A plane is divided into unit squares by two collections of parallel lines. For any $n\times n$ square with sides on the division lines, we define its frame as the set of those unit squares which internally touch the boundary of the $n\times n$ square. Prove that there exists only one way of covering a given $100\times 100$ square whose sides are on the division lines with frames of $50$ squares (not necessarily contained in the $100\times 100$ square). (A. Perlin)

The cube $ABCDA' B' C' D' $has of length a. Consider the points $K \in [AB], L \in [CC' ], M \in [D'A']$. a) Show that $\sqrt3 KL \ge KB + BC + CL$ b) Show that the perimeter of triangle $KLM$ is strictly greater than $2a\sqrt3$.

Prove that one of the digits 1, 2 and 9 must appear in the base-ten expression of $n$ or $3n$ for any positive integer $n$. [i](4 points)[/i]

Let $S = \{0, 1, 2, .., 1994\}$. Let $a$ and $b$ be two positive numbers in $S$ which are relatively prime. Prove that the elements of $S$ can be arranged into a sequence $s_1, s_2, s_3,... , s_{1995}$ such that $s_{i+1} - s_i \equiv \pm a$ or $\pm b$ (mod $1995$) for $i = 1, 2, ... , 1994$

In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies \[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\] Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.

Three points are randomly placed on a circle. What is the probability that they lie on the same semicircle

Let $ABC$ be a triangle with sides $a$, $b$, $c$ and (corresponding) angles $A$, $B$, $C$. Prove that if $3A + 2B = 180^{\circ}$, then $a^2+bc=c^2$. [b]Additional problem:[/b] Prove that the converse also holds, i. e. prove the following: Let $ABC$ be an arbitrary triangle. Then, $3A + 2B = 180^{\circ}$ if and only if $a^2+bc=c^2$. [b]Similar problem:[/b] Let $ABC$ be an arbitrary triangle. Then, $3A + 2B = 360^{\circ}$ if and only if $a^2-bc=c^2$.

Rays $\ell, \ell_1, \ell_2$ have the same starting point $O$, such that the angle between $\ell$ and $\ell_2$ is acute and the ray $\ell_1$ lies inside this angle. The ray $\ell$ contains a fixed point of $F$ and an arbitrary point $L$. Circles passing through $F$ and $L$ and tangent to $\ell_1$ at $L_1$, and passing through $F$ and $L$ and tangent to $\ell_2$ at $L_2$. Prove that the circumcircle of $\triangle FL_1L_2$ passes through a fixed point other than $F$ independent on $L$.

Let $x, y, z$ be positive integers such that $xy=z^2+2$. Prove that there exist integers $a, b, c, d$ such that the following equalities are satisfied: \begin{eqnarray*} x=a^2+2b^2\\ y=c^2+d^2\\ z=ac+2bd\\ \end{eqnarray*} [i]Proposed by Isaac Jiménez[/i]

Altitudes $BE$ and $CF$ of triangle $ABC$ intersect in $H$. A perpendicular $HT$ from $H$ to $EF$ is drawn. Circumcircles $ABC$ and $BHT$ intersect at $B$ and $X$. Prove that $\angle TXA= \angle BAC$. [i]Vadzim Kamianetski[/i]

A closed container in the shape of a rectangular parallelepiped contains $1$ liter of water. If the container rests horizontally on three different sides, the water level is $2$ cm, $4$ cm and $5$ cm. Calculate the volume of the parallelepiped.

Joe and Kathryn are both on the school math team, which practices every Wednesday after school until $4$ PM for competitions. The team was preparing for the $ 2005$ iTest when Joe realized how crazy he was for not asking Kathryn out – the way she worked those iTest problems, solving question after question, almost made him go insane sitting there that day. He never felt the same way when she worked on preparing for other competitions – they just aren’t the same. Kathryn always beat Joe at competitions, too. Joe admired her resolve and unwillingness to make herself look stupid, when so many other girls he knew at school tried to pretend they were stupid in order to attract guys. So as time ticked away and that afternoon’s Wednesday practice neared an end, Joe was determined to strike up a conversation with Kathryn and ask her out. He really wanted to impress her, so he thought he’d ask her a really hard history of math question that she didn’t know. Naturally, she’d want the answer, and be so impressed with Joe’s brilliance that she’d go out with him on Friday night. Great plan. Seriously. When Joe asked Kathryn after class, “Who was the mathematician that died in approximately $200$ B.C. that developed a method for calculating all prime numbers?” Kathryn gave the correct response. What name did she say?

Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that \[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]

On the side $DE$ and on the diagonal $BE$ of the regular pentagon $ABCDE$ we consider the squares $DEFG$ and $BEHI$. [list=a] [*] Prove that $A,I,$ and $G$ are collinear. [*] Prove that on this line lies also the centre $O$ of the square $BDJK$. [/list]

Let $ v$, $ w$, $ x$, $ y$, and $ z$ be the degree measures of the five angles of a pentagon. Suppose $ v < w < x < y < z$ and $ v$, $ w$, $ x$, $ y$, and $ z$ form an arithmetic sequence. Find the value of $ x$. $ \textbf{(A)}\ 72 \qquad \textbf{(B)}\ 84 \qquad \textbf{(C)}\ 90 \qquad \textbf{(D)}\ 108 \qquad \textbf{(E)}\ 120$

A man with mass $m$ jumps off of a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance $H$ at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke's force law with a spring constant $k$, and stretches from an original length of $L_0$ to a final length $L = L_0 + h$. The maximum tension in the Bungee cord is four times the weight of the man. Determine the spring constant $k$. $\textbf{(A) } \frac{mg}{h}\\ \\ \textbf{(B) } \frac{2mg}{h}\\ \\ \textbf{(C) } \frac{mg}{H}\\ \\ \textbf{(D) } \frac{4mg}{H}\\ \\ \textbf{(E) } \frac{8mg}{H}$

The diagonals $ AC$ and $ BD$ of a convex quadrilateral $ ABCD$ intersect at point $ M$. The bisector of $ \angle ACD$ meets the ray $ BA$ at $ K$. Given that $ MA \cdot MC \plus{}MA \cdot CD \equal{} MB \cdot MD$, prove that $ \angle BKC \equal{} \angle CDB$.

Let $P,Q,R,S$ be the midpoints of the sides $BC,CD,DA,AB$ of a convex quadrilateral, respectively. Prove that \[4(AP^2+BQ^2+CR^2+DS^2)\le 5(AB^2+BC^2+CD^2+DA^2)\]

Which of the following represents the result when the figure shown below is rotated clockwise $120^\circ$ about its center? [asy] unitsize(6); draw(circle((0,0),5)); draw((-1,2.5)--(1,2.5)--(0,2.5+sqrt(3))--cycle); draw(circle((-2.5,-1.5),1)); draw((1.5,-1)--(3,0)--(4,-1.5)--(2.5,-2.5)--cycle); [/asy] [asy] unitsize(6); for (int i = 0; i < 5; ++i) { draw(circle((12*i,0),5)); } draw((-1,2.5)--(1,2.5)--(0,2.5+sqrt(3))--cycle); draw(circle((-2.5,-1.5),1)); draw((1.5,-1)--(3,0)--(4,-1.5)--(2.5,-2.5)--cycle); draw((14,-2)--(16,-2)--(15,-2+sqrt(3))--cycle); draw(circle((12,3),1)); draw((10.5,-1)--(9,0)--(8,-1.5)--(9.5,-2.5)--cycle); draw((22,-2)--(20,-2)--(21,-2+sqrt(3))--cycle); draw(circle((27,-1),1)); draw((24,1.5)--(22.75,2.75)--(24,4)--(25.25,2.75)--cycle); draw((35,2.5)--(37,2.5)--(36,2.5+sqrt(3))--cycle); draw(circle((39,-1),1)); draw((34.5,-1)--(33,0)--(32,-1.5)--(33.5,-2.5)--cycle); draw((50,-2)--(52,-2)--(51,-2+sqrt(3))--cycle); draw(circle((45.5,-1.5),1)); draw((48,1.5)--(46.75,2.75)--(48,4)--(49.25,2.75)--cycle); label("(A)",(0,5),N); label("(B)",(12,5),N); label("(C)",(24,5),N); label("(D)",(36,5),N); label("(E)",(48,5),N); [/asy]

Let $a$ and $b$ be real numbers such that $a + b = 2$. Show that: $$\min \{|a|,|b|\} < 1 < \max \{|a|,|b|\} \Leftrightarrow a, b \in (-3,1)$$

How many finite sequances $x_1,x_2,\cdots,x_m$ are there such that $x_i=1$ or 2 and $\sum \limits_{i=1}^mx_i=10$ ? [list=1] [*] 89 [*] 73 [*] 107 [*] 119 [/list]

What is the largest possible area of a triangle with largest side length $39$ and inradius $10$?

For a sequence of positive integers $\{x_n\}$ where $x_1 = 2$ and $x_{n + 1} - x_n \in \{0, 3\}$ for all positve integers $n$, then $\{x_n\}$ is called a "frog sequence". Find all real numbers $d$ that satisfy the following condition. [b](Condition)[/b] For two frog sequence $\{a_n\}, \{b_n\}$, if there exists a positive integer $n$ such that $a_n = 1000b_n$, then there exists a positive integer $m$ such that $a_m = d\cdot b_m$.

Given that a duck found that $5-2\sqrt{3}i$ is one of the roots of $-259 + 107x - 17x^2 + x^3$, what is the sum of the real parts of the other two roots? [i]2022 CCA Math Bonanza Lightning Round 2.1[/i]

Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression $$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$ obtains its minimum value.