Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.

Triangle $ABC$ has $\angle BAC=90^\circ$. A semicircle with diameter $XY$ is inscribed inside $\triangle ABC$ such that it is tangent to a point $D$ on side $BC$, with $X$ on $AB$ and $Y$ on $AC$. Let $O$ be the midpoint of $XY$. Given that $AB=3$, $AC=4$, and $AX=\tfrac{9}{4}$, compute the length of $AO$.

If a train carrying $27$ passengers leaves Grand Central Station at $8:00$ AM and travels $900$ miles due west to Chicago, arriving at $5:00$ PM, what is the average speed of the train in miles per hour? [i]2015 CCA Math Bonanza Lightning Round #5.2[/i]

Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is $50\%$ more than the regular. After both consume $\tfrac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and $2$ additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together? ${ \textbf{(A)}\ 30\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}}\ 40\qquad\textbf{(E)}\ 50 $

Given a circle, its diameter $[AB]$ and a point $C$ on it. Construct (with the help of compasses and ruler) two points $X$ and $Y$, that are symmetric with respect to $(AB)$ line, such that $(YC)$ is orthogonal to $(XA)$.

Is it possible to tile space using a combination of regular tetrahedra and regular octahedra? (A Belov)

Denote by $S(n)$ the sum of digits of integer $n.$ Find 1) $S(3)+S(6)+S(9)+\ldots+S(300);$ 2) $S(3)+S(6)+S(9)+\ldots+S(3000).$

Let a and b be non-negative integers such that $ab \ge c^{2}$ where $c$ is an integer. Prove that there is a positive integer n and integers $x_{1}$, $x_{2}$, $\cdots$, $x_{n}$, $y_{1}$, $y_{2}$, $\cdots$, $y_{n}$ such that \[{x_{1}}^{2}+\cdots+{x_{n}}^{2}=a,\;{y_{1}}^{2}+\cdots+{y_{n}}^{2}=b,\; x_{1}y_{1}+\cdots+x_{n}y_{n}=c\]

Given a (fixed) positive integer $N$, solve the functional equation \[f \colon \mathbb{Z} \to \mathbb{R}, \ f(2k) = 2f(k) \textrm{ and } f(N-k) = f(k), \ \textrm{for all } k \in \mathbb{Z}.\] [i](Dan Schwarz)[/i]

Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11.$ Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number? $\textbf{(A)} \text{ 11} \qquad \textbf{(B)} \text{ 12} \qquad \textbf{(C)} \text{ 13} \qquad \textbf{(D)} \text{ 14} \qquad \textbf{(E)} \text{ 15}$

Let $ a, b \in \mathbb{N}$ with $ 1 \leq a \leq b,$ and $ M \equal{} \left[\frac {a \plus{} b}{2} \right].$ Define a function $ f: \mathbb{Z} \mapsto \mathbb{Z}$ by \[ f(n) \equal{} \begin{cases} n \plus{} a, & \text{if } n \leq M, \\ n \minus{} b, & \text{if } n >M. \end{cases} \] Let $ f^1(n) \equal{} f(n),$ $ f_{i \plus{} 1}(n) \equal{} f(f^i(n)),$ $ i \equal{} 1, 2, \ldots$ Find the smallest natural number $ k$ such that $ f^k(0) \equal{} 0.$

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

Let $H$ be the orthocenter of an obtuse triangle $ABC$ and $A_1B_1C_1$ arbitrary points on the sides $BC,AC,AB$ respectively.Prove that the tangents drawn from $H$ to the circles with diametrs $AA_1,BB_1,CC_1$ are equal.

Jan rolls a fair six-sided die and calls the result $r$. Then, he picks real numbers $a$ and $b$ between 0 and 1 uniformly at random and independently. If the probability that the polynomial $\tfrac{x^2}{r} - x\sqrt{a} + b$ has a real root can be expressed as simplified fraction $\frac{p}{q}$, find $p$. Let $T = TNYWR$. Compute the number of ordered triples $(a,b,c)$ such that $a$, $b$, and $c$ are distinct positive integers and $a + b + c = T$.

Given the integer $n > 1$ and the real number $a > 0$ determine the maximum of $\sum_{i=1}^{n-1} x_i x_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a.$

Let $\vartriangle ABC$ be a triangle. Determine all points $P$ in the plane such that the triangles $\vartriangle ABP$, $\vartriangle ACP$ and $\vartriangle BCP$ all have the same area.

Let \( x \), \( y \), \( p \) be positive integers that satisfy the equation \( x^4 = p + 9y^4 \), where \( p \) is a prime number. Show that \( \frac{p^2 - 1}{3} \) is a perfect square and a multiple of 16.

Find all prime $p$ numbers such that $p^3-4p+9$ is perfect square.

Let $n$ be a positive integer. In an $n\times n$ garden, a fountain is to be built with $1\times 1$ platforms covering the entire garden. Ana places all the platforms at a different height. Afterwards, Beto places water sources in some of the platforms. The water in each platform can flow to other platforms sharing a side only if they have a lower height. Beto wins if he fills all platforms with water. Find the least number of water sources that Beto needs to win no matter how Ana places the platforms.

Let $d \geq 3$ be a positive integer. The binary strings of length $d$ are splitted into $2^{d-1}$ pairs, such that the strings in each pair differ in exactly one position. Show that there exists an $\textit{alternating cycle}$ of length at most $2d-2$, i.e. at most $2d-2$ binary strings that can be arranged on a circle so that any pair of adjacent strings differ in exactly one position and exactly half of the pairs of adjacent strings are pairs in the split.

Let $a$ and $b$ be real numbers bigger than $1$. Find maximal value of $c \in \mathbb{R}$ such that $$\frac{1}{3+\log _{a} b}+\frac{1}{3+\log _{b} a} \geq c$$

The limit of the sum of an infinite number of terms in a geometric progression is $ \frac {a}{1 \minus{} r}$ where $ a$ denotes the first term and $ \minus{} 1 < r < 1$ denotes the common ratio. The limit of the sum of their squares is: $ \textbf{(A)}\ \frac {a^2}{(1 \minus{} r)^2} \qquad\textbf{(B)}\ \frac {a^2}{1 \plus{} r^2} \qquad\textbf{(C)}\ \frac {a^2}{1 \minus{} r^2} \qquad\textbf{(D)}\ \frac {4a^2}{1 \plus{} r^2} \qquad\textbf{(E)}\ \text{none of these}$

Consider a positive integer $a > 1$. If $a$ is not a perfect square then at the next move we add $3$ to it and if it is a perfect square we take the square root of it. Define the trajectory of a number $a$ as the set obtained by performing this operation on $a$. For example the cardinality of $3$ is $\{3, 6, 9\}$. Find all $n$ such that the cardinality of $n$ is finite. The following part problems may attract partial credit. $\textbf{(a)}$Show that the cardinality of the trajectory of a number cannot be $1$ or $2$. $\textbf{(b)}$Show that $\{3, 6, 9\}$ is the only trajectory with cardinality $3$. $\textbf{(c)}$ Show that there for all $k \geq 3$, there exists a number such that the cardinality of its trajectory is $k$. $\textbf{(d)}$ Give an example of a number with cardinality of trajectory as infinity.

The International Mathematical Olympiad is being organized in Japan, where a folklore belief is that the number $4$ brings bad luck. The opening ceremony takes place at the Grand Theatre where each row has the capacity of $55$ seats. What is the maximum number of contestants that can be seated in a single row with the restriction that no two of them are $4$ seats apart (so that bad luck during the competition is avoided)?

The dimensions of a rectangular box in inches are all positive integers and the volume of the box is $2002\text{ in}^3$. Find the minimum possible sum in inches of the three dimensions. $\textbf{(A) }36\qquad\textbf{(B) }38\qquad\textbf{(C) }42\qquad\textbf{(D) }44\qquad\textbf{(E) }92$