A four-element set $\{a, b, c, d\}$ of positive integers is called [i]good[/i] if there are two of them such that their product is a mutiple of the greatest common divisor of the remaining two. For example, the set $\{2, 4, 6, 8\}$ is good since the greatest common divisor of $2$ and $6$ is $2$, and it divides $4\times 8=32$. Find the greatest possible value of $n$, such that any four-element set with elements less than or equal to $n$ is good. [i]Proposed by Victor and Isaías de la Fuente[/i]

Given the rational numbers $r$, $q$, and $n$, such that $\displaystyle\frac1{r+qn}+\frac1{q+rn}=\frac1{r+q}$, prove that $\displaystyle\sqrt{\frac{n-3}{n+1}}$ is a rational number.

Prove that for every integer $n$ greater than $1:$ $$\frac{3n+1}{2n+2} < \left( \frac{1}{n} \right)^{n} + \left( \frac{2}{n} \right)^{n}+ \ldots+\left( \frac{n}{n} \right)^{n} <2.$$

Prove that you can select two adjacent sides of any quadrilateral and supplement them in order to create a parallelogram, the resulting parallelogram contains the original quadrilateral .

A positive number, if its fractional part, integeral part, and itself are geometric series, then the number is________.

The point $P$, from the plane in which $\Delta ABC$ lies, is such that if $A_1,B_1$, and $C_1$ are the orthogonal projections of $P$ on the respective altitudes of $ABC$, then $AA_1=BB_1=CC_1=t$. Determine the locus of $P$ and length of $t$.

In triangle $ABC$, angle $B$ is not right. The circle inscribed in $ABC$ touches $AB$ and $BC$ at points $C_1$ and $A_1$, and the feet of the altitudes drawn to the sides $AB$ and $BC$ are points $C_2$ and $A_2$. Prove that the intersection point of the altitudes of triangle $A_1BC_1$ is the center of the circle inscribed in triangle $A_2BC_2$.

Let $a_1,a_2,\ldots ,a_n\ge 0.$ For all $1\le k\le n$ define $$b_k:=\min_{1\le i<j\le n,j-i\le 2}|2a_k-a_i-a_j|.$$ Here the index mod $n.$ Find the maximum value of $\frac{b_1+b_2+\cdots +b_n}{a_1+a_2+\cdots +a_n}.$ [i]Proposed by Zheng Wang[/i]

Let $ a$, $ b$, and $ c$ be positive real numbers such that $ ab + bc + ca = 3$. Prove the inequality \[ 3 + \sum_{\mathrm{\cyc}} (a - b)^2 \ge \frac {a + b^2c^2}{b + c} + \frac {b + c^2a^2}{c + a} + \frac {c + a^2b^2}{a + b} \ge 3. \]

Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \] cannot be divided by $5$.

The positive integers from 1 to 100 are painted into three colors: 50 integers are red, 25 integers are yellow and 25 integers are green. The red and yellow integers can be divided into 25 triples such that each triple includes two red integers and one yellow integer which is greater than one of the red integers and smaller than another one. The same assertion is valid for the red and green integers. Is it necessarily possible to divide all the 100 integers into 25 quadruples so that each quadruple includes two red integers, one yellow integer and one green integer such that the yellow and the green integer lie between the red ones? [i]Alexandr Gribalko[/i]

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

Given two positive integers $x,y$, we define $z=x\,\oplus\,y$ to be the bitwise XOR sum of $x$ and $y$; that is, $z$ has a $1$ in its binary representation at exactly the place values where $x,y$ have differing binary representations. It is known that $\oplus$ is both associative and commutative. For example, $20 \oplus 18 = 10100_2 \oplus 10010_2 = 110_2 = 6$. Given a set $S=\{a_1, a_2, \dots, a_n\}$ of positive integers, we let $f(S) = a_1 \oplus a_2 \oplus a_3\oplus \dots \oplus a_n$. We also let $g(S)$ be the number of divisors of $f(S)$ which are at most $2018$ but greater than or equal to the largest element in $S$ (if $S$ is empty then let $g(S)=2018$). Compute the number of $1$s in the binary representation of $\displaystyle\sum_{S\subseteq \{1,2,\dots, 2018\}} g(S)$. [i]Proposed by Brandon Wang and Vincent Huang

Find all pair of integers $(m,n)$ and $m \ge n$ such that there exist a positive integer $s$ and a) Product of all divisor of $sm, sn$ are equal. b) Number of divisors of $sm,sn$ are equal.

Let $n$ be a fixed natural number. [b]a)[/b] Find all solutions to the following equation : \[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \] [b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) : \[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]

Each morning of her five-day workweek, Jane bought either a $ 50$-cent muffin or a $ 75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Let $a$ be a real number. Find the minimum value of $\int_0^1 |ax-x^3|dx$. How many solutions (including University Mathematics )are there for the problem? Any advice would be appreciated.

Let $P$ be a convex planar polygon with equal angles. Let $l_1,\cdots, l_n$ be its sides. Show that a necessary and sufficient condition for $P$ to be regular is that the sum of the ratios $\frac{l_i}{l_{i+1}} (i = 1,\cdots, n; l_{n+1}= l_1)$ equals the number of sides.

Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent: [b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$ [b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.

Find all the triples of positive integers $(a,b,c)$ for which the number \[\frac{(a+b)^4}{c}+\frac{(b+c)^4}{a}+\frac{(c+a)^4}{b}\] is an integer and $a+b+c$ is a prime.

In each cell of an $n\times n$ board is a lightbulb. Initially, all of the lights are off. Each move consists of changing the state of all of the lights in a row or of all of the lights in a column (off lights are turned on and on lights are turned off). Show that if after a certain number of moves, at least one light is on, then at this moment at least $n$ lights are on.

Find all positive integer $ n$ satisfying the following condition: There exist positive integers $ m$, $ a_1$, $ a_2$, $ \cdots$, $ a_{m\minus{}1}$, such that $ \displaystyle n \equal{} \sum_{i\equal{}1}^{m\minus{}1} a_i(m\minus{}a_i)$, where $ a_1$, $ a_2$, $ \cdots$, $ a_{m\minus{}1}$ may not distinct and $ 1 \leq a_i \leq m\minus{}1$.

Let there be a scalene triangle $ABC$, and its incircle hits $BC, CA, AB$ at $D, E, F$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at $P, Q$, where $P$ is on the same side with $A$ with respect to $BC$. Let the line parallel to $AQ$ and passing through $D$ meet $EF$ at $R$. Prove that the intersection between $EF$ and $PQ$ lies on the circumcircle of $BCR$.

Rachel has $3$ children, all of which are at least $2$ years old. The ages of the children are all pairwise relatively prime, but Rachel’s age is a multiple of each of her children’s ages. What is Rachel’s minimum possible age?

Solve for $x$, given $36x^4 + 36x^3 - 7x^2 - 6x + 1 = 0$.