Let $a_1,a_2,\ldots,a_n$ be positive real numbers. Denote $$s=\sum_{k=1}^na_k\text{ and }s'=\sum_{k=1}^na_k^{1-\frac1k}.$$ (a) Let $\lambda>1$ be a real number. Show that $s'<\lambda s+\frac\lambda{\lambda-1}$. (b) Deduce that $\sqrt{s'}<\sqrt s+1$.

In $\vartriangle ABC$, we have $AC = BC = 7$ and $AB = 2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 8$. What is the length of the segment $BD$?

Point $P$ is located inside traingle $ABC$ so that angles $PAB, PBC,$ and $PCA$ are all congruent. The sides of the triangle have lengths $AB=13, BC=14,$ and $CA=15,$ and the tangent of angle $PAB$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Let $P(x)$ be a real polynomial with $P(x) \ge 0$ for $0 \le x \le 1$. Show that there exist polynomials $P_i (x) (i = 0, 1,2)$ with $P_i (x) \ge 0$ for all real x such that $P (x) = P_0 (x) + xP_1 (x)( 1- x)P_2 (x)$.

Find the minimum possible least common multiple of twenty natural numbers whose sum is $801$.

Find all postitive integers n such that $$\left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor=n^2$$ where $\lfloor x \rfloor$ represents the largest integer less than the real number $x$.

Let $N, P$ be the centers of the faces A$BB'A'$ and $ADD'A'$, respectively, of a right parallelepiped $ABCDA'B'C'D'$ and $M \in (A'C)$ such that $A'M= \frac13 A' C$. Prove that $MN \perp AB'$ and $ MP \perp AD' $ if and only if the parallelepiped is a cube.

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. [i]Author: Kei Irie, Japan[/i]

Among the coordinates $(x,y)$ $(1\leq x,y\leq 101)$, choose some points such that there does not exist $4$ points which form a isoceles trapezium with its base parallel to either the $x$ or $y$ axis(including rectangles). Find the maximum number of coordinate points that can be chosen.

For $n\in\mathbb{N}$ prove that \[\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n}\leq\frac{1}{\sqrt{2n+1}}.\]

Find all pairs $\left(a,b\right)$ of posive integers such that $\frac{a^{2}\left(b-a\right)}{b+a}$ is square of prime.

Prove that $\frac{1}{2} \frac{3}{4} \frac{5}{6} \frac{7}{8} ... \frac{99}{100 } <\frac{1}{10}$.

Suppose $a$ is a real number such that the equation $$a\cdot(\sin x+\sin(2x))=\sin(3x)$$ has more than one solution in the interval $(0,\pi)$. The set of all such $a$ can be written in the form $(p,q)\cup(q,r)$, where $p$, $q$, and $r$ are real numbers with $p<q<r$. What is $p+q+r$? $\textbf{(A) }-4\qquad\textbf{(B) }-1\qquad\textbf{(C) }0\qquad\textbf{(D) }1\qquad\textbf{(E) }4$

Find the least prime factor of the number $\frac{2016^{2016}-3}{3}$.

Distinct positive integers $A$ and $B$ are given$.$ Prove that there exist infinitely many positive integers that can be represented both as $x_{1}^2+Ay_{1}^2$ for some positive coprime integers $x_{1}$ and $y_{1},$ and as $x_{2}^2+By_{2}^2$ for some positive coprime integers $x_{2}$ and $y_{2}.$ [i](Golovanov A.S.)[/i]

Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that $\bullet$ all of his happy snakes can add $\bullet$ none of his purple snakes can subtract, and $\bullet$ all of his snakes that can’t subtract also can’t add Which of these conclusions can be drawn about Tom’s snakes? $\textbf{(A)}$ Purple snakes can add. $\textbf{(B)}$ Purple snakes are happy. $\textbf{(C)}$ Snakes that can add are purple. $\textbf{(D)}$ Happy snakes are not purple. $\textbf{(E)}$ Happy snakes can't subtract.

Let the lines $e$ and $f$ be perpendicular and intersect each other at $H$. Let $A$ and $B$ lie on $e$ and $C$ and $D$ lie on $f$, such that all five points $A,B,C,D$ and $H$ are distinct. Let the lines $b$ and $d$ pass through $B$ and $D$ respectively, perpendicularly to $AC$; let the lines $a$ and $c$ pass through $A$ and $C$ respectively, perpendicularly to $BD$. Let $a$ and $b$ intersect at $X$ and $c$ and $d$ intersect at $Y$. Prove that $XY$ passes through $H$.

Consider a triangle $\vartriangle ACE$ with $\angle ACE = 45^o$ and $\angle CEA = 75^o$. Define points $Q, R$, and $P$ such that $AQ$, $CR$, and $EP$ are the altitudes of $\vartriangle ACE$. Let $H$ be the intersection of $AQ$, $CR$, and $EP$. Next define points $B, D$, and $F$ as follows. Extend $EP$ to point $B$ such that $BP = HP$, extend $AQ$ to point $D$ such that $DQ = HQ$, and extend $CR$ to point $F$ such that $F R = HR$. Finally, lengths $CH = 2$, $AH =\sqrt2$, and $EH =\sqrt3 - 1$. Compute the area of hexagon $ABCDEF$.

Show that for each positive integer $n$, there is a positive integer $k$ such that the decimal representation of each of the numbers $k, 2k,\ldots, nk$ contains all of the digits $0, 1, 2,\ldots, 9$.

Let $\{a_i\}_{i=0}^\infty$ be a sequence of real numbers such that \[\sum_{n=1}^\infty\dfrac {x^n}{1-x^n}=a_0+a_1x+a_2x^2+a_3x^3+\cdots\] for all $|x|<1$. Find $a_{1000}$. [i]Proposed by David Altizio[/i]

Find the smallest integer $n$ for which it's possible to cut a square into $2n$ squares of two sizes: $n$ squares of one size, and $n$ squares of another size. [i](Proposed by Bogdan Rublov)[/i]

For a $\triangle ABC$ , let $A_1$ be the symmetric point of the intersection of angle bisector of $\angle BAC$ and $BC$ , where center of the symmetry is the midpoint of side $BC$, In the same way we define $B_1 $ ( on $AC$ ) and $C_1$ (on $AB$). Intersection of circumcircle of $\triangle A_1B_1C_1$ and line $AB$ is the set $\{Z,C_1 \}$, with $BC$ is the set $\{X,A_1\}$ and with $CA$ is the set $\{Y,B_1\}$. If the perpendicular lines from $X,Y,Z$ on $BC,CA$ and $ AB$ , respectively are concurrent , prove that $\triangle ABC$ is isosceles.

Let $m$ be a positive integer, and consider a $m\times m$ checkerboard consisting of unit squares. At the centre of some of these unit squares there is an ant. At time $0$, each ant starts moving with speed $1$ parallel to some edge of the checkerboard. When two ants moving in the opposite directions meet, they both turn $90^{\circ}$ clockwise and continue moving with speed $1$. When more than $2$ ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear. Considering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard, or prove that such a moment does not necessarily exist. [i]Proposed by Toomas Krips, Estonia[/i]

The [i]fibboican[/i] sequence $a_1,\ a_2,\ \dots$, is defined by $a_1 = a_2 = 1$, and for integers $k \geq 3$, [list] [*] $a_k = a_{k-1} + a_{k-2}$ if $k$ is odd [*] $\frac {1}{a_k} = \frac {1}{a_{k-1}} + \frac {1}{a_{k-2}}$ if $k$ is even. [/list] Prove that, for each integer $m\ge 1$, the numerator of $a_m$ (when written in simplest form) is a power of $2$. [i]Eric Shen (CAN)[/i]

What is the value of $1 + 7 + 21 + 35 + 35 + 21 + 7 + 1$?