Let the vertices of the square $ABCD$ are on a circle of radius $r$ and with center $O$. Let $P, Q, R$ and $S$ are the mid points of $AB, BC, CD$ and $DA$ respectively. Then; (a) Show that the quadrilateral $P QRS$ is a square. (b) Find the distance from the mid point of $P Q$ to $O$.

Call a positive rational number in simplest terms [i]coddly[/i] if its numerator and denominator are both odd. Consider the equation $$2017= x_1\text{ }\square\text{ }x_2\text{ }\square\text{ }x_3\text{ }\ldots \text{ }\square \text{ }x_{2016} \text{ }\square \text{ }x_{2017},$$ where there are $2016$ boxes. We fill up the boxes randomly with the operations $+$, $-$, and $\times$. Compute the probability that there exists a solution in [b]distinct[/b] coddly numbers $(x_1,x_2, \ldots x_{2017})$ to the resulting equation.

Find all positive integers that are within $250$ of exactly $15$ perfect squares.

Milo rolls five fair dice which have 4, 6, 8, 12, and 20 sides respectively (and each one is labeled $1$-$n$ for appropriate $n$. How many distinct ways can they roll a full house (three of one number and two of another)? The same numbers appearing on different dice are considered distinct full houses, so $(1,1,1,2,2)$ and $(2,2,1,1,1)$ would both be counted. [i]Proposed by Robert Trosten[/i]

Circles $k_1$ and $k_2$ with radii $r_1=6$ and $r_2=3$ are externally tangent and touch a circle $k$ with radius $r=9$ from inside. A common external tangent of $k_1$ and $k_2$ intersects $k$ at $P$ and $Q$. Determine the length of $PQ$.

Prove the identity: $$\cos \frac{\pi}{7} \cdot \cos \frac{4\pi}{7} \cdot \cos \frac{5\pi}{7} = \frac{1}{8}$$

Let $a_1,a_2,\dots,a_n$ be positive real numbers whose product is $1$. Show that the sum \[\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}\] is greater than or equal to $\frac{2^n-1}{2^n}$.

Let us call a positive integer $k{}$ interesting if the product of the first $k{}$ primes is divisible by $k{}$. For example the product of the first two primes is $2\cdot3 = 6$, it is divisible by 2, hence 2 is an interesting integer. What is the maximal possible number of consecutive interesting integers? [i]Boris Frenkin[/i]

Show that $$ \int_{0}^{1} x^{x} \, dx = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^n }.$$

Let $ u_1, u_2, \ldots, u_m$ be $ m$ vectors in the plane, each of length $ \leq 1,$ with zero sum. Show that one can arrange $ u_1, u_2, \ldots, u_m$ as a sequence $ v_1, v_2, \ldots, v_m$ such that each partial sum $ v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $ \sqrt {5}.$

Consider the square $ABCD$ and the point $E$ inside the angle $CAB$, such that $\angle BAE =15^o$, and the lines $BE$ and $BD$ are perpendicular. Prove that $AE = BD$.

$\textbf{Bake Sale}$ Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown. $\circ$ Art's cookies are trapezoids: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label("5 in", (2.5,0), S); label("3 in", (5,1.5), E); label("3 in", (3.5,3), N);[/asy] $\circ$ Roger's cookies are rectangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label("4 in", (2,0), S); label("2 in", (4,1), E);[/asy] $\circ$ Paul's cookies are parallelograms: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label("3 in", (1.5,0), S); label("2 in", (2.5,1), W);[/asy] $\circ$ Trisha's cookies are triangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label("3 in", (1.5,0), S); label("4 in", (3,2), E);[/asy] Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Art's cookies sell for 60 cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents? $ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 75\qquad\textbf{(E)}\ 90$

Let $O$ be the center and $R$ be the radius of circumcircle $\omega$ of triangle $ABC$. Circle $\omega_1$ with center $O_1$ and radius $R$ pass through points $A, O$ and intersects the side $AC$ at point $K$. Let $AF$ be the diameter of circle $\omega$ and points $F, K, O_1$ are collinear. Determine $\angle ABC$:

(a) $K_1,K_2,..., K_n$ are the feet of the perpendiculars from an arbitrary point $M$ inside a given regular $n$-gon to its sides (or sides produced). Prove that the sum $\overrightarrow{MK_1} + \overrightarrow{MK_2} + ... + \overrightarrow{MK_n}$ equals $\frac{n}{2}\overrightarrow{MO}$, where $O$ is the centre of the $n$-gon. (b) Prove that the sum of the vectors whose origin is an arbitrary point $M$ inside a given regular tetrahedron and whose endpoints are the feet of the perpendiculars from $M$ to the faces of the tetrahedron equals $\frac43 \overrightarrow{MO}$, where $O$ is the centre of the tetrahedron. (VV Prasolov, Moscow)

Let $n\geq 2$ be an integer and $a_1,a_2,\cdots,a_n>1$ be real numbers. Prove that the inequality below holds. $$\prod_{i=1}^n\left(a_ia_{i+1}-\frac{1}{a_ia_{i+1}}\right)\geq 2^n\prod_{i=1}^n\left(a_i-\frac{1}{a_i}\right)$$

Suppose that $a_1, a_2, a_3, \ldots$ is an infinite geometric sequence such that for all $i \ge 1$, $a_i$ is a positive integer. Suppose furthermore that $a_{20} + a_{21} = 20^{21}$. If the minimum possible value of $a_1$ can be expressed as $2^a 5^b$ for positive integers $a$ and $b$, find $a + b$. [i]Proposed by Andrew Wu[/i]

Let $ ABC$ be a triangle with $ AB \equal{} AC$. A ray $ Ax$ is constructed in space such that the three planar angles of the trihedral angle $ ABCx$ at its vertex $ A$ are equal. If a point $ S$ moves on $ Ax$, find the locus of the incenter of triangle $ SBC$.

Find the value of $x+y$ for which the expression \[\frac{6x^2}{y^6} + \frac{6y^2}{x^6}+9x^2y^2+\frac{4}{x^6y^6}\] is minimized.

Is there a closed self-intersecting broken line in space that intersects each of its links exactly once, and in its midpoint ?

In $\triangle ABC$, $P,Q,R$ divides the perimeter of $\triangle ABC$ into three equal parts. $P,Q\in AB$. Prove that $\frac{S_{\triangle PQR}}{S_{\triangle ABC}}>\frac{2}{9}$.

Given $\frac{\log{a}}{p}=\frac{\log{b}}{q}=\frac{\log{c}}{r}=\log{x}$, all logarithms to the same base and $x \not= 1$. If $\frac{b^2}{ac}=x^y$, then $y$ is: $ \text{(A)}\ \frac{q^2}{p+r}\qquad\text{(B)}\ \frac{p+r}{2q}\qquad\text{(C)}\ 2q-p-r\qquad\text{(D)}\ 2q-pr\qquad\text{(E)}\ q^2-pr$

The numbers $3$, $4$ ,$...$, $2019$ are written on a blackboard. David and Edgardo play alternately, starting with David. On their turn, each player must erase a number from the board and write two positive integers whose sum is equal to the number just deleted. The winner is the one who makes all the numbers on the board equal. Determine who has a winning strategy and describe it.

Suppose a $3\times 3$ matrix $A$ satisfies $\mathbf{v}^t A \mathbf{v} > 0$ for any vector $\mathbf{v} \in\mathbb{R}^3 -\{0\}$. (Note that $A$ may not be a symmetric matrix.) (1) Prove that $\det(A)>0$. (2) Consider diagonal matrix $D=\text{diag}(-1,1,1)$. Prove that there's exactly one negative real among eigenvalues of $AD$.

Find the minimum area bounded by the graphs of $ y\equal{}x^2$ and $ y\equal{}kx(x^2\minus{}k)\ (k>0)$.

If $a$ and $b$ are integers and if the solutions of the system of equations $$y - 2x - a = 0$$ $$y^2 - xy + x^2 - b = 0$$ are rational, prove that the solutions are integers.