Let $ a$, $ b$, $ c$ be positive real numbers. Prove the inequality \[ \frac{1}{a\left(b+1\right)}+\frac{1}{b\left(c+1\right)}+\frac{1}{c\left(a+1\right)}\geq \frac{3}{1+abc}. \]

What is the smallest positive integer $k$ such that $k(3^3 + 4^3 + 5^3) = a^n$ for some positive integers $a$ and $n$, with $n > 1$?

Let $p,q$ be two distinct odd primes. Calculate $\displaystyle \sum_{j=1}^{\frac{p-1}{2}}\left \lfloor \frac{qj}{p}\right \rfloor +\sum_{j=1}^{\frac{q-1}{2}}\left \lfloor \frac{pj}{q}\right\rfloor$.

The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle’s height to the base. What is the measure, in degrees, of the vertex angle of this triangle? $\textbf{(A)}\ 105 \qquad\textbf{(B)}\ 120 \qquad\textbf{(C)}\ 135 \qquad\textbf{(D)}\ 150 \qquad\textbf{(E)}\ 165$

Of $ 28$ students taking at least one subject the number taking Mathematics and English only equals the number taking Mathematics only. No student takes English only or History only, and six students take Mathematics and History, but not English. The number taking English and History only is five times the number taking all three subjects. If the number taking all three subjects is even and non-zero, the number taking English and Mathematics only is: $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Give a geometric series $(a_n)$ with common ratio of $q$, let $b_1=a_1+a_2+a_3,b_2=a_4+a_5+a_6,\cdots,b_n=a_{3n}+a_{3n+1}+a_{3n+2}$, then sequence $(b_n)$ $\text{(A)}$ is an arithmetic sequence $\text{(B)}$ is a geometric series with common ratio of $q$ $\text{(C)}$ is a geometric series with common ratio of $q^3$ $\text{(D)}$ is neither an arithmetic sequence nor a geometric series

Let $n$ be a positive integer. Tasty and Stacy are given a circular necklace with $3n$ sapphire beads and $3n$ turquoise beads, such that no three consecutive beads have the same color. They play a cooperative game where they alternate turns removing three consecutive beads, subject to the following conditions: [list] [*]Tasty must remove three consecutive beads which are turquoise, sapphire, and turquoise, in that order, on each of his turns. [*]Stacy must remove three consecutive beads which are sapphire, turquoise, and sapphire, in that order, on each of her turns. [/list] They win if all the beads are removed in $2n$ turns. Prove that if they can win with Tasty going first, they can also win with Stacy going first. [i]Yannick Yao[/i]

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)f(1-x))=f(x)$ and $f(f(x))=1-f(x)$, for all real $x$.

A circle $k$ with center $M$ and radius $r$ is given. Find the locus of the incenters of all obtuse-angled triangles inscribed in $k$.

If $x + y = 306$ and $\frac{x}{y} = \frac{7}{10}$, compute $y - x$.

Let $n > 1$ be an odd natural number. The squares of an $n \times n$ chessboard are alternately colored white and black so that the four corner squares are black. An $L$-triomino is an $L$-shaped piece that covers exactly three squares of the board. For which values ​​of $n$ is it possible to cover all black squares with $L$-triominoes, so that no two $L$-triominos overlap? For these values ​​of $n$ determine the smallest possible number of $L$-triominoes that are necessary for this.

Given a positive integer $n$. The number $2n$ has $28$ positive factors, while the number $3n$ has $30$ positive factors. Find the number of positive divisors of $6n$.

Let $n \geq 2$ be an integer and $x_1, x_2, \ldots, x_n$ be positive real numbers such that $\sum_{i=1}^nx_i=1$. Show that $$\bigg(\sum_{i=1}^n\frac{1}{1-x_i}\bigg)\bigg(\sum_{1 \leq i < j \leq n}x_ix_j\bigg) \leq \frac{n}{2}.$$

Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$

In $\triangle ABC$, $BC=4$ and $CA=6$. If the perimeter of the triangle is $4$ times the length of side $BC$, what is the length of $AB$?

In the plane, a straight line $ m $ is given and points $ A $ and $ B $ lie on opposite sides of the straight line $ m $. Find a point $ M $ on the line $ m $ such that the difference in distances of this point from points $ A $ and $ B $ is as large as possible.

Let be two real numbers $ a<b, $ a natural number $ n\ge 2, $ and a continuous function $ f:[a,b]\longrightarrow (0,\infty ) $ whose image contains $ 1 $ and that admits a primitive $ F:[a,b]\longrightarrow [a,b] . $ Prove that there is a real number $ c\in (a,b) $ such that $$ (\underbrace{F\circ\cdots\circ F}_{\text{n times}} )(b) -(\underbrace{F\circ\cdots\circ F}_{\text{n times}} )(a) =(f(c))^{n+1} (b-a) $$ [i]Vlad Mihaly[/i]

Let $n$ be an integer. If the tens digit of $n^2$ is 7, what is the units digit of $n^2$?

$ (a)$ Prove that every multiple of $ 6$ can be written as a sum of four cubes. $ (b)$ Prove that every integer can be written as a sum of five cubes.

Prove that there are infinitely many integers $m$, $n$, such that $1 < m < n$, and the greatest common divisors $(m, n)$, $(m, n+1)$, $(m+1, n)$ and $(m+1, n+1)$ are all greater than $\sqrt{n}/999$.

Four distinct circles $C,C_1, C_2$, C3 and a line L are given in the plane such that $C$ and $L$ are disjoint and each of the circles $C_1, C_2, C_3$ touches the other two, as well as $C$ and $L$. Assuming the radius of $C$ to be $1$, determine the distance between its center and $L.$

Find all real numbers$ x$ and $y$ such that $$x^2 + y^2 = 2$$ $$\frac{x^2}{2 - y}+\frac{y^2}{2 - x}= 2.$$

Prove that $$\frac{2}{2!}+\frac{7}{3!}+\frac{14}{4!}+\frac{23}{5!}+...+\frac{k^2-2}{k!}+...+\frac{9998}{100!}<3$$ where $n! = 1 \times 2 \times ... \times n.$ (V Senderov)

A particle is launched from the surface of a uniform, stationary spherical planet at an angle to the vertical. The particle travels in the absence of air resistance and eventually falls back onto the planet. Spaceman Fred describes the path of the particle as a parabola using the laws of projectile motion. Spacewoman Kate recalls from Kepler’s laws that every bound orbit around a point mass is an ellipse (or circle), and that the gravitation due to a uniform sphere is identical to that of a point mass. Which of the following best explains the discrepancy? (A) Because the experiment takes place very close to the surface of the sphere, it is no longer valid to replace the sphere with a point mass. (B) Because the particle strikes the ground, it is not in orbit of the planet and therefore can follow a nonelliptical path. (C) Kate disregarded the fact that motions around a point mass may also be parabolas or hyperbolas. (D) Kepler’s laws only hold in the limit of large orbits. (E) The path is an ellipse, but is very close to a parabola due to the short length of the flight relative to the distance from the center of the planet.

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.