of numbers from the set $\{1,2,...,1982\}$ should be deleted to provide the property: [i]none of the remained numbers equals to the product of two other remained numbers[/i]?

Are there $n$-digit numbers M and N such that all digits $M$ are even, all $N$ digits are odd, every digit from $0$ to $9$ occurs in decimal notation M or N at least once, and $M$ is divisible by $N$?

Point $D$ is chosen on the Euler line of triangle $ABC$ and it is inside of the triangle. Points $E,F$ are were the line $BD,CD$ intersect with $AC,AB$ respectively. Point $X$ is on the line $AD$ such that $\angle EXF =180 - \angle A$, also $A,X$ are on the same side of $EF$. If $P$ is the second intersection of circumcircles of $CXF,BXE$ then prove the lines $XP,EF$ meet on the altitude of $A$ Proposed by [i]Alireza Danaie[/i]

Let $a,n$ be integer numbers, $p$ a prime number such that $p>|a|+1$. Prove that the polynomial $f(x)=x^n+ax+p$ cannot be represented as a product of two integer polynomials. [i]Laurentiu Panaitopol[/i]

A positive integer $N$ is [i]rioplatense[/i] if it satifies the following conditions: 1 -There exist $34$ consecutive integers such that its product is divisible by $N$, but none of them is divisible by $N$. 2 - There [b]not[/b] exist $30$ consecutive integers such that its product is divisible by $N$, but none of them is divisible by $N$. Determine all rioplatense numbers.

If the product of $(\sqrt2 +\sqrt3+\sqrt5) (\sqrt2 +\sqrt3-\sqrt5) (\sqrt2 -\sqrt3+\sqrt5) (-\sqrt2 +\sqrt3+\sqrt5)$ is $12\sqrt6+ 6\sqrt{x}$ , find $x$. ([i]0 points[/i] - [b]THROWN OUT[/b])

Let $0<x_i<1$ and $x_i+y_i=1$ for $i=1,2,\ldots,n$. Prove that $$(1-x_1x_2\cdots x_n)^m+(1-y_1^m)(1-y_2^m)\cdots(1-y_n^m)>1$$for any natural numbers $m$ and $n$.

How many functions $f : \mathbb{Z} \rightarrow \{0, 1, 2, \cdots, 2020 \}$ are there such that $f(n) = f(n+2021)$ and $2021 \mid f(2n) - f(n) - f(n-1)$ for all integers $n$?

There are two prime numbers $p$ so that $5p$ can be expressed in the form $\left\lfloor \dfrac{n^2}{5}\right\rfloor$ for some positive integer $n.$ What is the sum of these two prime numbers?

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]B16 / G11[/b] Let triangle $ABC$ be an equilateral triangle with side length $6$. If point $D$ is on $AB$ and point $E$ is on $BC$, find the minimum possible value of $AD + DE + CE$. [b]B17 / G12[/b] Find the smallest positive integer $n$ with at least seven divisors. [b]B18 / G13[/b] Square $A$ is inscribed in a circle. The circle is inscribed in Square $B$. If the circle has a radius of $10$, what is the ratio between a side length of Square $A$ and a side length of Square $B$? [b]B19 / G14[/b] Billy Bob has $5$ distinguishable books that he wants to place on a shelf. How many ways can he order them if he does not want his two math books to be next to each other? [b]B20 / G15[/b] Six people make statements as follows: Person $1$ says “At least one of us is lying.” Person $2$ says “At least two of us are lying.” Person $3$ says “At least three of us are lying.” Person $4$ says “At least four of us are lying.” Person $5$ says “At least five of us are lying.” Person $6$ says “At least six of us are lying.” How many are lying? [u]Set 5[/u] [b]B21 / G16[/b] If $x$ and $y$ are between $0$ and $1$, find the ordered pair $(x, y)$ which maximizes $(xy)^2(x^2 - y^2)$. [b]B22 / G17[/b] If I take all my money and divide it into $12$ piles, I have $10$ dollars left. If I take all my money and divide it into $13$ piles, I have $11$ dollars left. If I take all my money and divide it into $14$ piles, I have $12$ dollars left. What’s the least amount of money I could have? [b]B23 / G18[/b] A quadratic equation has two distinct prime number solutions and its coefficients are integers that sum to a prime number. Find the sum of the solutions to this equation. [b]B24 / G20[/b] A regular $12$-sided polygon is inscribed in a circle. Gaz then chooses $3$ vertices of the polygon at random and connects them to form a triangle. What is the probability that this triangle is right? [b]B25 / G22[/b] A book has at most $7$ chapters, and each chapter is either $3$ pages long or has a length that is a power of $2$ (including $1$). What is the least positive integer $n$ for which the book cannot have $n$ pages? [u]Set 6[/u] [b]B26 / G26[/b] What percent of the problems on the individual, team, and guts rounds for both divisions have integer answers? [b]B27 / G27[/b] Estimate $12345^{\frac{1}{123}}$. [b]B28 / G28[/b] Let $O$ be the center of a circle $\omega$ with radius $3$. Let $A, B, C$ be randomly selected on $\omega$. Let $M$, $N$ be the midpoints of sides $BC$, $CA$, and let $AM$, $BN$ intersect at $G$. What is the probability that $OG \le 1$? [b]B29 / G29[/b] Let $r(a, b)$ be the remainder when $a$ is divided by $b$. What is $\sum^{100}_{i=1} r(2^i , i)$? [b]B30 / G30[/b] Bongo flips $2023$ coins. Call a run of heads a sequence of consecutive heads. Say a run is maximal if it isn’t contained in another run of heads. For example, if he gets $HHHT T HT T HHHHT H$, he’d have maximal runs of length $3, 1, 4, 1$. Bongo squares the lengths of all his maximal runs and adds them to get a number $M$. What is the expected value of $M$? - - - - - - [b]G19[/b] Let $ABCD$ be a square of side length $2$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AD$. Let the intersection of $BN$ and $CM$ be $E$. Find the area of quadrilateral $NECD$. [b]G21[/b] Quadrilateral $ABCD$ has $\angle A = \angle D = 60^o$. If $AB = 8$, $CD = 10$, and $BC = 3$, what is length $AD$? [b]G23[/b] $\vartriangle ABC$ is an equilateral triangle of side length $x$. Three unit circles $\omega_A$, $\omega_B$, and $\omega_C$ lie in the plane such that $\omega_A$ passes through $A$ while $\omega_B$ and $\omega_C$ are centered at $B$ and $C$, respectively. Given that $\omega_A$ is externally tangent to both $\omega_B$ and $\omega_C$, and the center of $\omega_A$ is between point $A$ and line $\overline{BC}$, find $x$. [b]G24[/b] For some integers $n$, the quadratic function $f(x) = x^2 - (6n - 6)x - (n^2 - 12n + 12)$ has two distinct positive integer roots, exactly one out of which is a prime and at least one of which is in the form $2^k$ for some nonnegative integer $k$. What is the sum of all possible values of $n$? [b]G25[/b] In a triangle, let the altitudes concur at $H$. Given that $AH = 30$, $BH = 14$, and the circumradius is $25$, calculate $CH$ PS. You should use hide for answers. Rest problems have been posted [url=https://artofproblemsolving.com/community/c3h3132167p28376626]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the minimum number $n$ such that for any coloring of the integers from $1$ to $n$ into two colors, one can find monochromatic $a$, $b$, $c$, and $d$ (not necessarily distinct) such that $a+b+c=d$.

From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of $N$ cents, where $N$ is a positive integer. He uses the so-called greedy algorithm, successively choosing the coin of greatest value that does not cause the value of his collection to exceed $N.$ For example, to get 42 cents, Silas will choose a 25-cent coin, then a 10-cent coin, then 7 1-cent coins. However, this collection of 9 coins uses more coins than necessary to get a total of 42 cents; indeed, choosing 4 10-cent coins and 2 1-cent coins achieves the same total value with only 6 coins. In general, the greedy algorithm succeeds for a given $N$ if no other collection of 1-cent, 10-cent, and 25-cent coins gives a total value of $N$ cents using strictly fewer coins than the collection given by the greedy algorithm. Find the number of values of $N$ between $1$ and $1000$ inclusive for which the greedy algorithm succeeds.

Let $\triangle ABC$ be a right triangle with right angle at $C$. Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C$. If $\frac{DE}{BE} = \frac{8}{15}$, then $\tan B$ can be written as $\frac{m\sqrt{p}}{n}$, where $m$ and $n$ are relatively prime positive integers, and $p$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.

Find the integral solutions of the equation $7(x+y)=3(x^2-xy+y^2)$