Let $n$ be a fixed positive integer. An $n$-staircase is a polyomino with $\frac{n(n+1)}{2}$ cells arranged in the shape of a staircase, with arbitrary size. Here are two examples of $5$-staircases: [asy] int s = 5; for(int i = 0; i < s; i=i+1) { draw((0,0)--(0,i+1)--(s-i,i+1)--(s-i,0)--cycle); } for(int i = 0; i < s; i=i+1) { draw((10,.67*s)--(10,.67*(s-i-1))--(.67*(s-i)+10,.67*(s-i-1))--(.67*(s-i)+10,.67*s)--cycle); } [/asy] Prove that an $n$-staircase can be dissected into strictly smaller $n$-staircases. [i]Proposed by James Lin.[/i]

In the triangle $ABC$ it is known that $2AC=AB$ and $\angle A = 2\angle B$. In this triangle draw the angle bisector $AL$, and mark point $M$, the midpoint of the side $AB$. It turned out that $CL = ML$. Prove that $\angle B= 30^o$. (Hilko Danilo)

Let $S$ be an infinite set of real numbers such that $|x_1+x_2+\cdots + x_n | \leq 1 $ for all finite subsets $\{x_1,x_2,\ldots,x_n\} \subset S$. Show that $S$ is countable.

Let $A_1=0$ and $A_2=1$. For $n>2$, the number $A_n$ is defined by concatenating the decimal expansions of $A_{n-1}$ and $A_{n-2}$ from left to right. For example $A_3=A_2A_1=10$, $A_4=A_3A_2=101$, $A_5=A_4A_3=10110$, and so forth. Determine all $n$ such that $11$ divides $A_n$.

For each fixed positiver integer $n$, $n\geq 4$ and $P$ an integer, let $(P)_n \in [1, n]$ be the smallest positive residue of $P$ modulo $n$. Two sequences $a_1, a_2, \dots, a_k$ and $b_1, b_2, \dots, b_k$ with the terms in $[1, n]$ are defined as equivalent, if there is $t$ positive integer, gcd$(t,n)=1$, such that the sequence $(ta_1)_n, \dots, (ta_k)_n$ is a permutation of $b_1, b_2, \dots, b_k$. Let $\alpha$ a sequence of size $n$ and your terms are in $[1, n]$, such that each term appears $h$ times in the sequence $\alpha$ and $2h\geq n$. Show that $\alpha$ is equivalent to some sequence $\beta$ which contains a subsequence such that your size is(at most) equal to $h$ and your sum is exactly equal to $n$.

Let $x,y,n$ be positive integers with $n>1$. How many ordered triples $(x, y, n)$ of solutions are there to the equation $x^n-y^n=2^{100}$?

Three equal regular tetrahedrons have the common center. Is it possible that all faces of the polyhedron that forms their intersection are equal?

$ABCD$ is a cyclic quadrilateral with $\angle BAD=90^\circ$ and $\angle ABC>90^\circ$. $AB$ is extended to a point $E$ such that $\angle AEC=90^\circ$.If $AB=7,BE=9,$ and $EC=12$,calculate $AD$.

Find the last two digits of $2023+202^3+20^{23}$. [i]Proposed by Anthony Yang[/i]

An airplane accelerates at $10$ mph per second and decelerates at $15$ mph/sec. Given that its takeoff speed is $180$ mph, and the pilots want enough runway length to safely decelerate to a stop from any speed below takeoff speed, what’s the shortest length that the runway can be allowed to be? Assume the pilots always use maximum acceleration when accelerating. Please give your answer in miles.

An object moves in two dimensions according to \[\vec{r}(t) = (4.0t^2 - 9.0)\vec{i} + (2.0t - 5.0)\vec{j}\] where $r$ is in meters and $t$ in seconds. When does the object cross the $x$-axis? $ \textbf{(A)}\ 0.0 \text{ s}\qquad\textbf{(B)}\ 0.4 \text{ s}\qquad\textbf{(C)}\ 0.6 \text{ s}\qquad\textbf{(D)}\ 1.5 \text{ s}\qquad\textbf{(E)}\ 2.5 \text{ s}$

Compute $$\frac{\sum_{k=0}^{2022}\sin\frac{k\pi}{3033}}{\sum_{k=0}^{2022}\cos\frac{k\pi}{3033}}.$$

Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that [b](a)[/b] $\triangle ABC$ is acute. [b](b)[/b] $a+b+c > r+r_a+r_b+r_c$.

Determine the largest positive integer $n$ such that the following statement is true: There exists $n$ real polynomials, $P_1(x),\ldots,P_n(x)$ such that the sum of any two of them have no real roots but the sum of any three does.

Outside the square $ABCD$ is constructed the right isosceles triangle $ABD$ with hypotenuse $[AB]$. Let $N$ be the midpoint of the side $[AD]$ and ${M} = CE \cap AB$, ${P} = CN \cap AB$ , ${F} = PE \cap MN$. On the line $FP$ the point $Q$ is considered such that the $[CE$ is the bisector of the angle $QCB$. Prove that $MQ \perp CF$.

Let $ABC$ be a triangle with $\angle A = 90$ and let $D$ be the orthogonal projection of $A$ onto $BC$. The midpoints of $AD$ and $AC$ are called $E$ and $F$, respectively. Let $M$ be the circumcentre of $BEF$. Prove that $AC$ and $ BM$ are parallel.

Let $A_1, A_2, A_3, \ldots , A_{12}$ be the vertices of a regular $12-$gon (dodecagon). Find the number of points in the plane that are equidistant to at least $3$ distinct vertices of this $12-$gon.

$ABCD$ is a convex quadrilateral. $X$ lies on the segment $AB$ with $\frac{AX}{XB} = \frac{m}{n}$. $Y$ lies on the segment $CD$ with $\frac{CY}{YD} = \frac{m}{n}$. $AY$ and $DX$ intersect at $P$, and $BY$ and $CX$ intersect at $Q$. Show that $\frac{S_{XQYP}}{S_{ABCD}} < \frac{mn}{m^2 + mn + n^2}$.

Let $f(x) = x^2 + ax + b$, where $a$ and $b$ are real numbers. If $f(f(1)) = f(f(2)) = 0$, then find $f(0)$.

The circles $\mathcal{K}_1,\mathcal{K}_2$ and $\mathcal{K}_3$ are pairwise externally tangent to each other; the point of tangency betwwen $\mathcal{K}_1$ and $\mathcal{K}_2$ is $T$. One of the external common tangents of $\mathcal{K}_1$ and $\mathcal{K}_2$ meets $\mathcal{K}_3$ at points $P$ and $Q$. Prove that the internal common tangent of $\mathcal{K}_1$ and $\mathcal{K}_2$ bisects the arc $PQ$ of $\mathcal{K}_3$ which is closer to $T$.

Define the sequnce ${(a_n)}_{n\ge1}$ by $a_1=1$ and $a_n=5a_{n-1}+3^{n-1}$ for $n\ge2$. Find the greatest power of $2$ that divides $a_{2^{2019}}$.

There are $10$ objects of total weight $20$, each of the weights being a positive integers. Given that none of the weights exceeds $10$ , prove that the ten objects can be divided into two groups that balance each other when placed on 2 pans of a balance.

If $a,b,c$ are the roots of $x^3+20x^2+1x+5$, compute $(a^2+1)(b^2+1)(c^2+1)$. [i]2015 CCA Math Bonanza Tiebreaker Round #2[/i]

Farmer James has three types of cows on his farm. A cow with zero legs is called a $\textit{ground beef}$, a cow with one leg is called a $\textit{steak}$, and a cow with two legs is called a $\textit{lean beef}$. Farmer James counts a total of $20$ cows and $18$ legs on his farm. How many more $\textit{ground beef}$s than $\textit{lean beef}$s does Farmer James have? [i]Proposed by James Lin[/i]

We divide $999\times 999$ square into the angles with $3$ cells. Prove, that number of ways is divided by $2^7$.( Angle is a figure, that we can get if we remove one cell from $2 \times 2$ square).