$\triangle ABC$, $\angle A=23^{\circ},\angle B=46^{\circ}$. Let $\Gamma$ be a circle with center $C$, radius $AC$. Let the external angle bisector of $\angle B$ intersects $\Gamma$ at $M,N$. Find $\angle MAN$.

Let $ABC$ be a triangle and $D$ be a point on $BC$ such that $AB+BD=AC+CD$. The line $AD$ intersects the incircle of triangle $ABC$ at $X$ and $Y$ where $X$ is closer to $A$ than $Y$ i. Suppose $BC$ is tangent to the incircle at $E$, prove that: 1) $EY$ is perpendicular to $AD$; 2) $XD=2IM$ where $I$ is the incentre and $M$ is the midpoint of $BC$.

if $x,y,z>0$ are postive real numbers such that $x^{2}+y^{2}+z^{2}=x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}$ prove that \[((x-y)(y-z)(z-x))^{2}\leq 2((x^{2}-y^{2})^{2}+(y^{2}-z^{2})^{2}+(z^{2}-x^{2})^{2})\]

Let $m$ be an integer, $m \ge 2$. Each student in a school is practising $m$ hobbies the most. Among any $m$ students there exist two students who have a common hobby. Find the smallest number of students for which there must exist a hobby which is practised by at least $3$ students .

In a triangle $ABC$, $I$ is the incentre and $D$ the intersection point of $AI$ and $BC$. Show that $AI+CD=AC$ if and only if $\angle B=60^{\circ}+\frac{_1}{^3}\angle C$.

Let $\ell$ be a real number satisfying the equation $\tfrac{(1+\ell)^2}{1+\ell^2}=\tfrac{13}{37}$. Then \[\frac{(1+\ell)^3}{1+\ell^3}=\frac mn,\] where $m$ and $n$ are positive coprime integers. Find $m+n$.

Let $ ABCDEF$ be a convex hexagon with $ AB \equal{} BC \equal{} CD$ and $ DE \equal{} EF \equal{} FA$, such that $ \angle BCD \equal{} \angle EFA \equal{} \frac {\pi}{3}$. Suppose $ G$ and $ H$ are points in the interior of the hexagon such that $ \angle AGB \equal{} \angle DHE \equal{} \frac {2\pi}{3}$. Prove that $ AG \plus{} GB \plus{} GH \plus{} DH \plus{} HE \geq CF$.

For a positive integer $a$, define $F_1 ^{(a)}=1$, $F_2 ^{(a)}=a$ and for $n>2$, $F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}$. A positive integer is fibonatic when it is equal to $F_n ^{(a)}$ for a positive integer $a$ and $n>3$. Prove that there are infintely many not fibonatic integers.

Two positive integers $m$ and $n$ are called [i]similar [/i] if one of them can be obtained from the other one by swapping two digits (note that a $0$-digit cannot be swapped with the leading digit). Find the greatest integer $N$ such that N is divisible by $13$ and any number similar to $N$ is not divisible by $13$.

Let $ABC$ be a triangle and a circle $\Gamma'$ be drawn lying outside the triangle, touching its incircle $\Gamma$ externally, and also the two sides $AB$ and $AC$. Show that the ratio of the radii of the circles $\Gamma'$ and $\Gamma$ is equal to $\tan^ 2 { \left( \dfrac{ \pi - A }{4} \right) }.$

The length of the hypotenuse $BC$ of a right triangle $ABC$ is $a$, and on it the points $M$ and $N$ are taken such that $BM = NC = k$, with $k < a/2$. Assuming that (only) the data $a$ and $k$ are known, calculate: a) The value of the sum of the squares of the lengths $AM$ and $AN$. b) The ratio of the areas of triangles $ABC$ and $AMN$. c) The area enclosed by the circle that passes through the points $A, M' , N'$ , where $M'$ is the orthogonal projection of $M$ onto $AC$ and $N'$ that of $N$ onto $AB$.

Let $x_0,x_1,x_2,\dots$ be the sequence such that $x_0=1$ and for $n\ge 0,$ \[x_{n+1}=\ln(e^{x_n}-x_n)\] (as usual, the function $\ln$ is the natural logarithm). Show that the infinite series \[x_0+x_1+x_2+\cdots\] converges and find its sum.

Let $h_a, h_b, h_c$ be the lengths of the altitudes of a triangle $ABC$ from $A, B, C$ respectively. Let $P$ be any point inside the triangle. Show that \[\frac{PA}{h_b+h_c} + \frac{PB}{h_a+h_c} + \frac{PC}{h_a+h_b} \ge 1.\]

Let $\vartriangle ABC$ have $AB = 14$, $BC = 30$, $AC = 40$ and $\vartriangle AB'C'$ with $AB' = 7\sqrt6$, $B'C' = 15\sqrt6$, $AC' = 20\sqrt6$ such that $\angle BAB' = \frac{5\pi}{12}$ . The lines $BB'$ and $CC'$ intersect at point $D$. Let $O$ be the circumcenter of $\vartriangle BCD$, and let $O' $ be the circumcenter of $\vartriangle B'C'D$. Then the length of segment $OO'$ can be expressed as $\frac{a+b \sqrt{c}}{ d}$ , where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$

Show that for every real $x \ge \frac12$ there is an integer $n$ such that $|x - n^2| \le \sqrt{x-\frac{1}{4}}$.

Let $a,b,c$ and $m$ ($0 \le m \le 26$) be integers such that $a + b + c = (a - b)(b- c)(c - a) = m$ (mod $27$) then $m$ is (A): $0$, (B): $1$, (C): $25$, (D): $26$ (E): None of the above.

Jasper and Rose are playing a game. Twenty-six $32$-ounce jugs are in a line, labeled Quart $\text{A}$ through Quart $\text{Z}$ from left to right. All twenty-six jugs are initially full. Jasper and Rose take turns making one of the following two moves: [list] [*] remove a positive integer number of ounces (possibly all) from the leftmost nonempty jug, or [*] remove an [i]equal[/i] positive integer number of ounces from the two leftmost nonempty jugs, possibly emptying them. Neither player may remove more ounces from a jug than it currently contains. [/list] Jasper plays first. A player’s score is the number of ounces they take from Quart $\text{Z}.$ If both players play to maximize their score, compute the maximum score that Jasper can guarantee.

Acute triangle $ABC$ satisfies $\angle A > \angle C$. Let $D, E, F$ be the points that the triangle's incircle intersects with $BC, CA, AB$, respectively, and $P$ some point on $AF$ different from $F$. The angle bisector of $\angle ABC$ meets $PQR$'s circumcircle $O$ at $L, R$. $L$ is the point closer to $B$ than $R$. $O$ meets $DF, DR$ at point $Q(\neq F, L), S(\neq R)$ respectively, and $PS$ hits segment $BC$ at $T$. Show that $T, Q, L$ are collinear.

Michael the Mouse finds a block of cheese in the shape of a regular tetrahedron (a pyramid with equilateral triangles for all faces). He cuts some cheese off each corner with a sharp knife. How many faces does the resulting solid have? [i]2015 CCA Math Bonanza Individual Round #1[/i]

On the plane are chosen six points. Prove that the ratio of the longest distance between two points to the shortest is at least $\sqrt3$.

Let $\mathbb{N} = \{1, 2, 3, \dots\}$ be the set of all positive integers, and let $f$ be a bijection from $\mathbb{N}$ to $\mathbb{N}$. Must there exist some positive integer $n$ such that $(f(1), f(2), \dots, f(n))$ is a permutation of $(1, 2, \dots, n)$?

Externally tangent circles with centers at points $A$ and $B$ have radii of lengths $5$ and $3$, respectively. A line externally tangent to both circles intersects ray $AB$ at point $C$. What is $BC$? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 4.8 \qquad\textbf{(C)}\ 10.2 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 14.4 $

An enormous party has an infinite number of people. Each two people either know or don't know each other. Given a positive integer $n$, prove there are $n$ people in the party such that either they all know each other, or nobody knows each other (so the first possibility means that if $A$ and $B$ are any two of the $n$ people, then $A$ knows $B$, whereas the second possibility means that if $A$ and $B$ are any two of the $n$ people, then $A$ does not know $B$).

The number of real roots of the equation $\lg^2x-[\lg x]-2=0$ is________.

Is it possible to split all straight lines in a plane into the pairs of perpendicular lines, so that every line belongs to a single pair?