In triangle $ABC$ the angular bisector from $A$ intersects the side $BC$ at the point $D$, and the angular bisector from $B$ intersects the side $AC$ at the point $E$. Furthermore $|AE| + |BD| = |AB|$. Prove that $\angle C = 60^o$ [img]https://1.bp.blogspot.com/-8ARqn8mLn24/XzP3P5319TI/AAAAAAAAMUQ/t71-imNuS18CSxTTLzYXpd806BlG5hXxACLcBGAsYHQ/s0/2018%2BMohr%2Bp5.png[/img]

Let $N$ be the number of functions $f:\{1,2,3,4,5,6,7,8,9,10\} \rightarrow \{1,2,3,4,5\}$ that have the property that for $1\leq x\leq 5$ it is true that $f(f(x))=x$. Given that $N$ can be written in the form $5^a\cdot b$ for positive integers $a$ and $b$ with $b$ not divisible by $5$, find $a+b$. [i]Proposed by Nathan Ramesh

Let $ABC$ be a triangle with $\angle C = 90^\circ$ and $CA \neq CB$. Let $CH$ be an altitude and $CL$ be an interior angle bisector. Show that for $X \neq C$ on the line $CL$, we have $\angle XAC \neq \angle XBC$. Also show that for $Y \neq C$ on the line $CH$ we have $\angle YAC \neq \angle YBC$. [i]Bulgaria[/i]

A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected? $\textbf{(A) }\frac{1}{10}\qquad\textbf{(B) }\frac{1}{5}\qquad\textbf{(C) }\frac{3}{10}\qquad\textbf{(D) }\frac{2}{5}\qquad\textbf{(E) }\frac{1}{2}$

Find the smallest constant $ C$ such that for all real $ x,y$ \[ 1\plus{}(x\plus{}y)^2 \leq C \cdot (1\plus{}x^2) \cdot (1\plus{}y^2)\] holds.

The bisector of $\angle{BAD}$ in the parallellogram $ABCD$ intersects the lines $BC$ and $CD$ at the points $K$ and $L$ respectively. Prove that the centre of the circle passing through the points $C,\ K$ and $L$ lies on the circle passing through the points $B,\ C$ and $D$.

Gabriela found an encyclopedia with $2023$ pages, numbered from $1$ to $2023$. She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors?

Let $p$ be the product of $n$ real numbers $x_1$, $x_2$,$...$, $x_n$. Prove that if $p - x_k$ is an odd integer for $k = 1, 2,..., n$, then each of the numbers $x_1$, $x_2$,$...$, $x_n$is irrational. (G Galperin)

How many of the first $2018$ numbers in the sequence $101, 1001, 10001, 100001, \dots$ are divisible by $101$? $ \textbf{(A) }253 \qquad \textbf{(B) }504 \qquad \textbf{(C) }505 \qquad \textbf{(D) }506 \qquad \textbf{(E) }1009 \qquad $

Given is a regular $n$-sided pyramid with top $T$ and base $A_1A_2A_3... A_n$. The line perpendicular to the ground plane through a point $B$ of the ground plane within $A_1A_2A_3... A_n$ intersects the plane $TA_1A_2$ at $C_1$, the plane $TA_2A_3$ at $C_2$, and so on, and finally the plane $TA_nA_1$ at $C_n$. Prove that $BC_1 + BC_2 + ... + BC_n$ is independent of choice of $B$'s.

Define $f\left(n\right)=\textrm{LCM}\left(1,2,\ldots,n\right)$. Determine the smallest positive integer $a$ such that $f\left(a\right)=f\left(a+2\right)$. [i]2017 CCA Math Bonanza Lightning Round #2.4[/i]

Let $n\ge 3$ be an integer. Annalena has infinitely many cowbells in each of $n$ different colours. Given an integer $m \ge n + 1$ and a group of $m$ cows standing in a circle, she is tasked with tying one cowbell around the neck of every cow so that every group of $n + 1$ consecutive cows have cowbells of all the possible $n$ colours. Prove that there are only finitely many values of $m$ for which this is not possible and determine the largest such $m$ in terms of $n$.

Denote $U$ as the set of $20$ diagonals of the regular polygon $P_1P_2P_3P_4P_5P_6P_7P_8$. Find the number of sets $S$ which satisfies the following conditions. 1. $S$ is a subset of $U$. 2. If $P_iP_j \in S$ and $P_j P_k \in S$, and $i \neq k$, $P_iP_k \in S$.

Points $M$ and $N$ are given on sides $AD$ and $BC$ of rhombus $ABCD$, respectively. Line $MC$ intersects line $BD$ in point $T$, line $MN$ intersects line $BD$ in point $U$, line $CU$ intersects line $AB$ in point $Q$ and line $QT$ intersects line $CD$ in $P$. Prove that triangles $QCP$ and $MCN$ have equal area

Find all possible values of $\dfrac{d}{a}$ where $a^2-6ad+8d^2=0$, $a\neq 0$.

A circle with radius $1$ is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares? [asy] filldraw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle,mediumgray,black); filldraw(Circle((0,0),1), mediumgray,black); filldraw((-1,0)--(0,1)--(1,0)--(0,-1)--cycle,white,black);[/asy] $ \textbf{(A)}\ \frac{1}2\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{3}2\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \frac{5}2 $

$x$, $y$, and $z$ are positive real numbers that satisfy $x^3+2y^3+6z^3=1$. Let $k$ be the maximum possible value of $2x+y+3z$. Let $n$ be the smallest positive integer such that $k^n$ is an integer. Find the value of $k^n+n$.

Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence. Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths. [asy] size(4cm); draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin); draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin); draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin); draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin); draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin); draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin); draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin); [/asy] [i]Proposed by Zixiang Zhou, Canada[/i]

Show that amongst any $ 8$ points in the interior of a $7 \times 12$ rectangle, there exists a pair whose distance is less than $5$. Note: The interior of a rectangle excludes points lying on the sides of the rectangle.

Let $F_1=F_2=1$. We define inductively $F_{n+1}=F_n+F_{n-1}$ for all $n\geq 2$. Then the sum $$F_1+F_2+\cdots+F_{2017}$$ is [list=1] [*] Even but not divisible by 3 [*] Odd but divisible by 3 [*] Odd and leaves remainder 1 when divisible by 3 [*] None of these [/list]

Solve the equation $$2(x-6)=\dfrac{x^2}{(1+\sqrt{x+1})^2}$$

A knight is placed at the origin of the Cartesian plane. Each turn, the knight moves in an chess $\text{L}$-shape ($2$ units parallel to one axis and $1$ unit parallel to the other) to one of eight possible location, chosen at random. After $2016$ such turns, what is the expected value of the square of the distance of the knight from the origin?

If $x, y, z, w$ are nonnegative real numbers satisfying \[\left\{ \begin{array}{l}y = x - 2003 \\ z = 2y - 2003 \\ w = 3z - 2003 \\ \end{array} \right. \] find the smallest possible value of $x$ and the values of $y, z, w$ corresponding to it.

For a convex polygon (i.e. all angles less than $180^\circ$) call a diagonal [i]bisector[/i] if its bisects both area and perimeter of the polygon. What is the maximum number of bisector diagonals for a convex pentagon? [i]Proposed by Morteza Saghafian[/i]

Positive integer divisors $a$ and $b$ of $n$ are called [i]complementary[/i] if $ab=n$. Given that $N$ has a pair of complementary divisors that differ by $20$ and a pair of complementary divisors that differ by $23$, find the sum of the digits of $N$. $\textbf{(A) } 11 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$