Consider the problem of expressing $42$ as \(42 = x^3 + y^3 + z^3 - w^2\), where \(x, y, z, w\) are integers. Determine the number of ways to represent $42$ in this form and prove your conclusion.

Find the area of a triangle with side lengths $\sqrt{13},\sqrt{29},$ and $\sqrt{34}.$ The area can be expressed as $\frac{m}{n}$ for $m,n$ relatively prime positive integers, then find $m+n.$ [i]Proposed by Kaylee Ji[/i]

Let $O$ be the circumcenter of an acute $\vartriangle ABC$ which has altitude $AD$. Let $AO$ intersect the circumcircle of $\vartriangle BOC$ again at $X$. If $E$ and $F$ are points on lines $AB$ and $AC$ such that $\angle XEA = \angle XFA = 90^o$ , then prove that the line $DX$ bisects the segment $EF$.

Andrew the ant starts at vertex $A$ of square $ABCD$. Each time he moves, he chooses the clockwise vertex with probability $\frac{2}{3}$ and the counter-clockwise vertex with probability $\frac{1}{3}$. What is the probability that he ends up on vertex $A$ after $6$ moves? [i]2015 CCA Math Bonanza Lightning Round #4.3[/i]

Find the number of integers $n$ with $n \ge 2$ such that the remainder when $2013$ is divided by $n$ is equal to the remainder when $n$ is divided by $3$. [i]Proposed by Michael Kural[/i]

Let $BE$ and $CF$ be altitudes in triangle $ABC$. If $AE = 24$, $EC = 60$, and $BF = 31$, determine $AF$.

Given $ n \geq 3$, $ n$ is a integer. Prove that: \[ (2^n \minus{} 2) \cdot \sqrt{2i\minus{}1} \geq \left( \sum_{j\equal{}0}^{i\minus{}1}C_n^j \plus{} C_{n\minus{}1}^{i\minus{}1} \right) \cdot \sqrt{n}\] where if $ n$ is even, then $ \displaystyle 1 \leq i \leq \frac{n}{2}$; if $ n$ is odd, then $ \displaystyle 1 \leq i \leq \frac{n\minus{}1}{2}$.

Let the circle $ (I; r)$ be inscribed in the triangle $ ABC$. Let $ D$ be the point of contact of this circle with $ BC$. Let $ E$ and $ F$ be the midpoints of $ BC$ and $ AD$, respectively. Prove that the three points $ I$, $ E$, $ F$ are collinear.

The incircle of a right-angled triangle $ABC$ touches hypotenus $AB$ at $P$, $BC$ and $AC$ at $R$ and $Q$ respectively. $C_1$ and $C_2$ are reflections of $C$ in $PQ$ and $PR$. Find the angle $C_1IC_2$, where $I$ is the incenter of $ABC$.

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\Omega$ and let diagonals $AC$ and $BD$ intersect at $X$. Suppose that $AEFB$ is inscribed in a circumcircle of triangle $ABX$ such that $EF$ and $AB$ are parallel. $FX$ meets the circumcircle of triangle $CDX$ again at $G$. Let $EX$ meets $AB$ at $P$, and $XG$ meets $CD$ at $Q$. Denote by $S$ the intersection of the perpendicular bisector of $\overline{EG}$ and $\Omega$ such that $S$ is closer to $A$ than $B$. Prove that line through $S$ parallel to $PQ$ is tangent to $\Omega$.

In the plane there is a quadrilateral $ ABCD $ and a point $ M $. Construct a parallelogram with center $ M $ and its vertices lying on the lines $ AB $, $ BC $, $ CD $, $ DA $.

The triangle $ABC$ is such that $\angle BAC=30^{\circ},\angle ABC=45^{\circ}$. Prove that if $X$ lies on the ray $AC$, $Y$ lies on the ray $BC$ and $OX=BY$, where $O$ is the circumcentre of triangle $ABC$, then $S_{XY}$ passes through a fixed point. [i]Emil Kolev [/i]

An array $n \times n$ is given, consisting of $n^2$ unit squares. A pawn is placed arbitrarily on a unit square. A [i]move[/i] of the pawn means a jump from a square of the $k$th column to any square of the $k$th row. Show that there exists a sequence of $n^2$ moves of the pawn so that all the unit squares of the array are visited once and, in the end, the pawn returns to the original position.

Show that the equation \[ (9 - x^2) \left (\frac{\mathrm dy}{\mathrm dx} \right)^2 = (9 - y^2)\] characterizes a family of conics touching the four sides of a fixed square.

In an isosceles triangle $ABC$ with base $AC$, on side $BC$ is selected point $K$ so that $\angle BAK = 24^o$. On the segment $AK$ the point $M$ is chosen so that $\angle ABM = 90^o$, $AM=2BK$. Find the values ​​of all angles of triangle $ABC$.

One million [i]bucks [/i] (i.e. one million male deer) are in different cells of a $1000 \times 1000$ grid. The left and right edges of the grid are then glued together, and the top and bottom edges of the grid are glued together, so that the grid forms a doughnut-shaped torus. Furthermore, some of the bucks are [i]honest bucks[/i], who always tell the truth, and the remaining bucks are [i]dishonest bucks[/i], who never tell the truth. Each of the million [i]bucks [/i] claims that “at most one of my neighboring bucks is an [i]honest buck[/i].” A pair of [i]neighboring bucks[/i] is said to be [i]buckaroo[/i] if exactly one of them is an [i]honest buck[/i] . What is the minimum possible number of [i]buckaroo [/i] pairs in the grid? Note: Two [i]bucks [/i] are considered to be [i]neighboring [/i] if their cells $(x_1, y_1)$ and $(x_2, y_2)$ satisfy either: $x_1 = x_2$ and $y_1 - y_2 \equiv \pm1$ (mod $1000$), or $x_1 - x_2 \equiv \pm 1$ (mod $1000$) and $y_1 = y_2$.

Do there exist two real monic polynomials $P(x)$ and $Q(x)$ of degree 3,such that the roots of $P(Q(X))$ are nine pairwise distinct nonnegative integers that add up to $72$? (In a monic polynomial of degree 3, the coefficient of $x^{3}$ is $1$.)

Points $A, B, C$ and $D$ lie on line $l$ in this order. Two circular arcs $C_1$ and $C_2$, which both lie on one side of line $l$, pass through points $A$ and $B$ and two circular arcs $C_3$ and $C_4$ pass through points $C$ and $D$ such that $C_1$ is tangent to $C_3$ and $C_2$ is tangent to $C_4$. Prove that the common external tangent of $C_2$ and $C_3$ and the common external tangent of $C_1$ and $C_4$ meet each other on line $l$. [i]Proposed by Ali Khezeli[/i]

[b]p1.[/b] What is the largest distance between any two points on a regular hexagon with a side length of one? [b]p2.[/b] For how many integers $n \ge 1$ is $\frac{10^n - 1}{9}$ the square of an integer? [b]p3.[/b] A vector in $3D$ space that in standard position in the first octant makes an angle of $\frac{\pi}{3}$ with the $x$ axis and $\frac{\pi}{4}$ with the $y$ axis. What angle does it make with the $z$ axis? [b]p4.[/b] Compute $\sqrt{2012^2 + 2012^2 \cdot 2013^2 + 2013^2} - 2012^2$. [b]p5.[/b] Round $\log_2 \left(\sum^{32}_{k=0} {{32} \choose k} \cdot 3^k \cdot 5^k\right)$ to the nearest integer. [b]p6.[/b] Let $P$ be a point inside a ball. Consider three mutually perpendicular planes through $P$. These planes intersect the ball along three disks. If the radius of the ball is $2$ and $1/2$ is the distance between the center of the ball and $P$, compute the sum of the areas of the three disks of intersection. [b]p7.[/b] Find the sum of the absolute values of the real roots of the equation $x^4 - 4x - 1 = 0$. [b]p8.[/b] The numbers $1, 2, 3, ..., 2013$ are written on a board. A student erases three numbers $a, b, c$ and instead writes the number $$\frac12 (a + b + c)\left((a - b)^2 + (b - c)^2 + (c - a)^2\right).$$ She repeats this process until there is only one number left on the board. List all possible values of the remainder when the last number is divided by 3. [b]p9.[/b] How many ordered triples of integers $(a, b, c)$, where $1 \le a, b, c \le 10$, are such that for every natural number $n$, the equation $(a + n)x^2 + (b + 2n)x + c + n = 0$ has at least one real root? Problems' source (as mentioned on official site) is Gator Mathematics Competition. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Positive integers $a, b$ satisfy equality $b^2 = a^2 + ab + b$. Prove that $b$ is a square of a positive integer. (Patrik Bak)

$ABCD$ is a parallelogram, and circle $S$ (with radius $2$) is inscribed insider $ABCD$ such that $S$ is tangent to all four line segments $AB$, $BC$, $CD$, and $DA$. One of the internal angles of the parallelogram is $60^\circ$. What is the maximum possible area of $ABCD$?

Show that for all $n\ge 3$ there are $n$ different positive integers $x_1,x_2, ...,x_n$ such that $$\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= 1.$$

Kevin has $255$ cookies, each labeled with a unique nonempty subset of $\{1,2,3,4,5,6,7,8\}$. Each day, he chooses one cookie uniformly at random out of the cookies not yet eaten. Then, he eats that cookie, and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses the cookie labeled with $\{1,2\}$, he eats that cookie as well as the cookies with $\{1\}$ and $\{2\}$). The expected value of the number of days that Kevin eats a cookie before all cookies are gone can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [i]Proposed by Ray Li[/i]

A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.

You shoot an arrow in the air. It falls to earth, you know not where. But you do know that the arrow’s height in feet after ${t}$ seconds is $-16t^2 + 80t + 96.$ After how many seconds does the arrow hit the ground? (the ground has height 0) $$ \mathrm a. ~ 2\qquad \mathrm b.~3\qquad \mathrm c. ~4 \qquad \mathrm d. ~5 \qquad \mathrm e. ~6 $$