Let $ABCD$ be a cyclic quadrilateral and let $K,L,M,N,S,T$ the midpoints of $AB, BC, CD, AD, AC, BD$ respectively. Prove that the circumcenters of $KLS, LMT, MNS, NKT$ form a cyclic quadrilateral which is similar to $ABCD$.

Let \[T=\{(tq,1-t) \in\mathbb R^{2}| t \in [0,1],q\in\mathbb Q\}\]Prove that each continuous function $f: T\longrightarrow T$ has a fixed point.

Each face of a cube is painted either red or blue, each with probability $ 1/2$. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color? $ \textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac{5}{16} \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac{7}{16} \qquad \textbf{(E)}\ \frac12$

Evaluate $\int_0^{\frac{\pi}{2}} \sqrt{1-2\sin 2x+3\cos ^ 2 x}\ dx.$ [i]2011 University of Occupational and Environmental Health/Medicine　entrance exam[/i]

Find all functions $f:\mathbb R\to\mathbb R$ such that $$f\left( x^2+xf(y)\right)=xf(x+y)$$ for all reals $x,y$.

Find all second degree polynomials $P(x)$ such that for all $a \in\mathbb{R} , a \geq 1$, then $P(a^2+a) \geq a.P(a+1)$

Points $X_1$ and $X_2$ move along fixed circles with centers $O_1$ and $O_2$, respectively, so that $O_1X_1 \parallel O_2X_2$. Find the locus of the intersection point of lines $O_1X_2$ and $O_2X_1$.

Let $ABC$ be a non isosceles triangle with circumcircle $(O)$ and incircle $(I)$. Denote $(O_1)$ as the circle internal tangent to $(O)$ at $A_1$ and also tangent to segments $AB,AC$ at $A_b,A_c$ respectively. Define the circles $(O_2), (O_3)$ and the points $B_1, C_1, B_c , B_a, C_a, C_b$ similarly. 1. Prove that $AA_1, BB_1, CC_1$ are concurrent at the point $M$ and $3$ points $I,M,O$ are collinear. 2. Prove that the circle $(I)$ is inscribed in the hexagon with $6$ vertices $A_b,A_c , B_c , B_a, C_a, C_b$.

Two subsets $A$ and $B$ of the $(x,y)$-plane are said to be [i]equivalent[/i] if there exists a function $f: A\to B$ which is both one-to-one and onto. (i) Show that any two line segments in the plane are equivalent. (ii) Show that any two circles in the plane are equivalent.

The triplet of positive integers $(a,b,c)$ with $a<b<c$ is called a [i]fatal[/i] triplet if there exist three nonzero integers $p,q,r$ which satisfy the equation $a^p b^q c^r = 1$. As an example, $(2,3,12)$ is a fatal triplet since $2^2 \cdot 3^1 \cdot (12)^{-1} = 1$. The positive integer $N$ is called [i]fatal[/i] if there exists a fatal triplet $(a,b,c)$ satisfying $N=a+b+c$. (a) Prove that 16 is not [i]fatal[/i]. (b) Prove that all integers bigger than 16 which are [b]not[/b] an integer multiple of 6 are fatal.

Given a triangle $ABC$ isosceles at $A.$ A point $P$ lying inside the triangle such that $\angle PBC=\angle PCA$ and let $M$ be the midpoint of $BC.$ Prove that: $\angle APB+ \angle MPC =180^{\circ}.$

Let $\omega_1$ be a circle of radius 6, and let $\omega_2$ be a circle of radius 5 that passes through the center $O$ of $\omega_1$. Let $A$ and $B$ be the points of intersection of the two circles, and let $P$ be a point on major arc $AB$ of $\omega_2$. Let $M$ and $N$ be the second intersections of $PA$ and $PB$ with $\omega_1$, respectively. Let $S$ be the midpoint of $MN$. As $P$ ranges over major arc $AB$ of $\omega_2$, the minimum length of segment $SA$ is $a/b$, where $a$ and $b$ are positive integers and $\gcd(a, b) = 1$. Find $a+b$.

Find all pairs of relatively prime numbers ($x, y$) such that $x^2(x + y)$ is divisible by $y^2(y - x)^2$. .

Let $AD$ be an altitude of triangle $ABC$, and let $M$, $N$ and $P$ be midpoints of $AB$, $AD$ and $BC$, respectively. Furthermore let $K$ be a foot of perpendicular from point $D$ to line $AC$, and let $T$ be point on extension of line $KD$ (over point $D$) such that $\mid DT \mid = \mid MN \mid + \mid DK \mid$. If $\mid MP \mid = 2 \cdot \mid KN \mid$, prove that $\mid AT \mid = \mid MC \mid$.

Let $ R $ be the circumradius of a triangle $ ABC. $ The points $ B,C, $ lie on a circle of radius $ \rho $ that intersects $ AB,AC $ at $ E,D, $ respectively. $ \rho' $ is the circumradius of $ ADE. $ Show that there exists a triangle with sides $ R,\rho ,\rho' , $ and having an angle whose value doesn't depend on $ \rho . $ [i]Laurențiu Panaitopol[/i]

We say that a point inside a triangle is good if the lengths of the cevians passing through this point are inversely proportional to the respective side lengths. Find all the triangles for which the number of good points is maximal. (A.Zaslavsky, B.Frenkin)

Let two natural number $n > 1$ and $m$ be given. Find the least positive integer $k$ which has the following property: Among $k$ arbitrary integers $a_1, a_2, \ldots, a_k$ satisfying the condition $a_i - a_j$ ( $1 \leq i < j \leq k$) is not divided by $n$, there exist two numbers $a_p, a_s$ ($p \neq s$) such that $m + a_p - a_s$ is divided by $n$.

Find all real parameters $a$ for which all the roots of the equation $$x^6+3x^5+(6-a)x^4+(7-2a)x^3+(6-a)x^2+3x+1$$are real.

How many two-digit positive integers have at least one $ 7$ as a digit? $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 18\qquad \textbf{(C)}\ 19\qquad \textbf{(D)}\ 20\qquad \textbf{(E)}\ 30$

In an acute-angled triangle $ ABC$, $ D$ is the foot of the altitude from $ A$, and $ P$ a point on segment $ AD$. The lines $ BP$ and $ CP$ meet $ AC$ and $ AB$ at $ E$ and $ F$ respectively. Prove that $ AD$ bisects the angle $ EDF$.

Let $n$ be a positive integer and let $(a_1,a_2,\ldots ,a_{2n})$ be a permutation of $1,2,\ldots ,2n$ such that the numbers $|a_{i+1}-a_i|$ are pairwise distinct for $i=1,\ldots ,2n-1$. Prove that $\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\}$ if and only if $a_1-a_{2n}=n$.

Find all positive integers $n$ that have precisely $\sqrt{n+1}$ natural divisors.

Mellon Game Lab has come up with a concept for a new game: Square Finder. The premise is as follows. You are given an $n\times n$ grid of squares (for integer $n\geq 2$), each of which is either blank or has an arrow pointing up, down, left, or right. You are also given a $2\times 2$ grid of squares that appears somewhere in this grid, possibly rotated. For example, see if you can find the following $2\times 2$ grid inside the larger $4\times 4$ grid. [asy] size(2cm); defaultpen(fontsize(16pt)); string b = ""; string u = "$\uparrow$"; string d = "$\downarrow$"; string l = "$\leftarrow$"; string r = "$\rightarrow$"; // input should be n x n string[][] input = {{b,u},{r,l}}; int n = input.length; // draw table for (int i=0; i<=n; ++i) { draw((i,0)--(i,n)); draw((0,i)--(n,i)); } // fill table for (int i=1; i<=n; ++i) { for (int j=1; j<=n; ++j) { label(input[i-1][j-1], (j-0.5,n-i+0.5)); } } [/asy] [asy] size(4cm); defaultpen(fontsize(16pt)); string b = ""; string u = "$\uparrow$"; string d = "$\downarrow$"; string l = "$\leftarrow$"; string r = "$\rightarrow$"; // input should be n x n string[][] input = {{u,b,b,r},{b,r,u,d},{d,b,u,b},{u,r,b,l}}; int n = input.length; // draw table for (int i=0; i<=n; ++i) { draw((i,0)--(i,n)); draw((0,i)--(n,i)); } // fill table for (int i=1; i<=n; ++i) { for (int j=1; j<=n; ++j) { label(input[i-1][j-1], (j-0.5,n-i+0.5)); } } [/asy] Did you spot it? It's in the bottom left, rotated by $90^\circ$ clockwise. To make the game as interesting as possible, Mellon Game Lab would like the grid to be as large as possible and for no $2\times 2$ grid to appear more than once in the big grid. The grid above doesn't work, as the following $2\times 2$ grid appears twice, once in the top left corner (rotated $90^\circ$ counterclockwise) and once directly below it (overlapping). [asy] size(2cm); defaultpen(fontsize(16pt)); string b = ""; string u = "$\uparrow$"; string d = "$\downarrow$"; string l = "$\leftarrow$"; string r = "$\rightarrow$"; // input should be n x n string[][] input = {{b,r},{d,b}}; int n = input.length; // draw table for (int i=0; i<=n; ++i) { draw((i,0)--(i,n)); draw((0,i)--(n,i)); } // fill table for (int i=1; i<=n; ++i) { for (int j=1; j<=n; ++j) { label(input[i-1][j-1], (j-0.5,n-i+0.5)); } } [/asy] Let's call a grid that avoids such repeats a [i]repeat-free grid[/i]. We are interested in finding out for which $n$ constructing an $n\times n$ repeat-free grid is possible. Here's what we know so far. [list] [*] Any $2\times 2$ grid is repeat-free, as there is only one subgrid to worry about, and there can't possibly be any repeats. [*] If we can construct an $n\times n$ repeat-free grid, we can also construct a $k\times k$ repeat-free grid for any $k\leq n$ by just taking the top left $k\times k$ of the original one we found. [*] By the previous observation, if it is impossible to construct such an $n\times n$ repeat-free grid, we cannot construct a $k\times k$ repeat-free grid for any $k\geq n$, as otherwise we could take the top left $n\times n$ to get one working for $n$. [/list] These three observations together tell us that either we can construct an $n\times n$ repeat-free grid for all $n\geq 2$, or there exists some upper limit $N\geq 2$ such that we can construct an $n\times n$ repeat-free grid for all $n\leq N$ but cannot construct one for any $n> N$. Your goal is to determine if such an $N$ exists, and if so, place bounds on its value. More precisely, this problem consists of two parts: a lower bound and an upper bound. For the lower bound, to show that $N\geq n$ for some $n$, you need to construct an $n\times n$ repeat-free grid (you do not need to prove your construction works). For the upper bound, to show that $N$ is at most some value $n$, you must prove that it is impossible to construct an $(n+1)\times (n+1)$ repeat-free grid. [i]Proposed by Connor Gordon and Eric Oh[/i]

Solve in positive integers the following equation $$\left [\sqrt{1}\right]+\left [\sqrt{2}\right]+\left [\sqrt{3}\right]+\ldots+\left [\sqrt{x^2-2}\right]+\left [\sqrt{x^2-1}\right]=125,$$ where $[a]$ is the integer part of the real number $a$.

Prove that for any natural numbers$a,b$ there exist infinity many prime numbers $p$ so that $Ord_p(a)=Ord_p(b)$(Proving that there exist infinity prime numbers $p$ so that $Ord_p(a) \ge Ord_p(b)$ will get a partial mark)