In isosceles triangle $ABC$ with $AB=AC$, consider point $D$ on the base $BC$ and point $E$ on side $AC$ such that $ \angle BAD = 2 \angle CDE$. Prove that $AD=AE$.

Find all positive integers n with the property that $n^3 - 1$ is a perfect square.

Let $ S$ be a finite family of squares on a plane such that every point on that plane is contained in at most $ k$ squares in $ S$. Prove that $ P$ can be divided into $ 4(k\minus{}1)\plus{}1$ sub-family such that in each sub-family, each pair of squares are disjoint.

Prove that for every positive integer $n,$ the set $\{2,3,4,\ldots,3n+1\}$ can be partitioned into $n$ triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle. [i]Proposed by Canada[/i]

[u]Round 1[/u] [b]p1.[/b] Let $ABCDEF$ be a regular hexagon. How many acute triangles have all their vertices among the vertices of $ABCDEF$? [b]p2.[/b] A rectangle has a diagonal of length $20$. If the width of the rectangle is doubled, the length of the diagonal becomes $22$. Given that the width of the original rectangle is $w$, compute $w^2$. [b]p3.[/b] The number $\overline{2022A20B22}$ is divisible by 99. What is $A + B$? [u]Round 2[/u] [b]p4.[/b] How many two-digit positive integers have digits that sum to at least $16$? [b]p5.[/b] For how many integers $k$ less than $10$ do there exist positive integers x and y such that $k =x^2 - xy + y^2$? [b]p6.[/b] Isosceles trapezoid $ABCD$ is inscribed in a circle of radius $2$ with $AB \parallel CD$, $AB = 2$, and one of the interior angles of the trapezoid equal to $110^o$. What is the degree measure of minor arc $CD$? [u]Round 3[/u] [b]p7.[/b] In rectangle $ALEX$, point $U$ lies on side $EX$ so that $\angle AUL = 90^o$. Suppose that $UE = 2$ and $UX = 12$. Compute the square of the area of $ALEX$. [b]p8.[/b] How many digits does $20^{22}$ have? [b]p9.[/b] Compute the units digit of $3 + 3^3 + 3^{3^3} + ... + 3^{3^{...{^3}}}$ , where the last term of the series has $2022$ $3$s. [u]Round 4[/u] [b]p10.[/b] Given that $\sqrt{x - 1} + \sqrt{x} = \sqrt{x + 1}$ for some real number $x$, the number $x^2$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p11.[/b] Eric the Chicken Farmer arranges his $9$ chickens in a $3$-by-$3$ grid, with each chicken being exactly one meter away from its closest neighbors. At the sound of a whistle, each chicken simultaneously chooses one of its closest neighbors at random and moves $\frac12$ of a unit towards it. Given that the expected number of pairs of chickens that meet can be written as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers, compute $p + q$. [b]p12.[/b] For a positive integer $n$, let $s(n)$ denote the sum of the digits of $n$ in base $10$. Find the greatest positive integer $n$ less than $2022$ such that $s(n) = s(n^2)$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2949432p26408285]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Andrew takes a square sheet of paper $ABCD$ of side length $1$ and folds a kite shape. To do this, he takes the corners at $B$ and $D$ and folds the paper such that both corners now rest at a point $E$ on $AC$. This fold results in two creases $CF$ and $CG$, respectively, where $F$ lies on $AB$ and $G$ lies on $AD$. Compute the length of $FG$.

Burrito Bear has a white unit square. She inscribes a circle inside of the square and paints it black. She then inscribes a square inside the black circle and paints it white. Burrito repeats this process indefinitely. The total black area can be expressed as $\frac{a\pi+b}{c}$. Find $a+b+c$. [i]2022 CCA Math Bonanza Individual Round #4[/i]

Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the $ 13$ visible numbers have the greatest possible sum. What is that sum? [asy]unitsize(.8cm); pen p = linewidth(.8pt); draw(shift(-2,0)*unitsquare,p); label("1",(-1.5,0.5)); draw(shift(-1,0)*unitsquare,p); label("2",(-0.5,0.5)); label("32",(0.5,0.5)); draw(shift(1,0)*unitsquare,p); label("16",(1.5,0.5)); draw(shift(0,1)*unitsquare,p); label("4",(0.5,1.5)); draw(shift(0,-1)*unitsquare,p); label("8",(0.5,-0.5));[/asy]$ \textbf{(A)}\ 154 \qquad \textbf{(B)}\ 159 \qquad \textbf{(C)}\ 164 \qquad \textbf{(D)}\ 167 \qquad \textbf{(E)}\ 189$

Compute the sum of all positive integers $n$ such that the median of the $n$ smallest prime numbers is $n$. [i]Proposed by Luke Robitaille[/i]

Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number.

Let $b_m$ be numbers of factors $2$ of the number $m!$ (that is, $2^{b_m}|m!$ and $2^{b_m+1}\nmid m!$). Find the least $m$ such that $m-b_m = 1990$.

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

We are given the set $$M = \{-2^{2022}, -2^{2021}, . . . , -2^{2}, -2, -1, 1, 2, 2^2, . . . , 2^{2021}, 2^{2022}\}.$$ Let $T$ be a subset of $M$, such that neighbouring numbers have the same difference when the elements are ordered by size. (a) Determine the maximum number of elements that such a set $T$ can contain. (b) Determine all sets $T$ with the maximum number of elements. [i](Walther Janous)[/i]

Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings: (1) $\sum_{v\in V} f(v)=|E|$; (2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the number of uncolored vertices adjacent to $v$. Let $m(G)$ be the number of good maps. Prove that if every vertex in $G$ is adjacent to at least one another vertex, then $n\leq m(G)\leq n!$.

Let $R=gS-4$. When $S=8$, $R=16$. When $S=10$, $R$ is equal to: $\textbf{(A)}\ 11 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ \text{None of these}$

In a triangle $ABC$ , $B_{1}$ and $C_{1}$ are the midpoints of $AC$ and $AB$ respectively, and $I$ is the incenter. The lines $B_{1}I$ and $C_{1}I$ meet $AB$ and $AC$ respectively at $C_{2}$ and $B_{2}$ . If the areas of $\Delta ABC$ and $\Delta AB_{2}C_{2}$ are equal, find $\angle{BAC}$ .

Let $ABC$ be an acute-angled triangle with $AC < AB$ and circumradius $R$. Furthermore, let $D$ be the foot ofthe altitude from $A$ on $BC$ and let $T$ denote the point on the line $AD$ such that $AT = 2R$ holds with $D$ lying between $A$ and $T$. Finally, let $S$ denote the mid-point of the arc $BC$ on the circumcircle that does not include $A$. Prove: $\angle AST = 90^\circ$. (Karl Czakler)

Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?

Let $A$ and $B$ be the two foci of an ellipse and let $P$ be a point on this ellipse. Prove that the focal radii of $P$ (that is, the segments $\overline{AP}$ and $\overline{BP}$ ) form equal angles with the tangent to the ellipse at $P$.

Let $C$ be the graph of $xy = 1$, and denote by $C^*$ the reflection of $C$ in the line $y = 2x$. Let the equation of $C^*$ be written in the form \[ 12x^2 + bxy + cy^2 + d = 0. \] Find the product $bc$.

Prove that for each positive integer $n$, the sum of the numbers of digits of $4^n$ and of $25^n$ (in the decimal system) is odd.

Let natural $n \ge 2$ be given. Let Laura be a student in a class of more than $n+2$ students, all of which participated in an olympiad and solved some problems. Additionally, it is known that: [list] [*] for every pair of students there is exactly one problem that was solved by both students; [*] for every pair of problems there is exactly one student who solved both of them; [*] one specific problem was solved by Laura and exactly $n$ other students. [/list] Determine the number of students in Laura's class.

Prove that for all primes $p>3$, $\binom{2p}{p}-2$ is divisible by $p^3$

On the sides $AD$ and $BC$ of a rhombus $ABCD$ we consider the points $M$ and $N$ respectively. The line $MC$ intersects the segment $BD$ in the point $T$, and the line $MN$ intersects the segment $BD$ in the point $U$. We denote by $Q$ the intersection between the line $CU$ and the side $AB$ and with $P$ the intersection point between the line $QT$ and the side $CD$. Prove that the triangles $QCP$ and $MCN$ have the same area.