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?

In a certain tournament bracket, a player must be defeated three times to be eliminated. If 512 contestants enter the tournament, what is the greatest number of games that could be played?

The hostess takes a piece of meat from the fridge; kittens gather around her. Each minute, the hostess cuts a part from the piece and feeds it to one of the kittens (on her choice). Each time, the cut part is in the same proportion to the current piece. At some moment, the hostess puts the rest of the meat into the fridge. Can the hostess give the same amount of meat in total to each kitten if a) the number of kittens equals two; (3 marks) b) the number of kittens equals three? (7 marks)

Patrick started walking at a constant rate along a straight road from school to the park. One hour after Patrick left, Tanya started running along the same road from school to the park. One hour after Tanya left, Jose started bicycling along the same road from school to the park. Tanya ran at a constant rate of $2$ miles per hour faster than Patrick walked, Jose bicycled at a constant rate of $7$ miles per hour faster than Tanya ran, and all three arrived at the park at the same time. The distance from the school to the park is $\frac{m}{n}$ miles, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Suppose that $n$ polygons of area $s = (n - 1)^2$ are placed on a polygon of area $S = \frac{n(n - 1)^2}{2}$. Prove that there exist two of the $n$ smaller polygons whose intersection has the area at least $1$.

Prove that all positive rational numbers can be written as a fraction, which numerator and denominator are products of factorials of not necessarily different prime numbers. For example $\frac{10}{9}=\frac{2!5!}{3!3!3!}$.

Points $G$ and $N$ are chosen on the interiors of sides $ED$ and $DO$ of unit square $DOME$, so that pentagon $GNOME$ has only two distinct side lengths. The sum of all possible areas of quadrilateral $NOME$ can be expressed as $\frac{a-b\sqrt{c}}{d}$, where $a,b,c,d$ are positive integers such that $\gcd(a,b,d) = 1$ and $c$ is square-free (i.e. no perfect square greater than $1$ divides $c$). Compute $1000a+100b+10c+d$.

Find largest possible constant $M$ such that, for any sequence $a_n$, $n=0,1,2,...$ of real numbers, that satisfies the conditions : i) $a_0=1$, $a_1=3$ ii) $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$ for any integer $n\ge 1$ to be true that $$\frac{a_{n+1}}{a_n} >M$$ for any integer $n\ge 0$.

Given a positive integer $n$, find all $n$-tuples of real number $(x_1,x_2,\ldots,x_n)$ such that \[ f(x_1,x_2,\cdots,x_n)=\sum_{k_1=0}^{2} \sum_{k_2=0}^{2} \cdots \sum_{k_n=0}^{2} \big| k_1x_1+k_2x_2+\cdots+k_nx_n-1 \big| \] attains its minimum.

We have $100$ cards with two sides, the [i]even[/i] and the [i]odd[/i]. In each side there are written two succesive integers, in the [i]odd[/i] side and odd integer and at the back in the [i]even[/i] side the even number that follows the odd number of the [i]odd[/i] side, such that all intgers from $1$ to $200$ are used. Student $A$ randomly choses $21$ cards and sums all the numbers of boths sides and announces as their sum the number $913$. Student $B$ randomly choses from the remaining cards $20$ cards and sums all the numbers of boths sides and announces as their sum the number $2400$. a) Explain why student $A$ has done an error in the addition. b) If the correct result for student $A$ is $903$, explain why also student $B$ has done an error in the addition.

Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz start on the ground, make a sequence of $6$ hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.) $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }12$