The diagram shows two intersecting line segments that form some of the sides of two squares with side lengths $3$ and $6$. Two line segments join vertices of these squares. Find the area of the region enclosed by the squares and segments.

Given a cyclic quadrilateral $ABCD$. Let $X$ be a point on segment $BC$ ($X \not= BC$) such that line $AX$ is perpendicular to the angle bisector of $\angle CBD$, and $Y$ be a point on segment $AD$ ($Y \not= D)$ such that $BY$ is perpendicular to the angle bisector of $\angle CAD$. Prove that $XY$ is parallel to $CD$.

Prove that a prime of the form $2^{2^n}+1$ cannot be the difference of two fifth powers of two positive integers. Pierre.

There are $ n \geq4 $ points on the plane, the distance between any two of which is an integer. Prove that there are at least $ \frac {1} {6} $ distances, each of which is divisible by $3$.

Let be two positive real numbers $ a,b $ whose product is $ 1$ and whose sum is irrational. Prove that for any natural number $ n\ge 2 $ the epression $ \sqrt[n]{a}+\sqrt[n]{b} $ is irrational. [i]Râmbu Gheorghe[/i]

Show that for every prime $p$ and integer $n$, there is an irreducible polynomial of degree $n$ in $\mathbb{Z}_p[x]$ and use that to show there is a field of size $p^n$.

Let $F_{n}$ be the $n-$ th Fibonacci number (where $F_{1}= F_{2}= 1$). Consider the functions $f_{n}(x)=\parallel . . . \parallel |x|-F_{n}|-F_{n-1}|-...-F_{2}|-F_{1}|, g_{n}(x)=| . . . \parallel x-1|-1|-...-1|$ ($F_{1}+...+F_{n}$ one’s). Show that $f_{n}(x) = g_{n}(x)$ for every real number $x.$

Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode? $\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad\textbf{(E) }13$

Vessel $A$ has liquids $X$ and $Y$ in the ratio $X:Y=8:7$. Vessel $B$ holds a mixture of $X$ and $Y$ in the ratio $X:Y=5:9$. What ratio should you mix the liquids in both vessels if you need the mixture to be $X:Y=1:1$? $\textbf{(A)}~4:3$ $\textbf{(B)}~30:7$ $\textbf{(C)}~17:25$ $\textbf{(D)}~7:30$

Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$ Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$

Show that there exist infinitely many pairs $(a, b)$ of positive integers with the property that $a+b$ divides $ab+1$, $a-b$ divides $ab-1$, $b>1$ and $a>b\sqrt{3}-1$

A given sequence $r_1, r_2, \dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more ``bubble passes''. A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then comparing the third term with the second term and exchanging them if and only if the third term is smaller, and so on in order, through comparing the last term, $r_n$, with its current predecessor and exchanging them if and only if the last term is smaller. The example below shows how the sequence 1, 9, 8, 7 is transformed into the sequence 1, 8, 7, 9 by one bubble pass. The numbers compared at each step are underlined. \[ \begin{array}{c} \underline{1 \quad 9} \quad 8 \quad 7 \\ 1 \quad \underline{9 \quad 8} \quad 7 \\ 1 \quad 8 \quad \underline{9 \quad 7} \\ 1 \quad 8 \quad 7 \quad 9 \end{array} \] Suppose that $n = 40$, and that the terms of the initial sequence $r_1, r_2, \dots, r_{40}$ are distinct from one another and are in random order. Let $p/q$, in lowest terms, be the probability that the number that begins as $r_{20}$ will end up, after one bubble pass, in the $30^{\text{th}}$ place. Find $p + q$.

In an isosceles, obtuse-angled triangle, the lengths of two internal angle bisectors are in a $2:1$ ratio. Find the obtuse angle of the triangle.

Semicircle $\stackrel{\frown}{AB}$ has center $C$ and radius $1$. Point $D$ is on $\stackrel{\frown}{AB}$ and $\overline{CD} \perp \overline{AB}$. Extend $\overline{BD}$ and $\overline{AD}$ to $E$ and $F$, respectively, so that circular arcs $\stackrel{\frown}{AE}$ and $\stackrel{\frown}{BF}$ have $B$ and $A$ as their respective centers. Circular arc $\stackrel{\frown}{EF}$ has center $D$. The area of the shaded "smile", $AEFBDA$, is [asy] size(200); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair A=(-1,0), B=(1,0), D=(0,-1), C=(0,0), E=(1-sqrt(2),-sqrt(2)), F=(-1+sqrt(2),-sqrt(2)); fill(Arc((1,0),2,180,225)--Arc((0,-1),(2-sqrt(2)),225,315)--Arc((-1,0),2,315,360)--Arc((0,0),1,360,180)--cycle,mediumgray); draw(A--B^^C--D^^A--F^^B--E); draw(Arc((1,0),2,180,225)--Arc((0,-1),(2-sqrt(2)),225,315)--Arc((-1,0),2,315,360)--Arc((0,0),1,360,180)); label("$A$",A,N); label("$B$",B,N); label("$C$",C,N); label("$D$",(-0.1,-.7)); label("$E$",E,SW); label("$F$",F,SE); [/asy] $ \textbf{(A)}\ (2 - \sqrt{2})\pi\qquad\textbf{(B)}\ 2\pi - \pi\sqrt{2} - 1\qquad\textbf{(C)}\ \left(1 - \frac{\sqrt{2}}{2}\right)\pi\qquad\textbf{(D)}\ \frac{5\pi}{2} - \pi\sqrt{2} - 1\qquad\textbf{(E)}\ (3 - 2\sqrt{2})\pi $

Prove that the set of all linearly combinations (with real coefficients) of the system of polynomials $ \{ x^n\plus{}x^{n^2} \}_{n\equal{}0}^{\infty}$ is dense in $ C[0,1]$. [i]J. Szabados[/i]

A sphere of radius $1$ is covered in ink and rolling around between concentric spheres of radii $3$ and $5$. If this process traces a region of area 1 on the larger sphere, what is the area of the region traced on the smaller sphere?

$k$ is a positive integer. $A$ company has a special method to sell clocks. Every customer can reason with two customers after he has bought a clock himself $;$ it's not allowed to reason with an agreed person. These new customers can reason with other two persons and it goes like this.. If both of the customers agreed by a person could play a role (it can be directly or not) in buying clocks by at least $k$ customers, this person gets a present. Prove that, if $n$ persons have bought clocks, then at most $\frac{n}{k+2}$ presents have been accepted.

In the figure below $\angle$LAM = $\angle$LBM = $\angle$LCM = $\angle$LDM, and $\angle$AEB = $\angle$BFC = $\angle$CGD = 34 degrees. Given that $\angle$KLM = $\angle$KML, find the degree measure of $\angle$AEF. This is #8 on the 2015 Purple comet High School. For diagram go to http://www.purplecomet.org/welcome/practice

A triple of positive integers $(a, b, c)$ is [i]tasty [/i] if $lcm (a, b, c) | a + b + c - 1$ and $a < b < c$. Find the sum of $a + b + c$ across all tasty triples.

Find all polynomials $W$ with integer coefficients satisfying the following condition: For every natural number $n, 2^n - 1$ is divisible by $W(n).$

Right triangle $ABC$ has hypotenuse $AB = 26$, and the inscribed circle of $ABC$ has radius $5$. The largest possible value of $BC$ can be expressed as $m + \sqrt{n}$, where $m$ and $n$ are both positive integers. Find $100m + n$. [i]Proposed by Jason Xia[/i]

Prove that from every set of $n+1$ natural numbers, whose prime factors are in a given set of $n$ prime numbers, one can select several distinct numbers whose product is a perfect square.

Knowing that the system \[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\] has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.

Half the people in a room left. One third of those remaining started to dance. There were then $12$ people who were not dancing. The original number of people in the room was $\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 42 \qquad \text{(E)}\ 72$

Solve the system of equations $ \begin{cases} x^3 \plus{} x(y \minus{} z)^2 \equal{} 2\\ y^3 \plus{} y(z \minus{} x)^2 \equal{} 30\\ z^3 \plus{} z(x \minus{} y)^2 \equal{} 16\end{cases}$.