Assume that $a$, $b$, $c$, and $d$ are positive integers such that $a^5 = b^4$, $c^3 = d^2$, and $c - a = 19$. Determine $d - b$.

[asy]size(8cm); real w = 2.718; // width of block real W = 13.37; // width of the floor real h = 1.414; // height of block real H = 7; // height of block + string real t = 60; // measure of theta pair apex = (w/2, H); // point where the strings meet path block = (0,0)--(w,0)--(w,h)--(0,h)--cycle; // construct the block draw(shift(-W/2,0)*block); // draws white block path arrow = (w,h/2)--(w+W/8,h/2); // path of the arrow draw(shift(-W/2,0)*arrow, EndArrow); // draw the arrow picture pendulum; // making a pendulum... draw(pendulum, block); // block fill(pendulum, block, grey); // shades block draw(pendulum, (w/2,h)--apex); // adds in string add(pendulum); // adds in block + string add(rotate(t, apex) * pendulum); // adds in rotated block + string dot("$\theta$", apex, dir(-90+t/2)*3.14); // marks the apex and labels it with theta draw((apex-(w,0))--(apex+(w,0))); // ceiling draw((-W/2-w/2,0)--(w+W/2,0)); // floor[/asy] A block of mass $m=\text{4.2 kg}$ slides through a frictionless table with speed $v$ and collides with a block of identical mass $m$, initially at rest, that hangs on a pendulum as shown above. The collision is perfectly elastic and the pendulum block swings up to an angle $\theta=12^\circ$, as labeled in the diagram. It takes a time $ t = \text {1.0 s} $ for the block to swing up to this peak. Find $10v$, in $\text{m/s}$ and round to the nearest integer. Do not approximate $ \theta \approx 0 $; however, assume $\theta$ is small enough as to use the small-angle approximation for the period of the pendulum. [i](Ahaan Rungta, 6 points)[/i]

Let $L$ be the answer to problem $9$. Find the solution to the equation $4x+\sqrt{L}=0$. Let $M$ be the answer to problem $7$. Let $f(x)=x^4+4x^3+6x^2+1$. Find $f(M)$. Let $T$ be the answer to problem $8$. Find the area of a square with side length $T$.

Let $n$ be a natural number $n \geq 2$ and matrices $A,B \in M_{n}(\mathbb{C}),$ with property $A^2 B = A.$ a) Prove that $(AB - BA)^2 = O_{n}.$ b) Show that for all natural number $k$, $k \leq \frac{n}{2}$ there exist matrices $A,B \in M_{n}(\mathbb{C})$ with property stated in the problem such that $rank(AB - BA) = k.$

What is the tens digit of $2015^{2016}-2017?$ $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 8$

Evaluate the following integrals. (1) $\int_{0}^{\frac{\pi}{4}}\frac{dx}{1+\sin x}.$ (2) $\int_{\frac{4}{3}}^{2}\frac{dx}{x^{2}\sqrt{x-1}}.$

Given $4$ lines in the plane such that there are not $2$ parallel to each other or no $3$ concurrent, we consider the following $ 8$ segments: in each line we have $2$ consecutive segments determined by the intersections with the other three lines. Prove that: a) The lengths of the $ 8$ segments cannot be the numbers $1, 2, 3,4, 5, 6, 7, 8$ in some order. b) The lengths of the $ 8$ segments can be $ 8$ different integers.

A circle that passes through the vertex $A$ of a rectangle $ABCD$ intersects the side $AB$ at a second point $E$ different from $B.$ A line passing through $B$ is tangent to this circle at a point $T,$ and the circle with center $B$ and passing through $T$ intersects the side $BC$ at the point $F.$ Show that if $\angle CDF= \angle BFE,$ then $\angle EDF=\angle CDF.$

The digits $2,3,4,5,6,7,8,9$ are written down in some order. When read in that order, the digits form an $8$-digit, base $10$ positive integer. if this integer is divisible by $44$, how many ways could the digits have been initially ordered? [i]Proposed by Evan Chang (squareman), USA[/i]

For non-negative integer $n$, the function $f$ is given by \[f(x)=\begin{cases} \frac{x}{2} & \text{if $n$ is even} \\ x-1 & \text{if $n$ is odd.} \end{cases} \] Furthermore, let $h(n)$ be the smallest $k$ for which $f^k(n)=0$. Compute \[\sum_{n=1}^{1024} h(n).\]

Suppose that \[\prod_{n=1}^{1996}(1+nx^{3^{n}}) = 1+a_{1}x^{k_{1}}+a_{2}x^{k_{2}}+\cdots+a_{m}x^{k_{m}}\] where $a_{1}$, $a_{2}$,..., $a_{m}$ are nonzero and $k_{1}< k_{2}< \cdots < k_{m}$. Find $a_{1996}$.

Show that $ \frac{1}{2^2} +\frac{1}{3^2} +...+\frac{1}{n^2} <\frac{2}{3}$ for all $n \ge 2 $.

The side $AC$ of an acute triangle $ABC$ is the diameter of the circle $c_1$ and side $BC$ is the diameter of the circle $c_2$. Let $E$ be the foot of the altitude drawn from the vertex $B$ of the triangle and $F$ the foot of the altitude drawn from the vertex $A$. In addition, let $L$ and $N$ be the points of intersection of the line $BE$ with the circle $c_1$ (the point $L$ lies on the segment $BE$) and the points of intersection of $K$ and $M$ of line $AF$ with circle $c_2$ (point $K$ is in section $AF$). Prove that $K LM N$ is a cyclic quadrilateral.

Determine all functions $f$ from the real numbers to the real numbers, different from the zero function, such that $f(x)f(y)=f(x-y)$ for all real numbers $x$ and $y$.

Suppose that $ I$ is incenter of triangle $ ABC$ and $ l'$ is a line tangent to the incircle. Let $ l$ be another line such that intersects $ AB,AC,BC$ respectively at $ C',B',A'$. We draw a tangent from $ A'$ to the incircle other than $ BC$, and this line intersects with $ l'$ at $ A_1$. $ B_1,C_1$ are similarly defined. Prove that $ AA_1,BB_1,CC_1$ are concurrent.

Let $\delta$ be a symbol such that $\delta \neq 0$ and $\delta^2 = 0$. Define $\mathbb R[\delta] = \{a + b \delta | a, b \in \mathbb R\}$, where $a+ b \delta = c+ d \delta$ if and only if $a = c$ and $b = d$, and define \[(a + b \delta) + (c + d \delta) = (a + c) + (b + d) \delta,\]\[(a + b \delta) \cdot (c + d \delta) = ac + (ad + bc) \delta.\] Let $P(x)$ be a polynomial with real coefficients. Show that $P(x)$ has a multiple real root if and only if $P(x)$ has a non-real root in $\mathbb R[\delta].$

Octavio writes an integer $n \geq 1$ on a blackboard and then he starts a process in which, at each step he erases the integer $k$ written on the blackboard and replaces it with one of the following numbers: $$3k-1, \quad 2k+1, \quad \frac{k}{2}.$$ provided that the result is an integer. Show that for any integer $n \geq 1$, Octavio can write on the blackboard the number $3^{2023}$ after a finite number of steps.

Let $m$ and $n$ be positive integers. Show that $25m+ 3n$ is divisible by $83$ if and only if so is $3m+ 7n$.

[b]p1.[/b] Alan leaves home when the clock in his cardboard box says $7:35$ AM and his watch says $7:41$ AM. When he arrives at school, his watch says $7:47$ AM and the $7:45$ AM bell rings. Assuming the school clock, the watch, and the home clock all go at the same rate, how many minutes behind the school clock is the home clock? [b]p2.[/b] Compute $$\left( \frac{2012^{2012-2013} + 2013}{2013} \right) \times 2012.$$ Express your answer as a mixed number. [b]p3.[/b] What is the last digit of $$2^{3^{4^{5^{6^{7^{8^{9^{...^{2013}}}}}}}}} ?$$ [b]p4.[/b] Let $f(x)$ be a function such that $f(ab) = f(a)f(b)$ for all positive integers $a$ and $b$. If $f(2) = 3$ and $f(3) = 4$, find $f(12)$. [b]p5.[/b] Circle $X$ with radius $3$ is internally tangent to circle $O$ with radius $9$. Two distinct points $P_1$ and $P_2$ are chosen on $O$ such that rays $\overrightarrow{OP_1}$ and $\overrightarrow{OP_2}$ are tangent to circle $X$. What is the length of line segment $P_1P_2$? [b]p6.[/b] Zerglings were recently discovered to use the same $24$-hour cycle that we use. However, instead of making $12$-hour analog clocks like humans, Zerglings make $24$-hour analog clocks. On these special analog clocks, how many times during $ 1$ Zergling day will the hour and minute hands be exactly opposite each other? [b]p7.[/b] Three Small Children would like to split up $9$ different flavored Sweet Candies evenly, so that each one of the Small Children gets $3$ Sweet Candies. However, three blind mice steal one of the Sweet Candies, so one of the Small Children can only get two pieces. How many fewer ways are there to split up the candies now than there were before, assuming every Sweet Candy is different? [b]p8.[/b] Ronny has a piece of paper in the shape of a right triangle $ABC$, where $\angle ABC = 90^o$, $\angle BAC = 30^o$, and $AC = 3$. Holding the paper fixed at $A$, Ronny folds the paper twice such that after the first fold, $\overline{BC}$ coincides with $\overline{AC}$, and after the second fold, $C$ coincides with $A$. If Ronny initially marked $P$ at the midpoint of $\overline{BC}$, and then marked $P'$ as the end location of $P$ after the two folds, find the length of $\overline{PP'}$ once Ronny unfolds the paper. [b]p9.[/b] How many positive integers have the same number of digits when expressed in base $3$ as when expressed in base $4$? [b]p10.[/b] On a $2 \times 4$ grid, a bug starts at the top left square and arbitrarily moves north, south, east, or west to an adjacent square that it has not already visited, with an equal probability of moving in any permitted direction. It continues to move in this way until there are no more places for it to go. Find the expected number of squares that it will travel on. Express your answer as a mixed number. PS. You had better use hide for answers.

Let $ABCD$ be a convex quadrilateral with $AB = AD$ and $CB = CD$. The bisector of $\angle BDC$ intersects $BC$ at $L$, and $AL$ intersects $BD$ at $M$, and it is known that $BL = BM$. Determine the value of $2\angle BAD + 3\angle BCD$.

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]

Let $n\ge 3$ be a positive integer. In the plane $n$ points which are not all collinear are marked. Find the least possible number of triangles whose vertices are all marked. (Recall that the vertices of a triangle are not collinear.)

Given an equilateral triangle $ABC$, let $M$ be an arbitrary point in space. $(\text{a})$ Prove that one can construct a triangle from the segments $MA, MB, MC$. $(\text{b})$ Suppose that $P$ and $Q$ are two points symmetric with respect to the center $O$ of $ABC$. Prove that the two triangles constructed from the segments $PA,PB,PC$ and $QA,QB,QC$ are of equal area.

Using $ AB$ and $ AC$ as diameters, two semi-circles are constructed respectively outside the acute triangle $ ABC$. $ AH \perp BC$ at $ H$, $ D$ is any point on side $ BC$ ($ D$ is not coinside with $ B$ or $ C$ ), through $ D$, construct $ DE \parallel AC$ and $ DF \parallel AB$ with $ E$ and $ F$ on the two semi-circles respectively. Show that $ D$, $ E$, $ F$ and $ H$ are concyclic.

Convert the following decimal to a common fraction in lowest terms: $ 0.92007200720072007...$ (or $ 0.9\overline{2007}$).