Find all triples $(a,b, c)$ of real numbers for which there exists a non-zero function $f: R \to R$, such that $$af(xy + f(z)) + bf(yz + f(x)) + cf(zx + f(y)) = 0$$ for all real $x, y, z$. (E. Barabanov)

Let $x_1,x_2,\ldots, x_{10}$ be non-negative real numbers such that $\frac{x_1}{1}+ \frac{x_2}{2} +\cdots+ \frac{x_{10}}{10}$ $\leq9$. Find the maximum possible value of $\frac{{x_1}^2}{1}+\frac{{x_2}^2}{2}+\cdots+\frac{{x_{10}}^2}{10}$.

Among all fractions (whose numerator and denominator are positive integers) strictly between $\tfrac{6}{17}$ and $\tfrac{9}{25}$, which one has the smallest denominator?

Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $ (x, y)$ such that [i](i)[/i] $ x$ and $ y$ are relatively prime; [i](ii)[/i] $ y$ divides $ x^2 \plus{} m$; [i](iii)[/i] $ x$ divides $ y^2 \plus{} m.$ [i](iv)[/i] $ x \plus{} y \leq m \plus{} 1\minus{}$ (optional condition)

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.

For $n$positive integers $ x_1,x2,...,x_n$, $a_n$ is their arithmetic and $g_n$ the geometric mean. Consider the statement $S_n$: If $a_n/g_n$ is a positive integer, then $x_1 = x_2 = ··· = x_n$. Prove $S_2$ and disprove $S_n$ for all even $n > 2$.

As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$. [asy] size(200); pair A=origin, B=(14,0), C=(9,12), D=foot(A, B,C), E=foot(B, A, C), F=foot(C, A, B), H=orthocenter(A, B, C); draw(F--C--A--B--C^^A--D^^B--E); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("84", centroid(H, C, E), fontsize(9.5)); label("35", centroid(H, B, D), fontsize(9.5)); label("30", centroid(H, F, B), fontsize(9.5)); label("40", centroid(H, A, F), fontsize(9.5));[/asy]

A bug is walking on the surface of a Rubik's cube(cube $3 \times 3 \times 3$). It can go to the adjacent cell on the same face or on the adjacent face. One day the bug started walking from some cell and returned to it, and visited all other cells exactly once. Prove that he made an even amount of moves that changed the face he is on.

Let be four distinct complex numbers $ a,b,c,d $ chosen such that $$ |a|=|b|=|c|=|d|=|b-c|=\frac{|c-d|}{2}=1, $$ and $$ \min_{\lambda\in\mathbb{C}} |a-\lambda d -(1-\lambda )c| =\min_{\lambda\in\mathbb{C}} |b-\lambda d -(1-\lambda )c| . $$ Calculate $ |a-c| $ and $ |a-d|. $ [i]Carmen Botea[/i]

Find the value(s) of $ x$ such that $ 8xy\minus{}12y\plus{}2x\minus{}3\equal{}0$ is true for all values of $ y$. $ \textbf{(A)}\ \frac{2}{3} \qquad \textbf{(B)}\ \frac{3}{2}\text{ or }\minus{}\frac{1}{4} \qquad \textbf{(C)}\ \minus{}\frac{2}{3}\text{ or }\minus{}\frac{1}{4} \qquad \textbf{(D)}\ \frac{3}{2} \qquad \textbf{(E)}\ \minus{}\frac{3}{2}\text{ or }\minus{}\frac{1}{4}$

Suppose that $p$ is an odd prime. Prove that \[\sum_{j=0}^{p}\binom{p}{j}\binom{p+j}{j}\equiv 2^{p}+1\pmod{p^{2}}.\]

If for the coefficients of equation $x^3+ax^2+bx+c=0$ whose roots are all real, holds, $a^2= 2(b+1)$ then $|a-c|\le 2$. Prove it.

Let $x$ be a positive real number and let $k$ be a positive integer. Assume that $x^k+\frac{1}{x^k}$ and $x^{k+1}+\frac{1}{x^{k+1}}$ are both rational numbers. Prove that $x+\frac{1}{x}$ is also a rational number.

A triangle and a circle are in the same plane. Show that the area of the intersection of the triangle and the circle is at most one third of the area of the triangle plus one half of the area of the circle. [i]Krit Boonsiriseth[/i]

[b]The [color = #833]R[/color]ELMO has concluded[/b]. Thanks to all participants! [rule] [center] [size = 250][b][color = #833]Revenge[/color][/b] ELMO, Year Three [/size] [img width = 40] https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQrox2Fm5Kg9-F5edUKOykXa6Bbtzr2Os00ZBlhhNx6YiXgyORoJUIFpVbjBdh4bUPwIYE&usqp=CAU[/img] [/center] [rule] [size = 150]Overview[/size] The [b]ELMO[/b] is an annual contest given to [b]new students[/b] at the USA Math Olympiad Program, written completely by the [b]returning students[/b].... The [b][color = #833]R[/color]ELMO[/b] is an annual contest given to [b]returning students[/b] at the USA Math Olympiad Program, written completely by the [b]new students[/b]! We are inviting everybody from AoPS to take the RELMO. On [b]June 16th[/b], while the returning MOPpers are taking the RELMO, we will publish the problems on [b]this thread[/b]. [rule] [size = 150]Rules And Procedures[/size] Find the test links here: [hide] [url=https://drive.google.com/file/d/1UkFM1WJn9vu6kf4aALIguE7gfdNxk527/view?usp=drive_link]The RELMO[/url] [hide = The S variants] [url=https://drive.google.com/file/d/10PBVMIWN6Fy4ooJlqpk9D7qo4IIEhrc8/view?usp=drive_link]The RELSMO[/url] [rule] [url=https://drive.google.com/file/d/1USsVgml7yveN5IlqnaAae9op0zLzG-_n/view?usp=drive_link]The RELBMO[/url] [url=https://drive.google.com/file/d/1dYlw8339D-tUfvM-9lsqDWN66XDASJOO/view?usp=drive_link]The RELMORZ[/url] [url=https://drive.google.com/file/d/10SIm5jb7aqQqN8x1vjlhCm88vRu-JgyN/view?usp=drive_link]The RELSSMO[/url] [url=https://drive.google.com/file/d/1YUJ6atkC2wU6x_DZmjszXV0Flzr_W5SX/view?usp=drive_link]The RELXMO[/url] [/hide][/hide] [hide = Test Errata] For problem 2 on the RELMO: assume that the quadrilateral is convex. [/hide] The [b][color = #833]RELMO[/color][/b] consists of [b]six problems[/b] to be solved in [b]four and a half hours[/b]. Online submissions will be [b]unofficially graded[/b] – when the test is released, there will be a google form to submit solutions. If you would like some practice before the test, we recommend that you take a look at the past two RELMO's: [url=https://artofproblemsolving.com/community/c5t32737f5h2870938](Year 1)[/url] [url = https://artofproblemsolving.com/community/c5h3098990](Year 2)[/url] [rule] We hope you enjoy the problems! [color = #833] - the new MOPpers[/color]

Evaluate \[\left \lfloor \ \prod_{n=1}^{1992} \frac{3n+2}{3n+1} \ \right \rfloor\]

The taxi fare in Gotham City is $\$2.40$ for the first $\frac12$ mile and additional mileage charged at the rate $\$ 0.20$ for each additional 0.1 mile. You plan to give the driver a $\$2$ tip. How many miles can you ride for $\$10?$ $ \textbf{(A)}3.0\qquad\textbf{(B)}3.25\qquad\textbf{(C)}3.3\qquad\textbf{(D)}3.5\qquad\textbf{(E)}3.75 $

There are some balls, on each of which a positive integer not exceeding $14$ (and not necessarily distinct) is written, and the sum of the numbers on all balls is $S$. Find the greatest possible value of $S$ such that, regardless of what the integers are, one can ensure that the balls can be divided into two piles so that the sum of the numbers on the balls in each pile does not exceed $129$.

Prove that among any $42$ numbers from the interval $[1,10^6]$, you can choose four numbers so that for any permutation $(a, b, c, d)$ of these numbers, the inequality $$25 (ab + cd) (ad + bc) \ge 16 (ac + bd)^ 2$$ holds.

Compute $\frac{664.02}{9.3}$.

Show that there are $1977$ non-similar triangles such that the angles $A, B, C$ satisfy $\frac{\sin A + \sin B + \sin C}{\cos A +\cos B + \cos C} = \frac{12}{7}$ and $\sin A \sin B \sin C = \frac{12}{25}$.

Given a non- isosceles triangle $ABC$. Let the points $B'$ and $C'$ be symmetric to the points $B$ and $C$ wrt $AC$ and $AB$ respectively. If the circles circumscribed around triangles $ABB'$ and $ACC'$ intersect at point $P$, prove that the line $AP$ passes through the center of the circumcircle of the triangle $ABC$.

For given integer $n \geq 3$, set $S =\{p_1, p_2, \cdots, p_m\}$ consists of permutations $p_i$ of $(1, 2, \cdots, n)$. Suppose that among every three distinct numbers in $\{1, 2, \cdots, n\}$, one of these number does not lie in between the other two numbers in every permutations $p_i$ ($1 \leq i \leq m$). (For example, in the permutation $(1, 3, 2, 4)$, $3$ lies in between $1$ and $4$, and $4$ does not lie in between $1$ and $2$.) Determine the maximum value of $m$.

For any integer $n$ greater than $1$, the number of prime numbers greater than $n!+1$ and less than $n!+n$ is: $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }\dfrac n2\text{ for }n\text{ even,}\,\dfrac{n+1}2\text{ for }n\text{ odd}$ $\textbf{(D) }n-1\qquad \textbf{(E) }n$

Determine all three-digit numbers $N$ such that the average of the six numbers that can be formed by permutation of its three digits is equal to $N$.