Let $a_n$ denote the angle opposite to the side of length $4n^2$ units in an integer right angled triangle with lengths of sides of the triangle being $4n^2, 4n^4+1$ and $4n^4-1$ where $n \in N$. Then find the value of $\lim_{p \to \infty} \sum_{n=1}^p a_n$ (A) $\pi/2$ (B) $\pi/4$ (C) $\pi $ (D) $\pi/3$

The polygon ABCDEFG (shown on the right) is a regular octagon. Prove that the area of the rectangle $ADEH$ is one half the area of the whole polygon $ABCDEFGH$. [asy] draw((0,1.414)--(1.414,0)--(3.414,0)--(4.828,1.414)--(4.828,3.414)--(3.414,4.828)--(1.414,4.828)--(0,3.414)--(0,1.414)); fill((0,1.414)--(0,3.414)--(4.828,3.414)--(4.828,1.414)--cycle, mediumgrey); label("$B$",(1.414,0),SW); label("$C$",(3.414,0),SE); label("$D$",(4.828,1.414),SE); label("$E$",(4.828,3.414),NE); label("$F$",(3.414,4.828),NE); label("$G$",(1.414,4.828),NW); label("$H$",(0,3.414),NW); label("$A$",(0,1.414),SW); [/asy]

Let $ABC$ be a triangle. $D$ is the midpoint of $AB$, $E$ is a point on the side $BC$ such that $BE = 2 EC$ and $\angle ADC = \angle BAE$. Find $\angle BAC$.

Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle? $ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 $

A non-isosceles trapezoid $ABCD$ ($AB // CD$) is given. An arbitrary circle passing through points $A$ and $B$ intersects the sides of the trapezoid at points $P$ and $Q$, and the intersect the diagonals at points $M$ and $N$. Prove that the lines $PQ, MN$ and $CD$ are concurrent.

If $ p$ is a positive integer, then $ \frac {3p \plus{} 25}{2p \minus{} 5}$ can be a positive integer, if and only if $ p$ is: $ \textbf{(A)}\ \text{at least }3 \qquad\textbf{(B)}\ \text{at least }3\text{ and no more than }35 \qquad\textbf{(C)}\ \text{no more than }35$ $ \textbf{(D)}\ \text{equal to }35 \qquad\textbf{(E)}\ \text{equal to }3\text{ or }35$

Prove that the following inequality holds for all positive real numbers $a$, $b$, $c$, $d$, $e$ and $f$ \[\sqrt[3]{\frac{abc}{a+b+d}}+\sqrt[3]{\frac{def}{c+e+f}} < \sqrt[3]{(a+b+d)(c+e+f)} \text{.}\] [i]Proposed by Dimitar Trenevski.[/i]

Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality.

Find the number of solutions in positive integers of the equation $\lfloor \frac{x}{2} \rfloor = \lfloor \frac{x}{11} \rfloor +1$ where $\lfloor A\rfloor$ denotes the integer part of the number $A$, e.g. $\lfloor 2.031\rfloor = 2$, $\lfloor 2\rfloor = 2$, etc.

Find all the triangles such that its side lenghts, area and its angles' measures (in degrees) are rational.

Point L is marked on side BC of triangle ABC so that AL is twice as long as the median CM. Given that angle ALC is equal to 45 degrees prove that AL is perpendicular to CM.

A plane is cut into unit squares, each of which is colored in black or white. It is known that each rectangle 3 x 4 or 4 x 3 contains exactly 8 white squares. In how many ways can this plane be colored?

Let $ S\equal{}\{1,2,\ldots,n\}$ be a set, where $ n\geq 6$ is an integer. Prove that $ S$ is the reunion of 3 pairwise disjoint subsets, with the same number of elements and the same sum of their elements, if and only if $ n$ is a multiple of 3.

Find all functions $f$ that map the set of real numbers into the set of real numbers, satisfying the following conditions: 1) $|f(x)|\ge 1$, 2) $f(x+y)=\frac{f(x)+f(y)}{1+f(x)f(y)}$ of all real values of $x $ and $y$.

Find all functions $ f : \mathbb R \rightarrow \mathbb R$ such that $ (x \plus{} y)(f(x) \minus{} f(y)) \equal{} (x \minus{}y)f(x \plus{} y)$ for all $ x, y\in \mathbb R$

Does there exist a number $n=\overline{a_1a_2a_3a_4a_5a_6}$ such that $\overline{a_1a_2a_3}+4 = \overline{a_4a_5a_6}$ (all bases are $10$) and $n=a^k$ for some positive integers $a,k$ with $k \geq 3 \ ?$

For every positive integer \( n \), do there exist pairwise distinct positive integers \( a_1, a_2, \dots, a_n \) that satisfy the following condition? For every \( 3 \leq m \leq n \), there exists an \( i \leq m-2 \) such that: $$ a_m = a_{\gcd(m-1, i)} + \gcd(a_{m-1}, a_i). $$ Proposed by Alireza Jannati

There are given two positive real number $a<b$. Show that there exist positive integers $p, \ q, \ r, \ s$ satisfying following conditions: $1$. $a< \frac{p}{q} < \frac{r}{s} < b$. $2.$ $p^2+q^2=r^2+s^2$.

The points $A, B, C$ and $D$ lie in this order on the circle $k$. Let $t$ be the tangent at $k$ through $C$ and $s$ the reflection of $AB$ at $AC$. Let $G$ be the intersection of the straight line $AC$ and $BD$ and $H$ the intersection of the straight lines $s$ and $CD$. Show that $GH$ is parallel to $t$.

For each positive number $x$, let \[f(x)=\displaystyle\frac{\left(x+\dfrac{1}{x}\right)^6-\left(x^6+\dfrac{1}{x^6}\right)-2}{\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}.\] The minimum value of $f(x)$ is $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }6$

Determine all pairs $(a, b)$ of positive integers for which the equation $x^3 - 17x^2 + ax - b^2 = 0$ has three integer roots (not necessarily different).

Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets $\Omega$ again at $K$. Prove that the line $KT$ is tangent to $\Gamma$. [i]Proposed by Charles Leytem, Luxembourg[/i]

Circles with centers $A ,~ B,$ and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, then length $B'C'$ equals $\textbf{(A) }3r-2\qquad\textbf{(B) }r^2\qquad\textbf{(C) }r+\sqrt{3(r-1)}\qquad$ $\textbf{(D) }1+\sqrt{3(r^2-1)}\qquad\textbf{(E) }\text{none of these}$ [asy] //Holy crap, CSE5 is freaking amazing! import cse5; pathpen=black; pointpen=black; dotfactor=3; size(200); pair A=(1,2),B=(2,0),C=(0,0); D(CR(A,1.5)); D(CR(B,1.5)); D(CR(C,1.5)); D(MP("$A$",A)); D(MP("$B$",B)); D(MP("$C$",C)); pair[] BB,CC; CC=IPs(CR(A,1.5),CR(B,1.5)); BB=IPs(CR(A,1.5),CR(C,1.5)); D(BB[0]--CC[1]); MP("$B'$",BB[0],NW);MP("$C'$",CC[1],NE); //Credit to TheMaskedMagician for the diagram [/asy]

Consider the equation $ y^2\minus{}k\equal{}x^3$, where $ k$ is an integer. Prove that the equation cannot have five integer solutions of the form $ (x_1,y_1),(x_2,y_1\minus{}1),(x_3,y_1\minus{}2),(x_4,y_1\minus{}3),(x_5,y_1\minus{}4)$. Also show that if it has the first four of these pairs as solutions, then $ 63|k\minus{}17$.

Find $f: \mathbb{Z}_+ \rightarrow \mathbb{Z}_+$, such that for any $x,y \in \mathbb{Z}_+$, $$f(f(x)+y)\mid x+f(y).$$