Let $1\le k\le n$. Suppose that the sequence $a_1, a_2, \ldots, a_n$ satisfies $0\le a_1 \le a_2 \le \ldots \le a_k$ and $0 \le a_n \le a_{n-1} \le \ldots \le a_k$. The sequence $b_1, b_2, \ldots, b_n$ is the nondecreasing permutation of $a_1, a_2, \ldots, a_n$. Prove that \[\sum_{i=1}^n \sum_{j=1}^n (j-i)^2a_ia_j \le \sum_{i=1}^n \sum_{j=1}^n (j-i)^2b_ib_j \]

Let $ABC$ be an acute triangle with orthocenter $H$ and altitudes $AA_1$, $BB_1$, $CC_1$. The lines $AB$ and $A_1B_1$ intersect at $C_2$ and $\ell_C$ is the line through the midpoint of $CH$, perpendicular to $CC_2$. The lines $\ell_A$ and $\ell_B$ are defined analogously. Prove that the lines $\ell_A$, $\ell_B$ and $\ell_C$ are concurrent.

The altitudes $AA_1$,$BB_1$,$CC_1$ of $\triangle{ABC}$ intersect at $H$.$O$ is the circumcenter of $\triangle{ABC}$.Let $A_2$ be the reflection of $A$ wrt $B_1C_1$.Prove that: a)$O$,$A_2$,$B_1$,$C$ are all on a circle b)$O$,$H$,$A_1$,$A_2$ are all on a circle

Prove that : \[\int_1^{\sqrt{e}} (\ln x)^n\ dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^{m}\]

You are given a standard kilogram mass and a tuning fork that is calibrated in Hz. You are also provided with a complete collection of laboratory equipment, but none of it is calibrated in SI units. You do not know the values of any fundamental constants. Which of the following quantities could you measure in SI units? (A) The acceleration due to gravity. (B) The speed of light in a vacuum. (C) The density of room temperature water. (D) The spring constant of a given spring. (E) The air pressure in the room.

The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD? [asy] size(250); defaultpen(linewidth(0.55)); pair A=(-6,0), B=origin, C=(0,6), D=(0,12); pair ac=C+2.828*dir(45), ca=A+2.828*dir(225), ad=D+2.828*dir(A--D), da=A+2.828*dir(D--A), ab=(2.828,0), ba=(-6-2.828, 0); fill(A--C--D--cycle, gray); draw(ba--ab); draw(ac--ca); draw(ad--da); draw((0,-1)--(0,15)); draw((1/3, -1)--(1/3, 15)); int i; for(i=1; i<15; i=i+1) { draw((-1/10, i)--(13/30, i)); } label("$A$", A, SE); label("$B$", B, SE); label("$C$", C, SE); label("$D$", D, SE); label("$3$", (1/3,3), E); label("$3$", (1/3,9), E); label("$3$", (-3,0), S); label("Main", (-3,0), N); label(rotate(45)*"Aspen", A--C, SE); label(rotate(63.43494882)*"Brown", A--D, NW); [/asy] $\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4.5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9$

Positive integers $1, 2, 3, ..., n$ are written on a blackboard ($n > 2$). Every minute two numbers are erased and the least prime divisor of their sum is written. In the end only the number $97$ remains. Find the least $n$ for which it is possible.

Yuri is looking at the great Mayan table. The table has $200$ columns and $2^{200}$ rows. Yuri knows that each cell of the table depicts the sun or the moon, and any two rows are different (i.e. differ in at least one column). Each cell of the table is covered with a sheet. The wind has blown aways exactly two sheets from each row. Could it happen that now Yuri can find out for at least $10000$ rows what is depicted in each of them (in each of the columns)? [i]Proposed by I. Bogdanov, K. Knop[/i]

The fields of the $8\times 8$ chessboard are numbered from $1$ to $64$ in the following manner: For $i=1,2,\cdots,63$ the field numbered by $i+1$ can be reached from the field numbered by $i$ by one move of the knight. Let us choose positive real numbers $x_{1},x_{2},\cdots,x_{64}$. For each white field numbered by $i$ define the number $y_{i}=1+x_{i}^{2}-\sqrt[3]{x_{i-1}^{2}x_{i+1}}$ and for each black field numbered by $j$ define the number $y_{j}=1+x_{j}^{2}-\sqrt[3]{x_{j-1}x_{j+1}^{2}}$ where $x_{0}=x_{64}$ and $x_{1}=x_{65}$. Prove that \[\sum_{i=1}^{64}y_{i}\geq 48\]

We are given convex quadrilateral $ABCD$. The midpoints of $BC$ and $DA$ are $M$ and $N$ respectively. The diagonal $AC$ divides $MN$ in half. Prove that the areas of triangles $ABC$ and $ACD$ are equal .

Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$. Points $X$ and $Y$ lie on sides $AB$ and $CD$, respectively. Suppose the circumcircles of $CDX$ and $ABY$ meet line $XY$ again at $P$ and $Q$ respectively. Show that $OP=OQ$. [i]Proposed by Evan Chang (squareman), USA[/i]

Let $J$ be the $A$-excenter of an acute triangle $ABC$. Let $X$, $Y$ be two points on the circumcircle of the triangle $ACJ$ such that $\overline{JX} = \overline{JY} < \overline{JC}$. Let $P$ be a point lies on $XY$ such that $PB$ is tangent to the circumcircle of the triangle $ABC$. Let $Q$ be a point lies on the circumcircle of the triangle $BXY$ such that $BQ$ is parallel to $AC$. Prove that $\angle BAP = \angle QAC$. [i]Proposed by Li4.[/i]

The points $A,B,$ and $C$ lies on the circumference of the unit circle. Furthermore, it is known that $AB$ is a diameter of the circle and \[\frac{|AC|}{|CB|}=\frac{3}{4}.\] The bisector of $ABC$ intersects the circumference at the point $D$. Determine the length of the $AD$.

The fuction $f(0,\infty)\rightarrow\mathbb{R}$ verifies $f(x)+f(y)=2f(\sqrt{xy}), \forall x,y>0$. Show that for every positive integer $n>2$ the following relation takes place $$f(x_1)+f(x_2)+\ldots+f(x_n)=nf(\sqrt[n]{x_1x_2\ldots x_n}),$$ for every positive integers $x_1,x_2,\ldots,x_n$.

Find the units of $R=\mathbb Z[t][\sqrt{t^2-1}]$.

Let $ABC$ be a triangle with $\angle C=90^\circ$. A line joining the midpoint of its altitude $CH$ and the vertex $A$ meets $CB$ at point $K$. Let $L$ be the midpoint of $BC$ ,and $T$ be a point of segment $AB$ such that $\angle ATK=\angle LTB$. It is known that $BC=1$. Find the perimeter of triangle $KTL$.

Let $ A$, $ B$, $ C$, $ A^{\prime}$, $ B^{\prime}$, $ C^{\prime}$, $ X$, $ Y$, $ Z$, $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$ and $ P$ be pairwise distinct points in space such that $ A^{\prime} \in BC;\ B^{\prime}\in CA;\ C^{\prime}\in AB;\ X^{\prime}\in YZ;\ Y^{\prime}\in ZX;\ Z^{\prime}\in XY;$ $ P \in AX;\ P\in BY;\ P\in CZ;\ P\in A^{\prime}X^{\prime};\ P\in B^{\prime}Y^{\prime};\ P\in C^{\prime}Z^{\prime}$. Prove that $ \frac {BA^{\prime}}{A^{\prime}C}\cdot\frac {CB^{\prime}}{B^{\prime}A}\cdot\frac {AC^{\prime}}{C^{\prime}B} \equal{} \frac {YX^{\prime}}{X^{\prime}Z}\cdot\frac {ZY^{\prime}}{Y^{\prime}X}\cdot\frac {XZ^{\prime}}{Z^{\prime}Y}$.

In $\triangle ABC$, $b\neq1$. If $\frac{C}{A}$ and $\frac{\sin B}{\sin A}$ are solutions to equation $\log_{\sqrt{b}}x=\log_{b}(4x-4)$, then $\triangle ABC$ $\text{(A)}$is an isosceles triangle, but not right-angled triangle $\text{(B)}$is a right-angled triangle, but not isosceles triangle $\text{(C)}$is an isosceles right-angled triangle $\text{(D)}$is neither a right-angled triangle nor an isosceles triangle

Let $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ be sequences of integers such that $a_1 = 20$, $b_1 = 15$, and for $n \ge 1$, \[\left\{\begin{aligned} a_{n+1}&=a_n^2-b_n^2, \\ b_{n+1}&=2a_nb_n-b_n^2 \end{aligned}\right.\] Let $G = a_{10}^2-a_{10}b_{10}+b_{10}^2$. Determine the number of positive integer factors of $G$. [i]Proposed by Michael Ren[/i]

Compute the infinite sum $\frac{1^3}{2^1}+\frac{2^3}{2^2}+\frac{3^3}{2^3}+\dots+\frac{n^3}{2^n}+\dots.$

Let $g$ and $h$ be two distinct elements of a group $G$, and let $n$ be a positive integer. Consider a sequence $w=(w_1,w_2,\dots)$ which is not eventually periodic and where each $w_i$ is either $g$ or $h$. Denote by $H$ the subgroup of $G$ generated by all elements of the form $w_kw_{k+1}\dotsc w_{k+n-1}$ with $k \ge 1$. Prove that $H$ does not depend on the choice of the sequence $w$ (but may depend on $n$).

Let $p$ be a prime number greater than 5. Suppose there is an integer $k$ satisfying that $k^2+5$ is divisible by $p$. Prove that there are positive integers $m$ and $n$ such that $p^2=m^2+5n^2$

Call a polynomial $f(x)$ [i]excellent[/i] if its coefficients are all in [0, 1) and $f(x)$ is an integer for all integers $x$. a) Compute the number of excellent polynomials with degree at most 3. b) Compute the number of excellent polynomials with degree at most $n$, in terms of $n$. c) Find the minimum $n\ge3$ for which there exists an excellent polynomial of the form $\frac{1}{n!}x^n+g(x)$, where $g(x)$ is a polynomial of degree at most $n-3$.

Let $V$ be the set of all continuous functions $f\colon [0,1]\to \mathbb{R}$, differentiable on $(0,1)$, with the property that $f(0)=0$ and $f(1)=1$. Determine all $\alpha \in \mathbb{R}$ such that for every $f\in V$, there exists some $\xi \in (0,1)$ such that \[f(\xi)+\alpha = f'(\xi)\]

(a) On a long pavement, a sequence of $999$ integers is written in chalk. The numbers need not be in increasing order and need not be distinct. Merlijn encircles $500$ of the numbers with red chalk. From left to right, the numbers circled in red are precisely the numbers $1, 2, 3, ...,499, 500$. Next, Jeroen encircles $500$ of the numbers with blue chalk. From left to right, the numbers circled in blue are precisely the numbers $500, 499, 498, ...,2,1$. Prove that the middle number in the sequence of $999$ numbers is circled both in red and in blue. (b) Merlijn and Jeroen cross the street and find another sequence of $999$ integers on the pavement. Again Merlijn circles $500$ of the numbers with red chalk. Again the numbers circled in red are precisely the numbers $1, 2, 3, ...,499, 500$ from left to right. Now Jeroen circles $500$ of the numbers, not necessarily the same as Merlijn, with green chalk. The numbers circled in green are also precisely the numbers $1, 2, 3, ...,499, 500$ from left to right. Prove: there is a number that is circled both in red and in green that is not the middle number of the sequence of $999$ numbers.