Find all natural numbers $a$ and $b$ such that \[a|b^2, \quad b|a^2 \mbox{ and } a+1|b^2+1.\]

The diagram shows a parabola, a line perpendicular to the parabola's axis of symmetry, and three similar isosceles triangles each with a base on the line and vertex on the parabola. The two smaller triangles are congruent and each have one base vertex on the parabola and one base vertex shared with the larger triangle. The ratio of the height of the larger triangle to the height of the smaller triangles is $\tfrac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$. [asy] size(200); real f(real x) {return 1.2*exp(2/3*log(16-x^2));} path Q=graph(f,-3.99999,3.99999); path [] P={(-4,0)--(-2,0)--(-3,f(-3))--cycle,(-2,0)--(2,0)--(0,f(0))--cycle,(4,0)--(2,0)--(3,f(3))--cycle}; for(int k=0;k<3;++k) { fill(P[k],grey); draw(P[k]); } draw((-6,0)--(6,0),linewidth(1)); draw(Q,linewidth(1));[/asy]

For a prime number $p$ and a positive integer $n$, denote by $f(p, n)$ the largest integer $k$ such that $p^k \mid n!$. Let $p$ be a given prime number and let $m$ and $c$ be given positive integers. Prove that there exist infinitely many positive integers $n$ such that $f(p, n) \equiv c \pmod m$.

Find out wich of the following polynomials are irreducible. a) $t^4+1$ over $\mathbb{R}$; b) $t^4+1$ over $\mathbb{Q}$; c) $t^3-7t^2+3t+3$ over $\mathbb{Q}$; d) $t^4+7$ over $\mathbb{Z}_{17}$; e) $t^3-5$ over $\mathbb{Z}_{11}$; f) $t^6+7$ over $\mathbb{Q}(i)$.

Let us call "fine" the $2n$-digit number if it is exact square itself and the two numbers represented by its first $n$ digits (first digit may not be zero) and last $n$ digits (first digit may be zero, but it may not be zero itself) are exact squares also. a) Find all two- and four-digit fine numbers. b) Is there any six-digit fine number? c) Prove that there exists $20$-digit fine number. d) Prove that there exist at least ten $100$-digit fine numbers. e) Prove that there exists $30$-digit fine number.

If $f$ is a function from the set of positive integers to itself such that $f(x) \leq x^2$ for all natural $x$, and $f\left( f(f(x)) f(f(y))\right) = xy$ for all naturals $x$ and $y$. Find the number of possible values of $f(30)$. [i]Author: Alex Zhu[/i]

Two circles intersect at points $M$ and $N$. Let $A$ be a point on the first circle, distinct from $M,N$. The lines $AM$ and $AN$ meet the second circle again at $B$ and $C$, respectively. Prove that the tangent to the first circle at $A$ is parallel to $BC$.

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? $\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21$

Given $2^x=8^{y+1}$ and $9^y=3^{x-9}$, find the value of $x+y$. ${{ \textbf{(A)}\ 18 \qquad\textbf{(B)}\ 21 \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 27 }\qquad\textbf{(E)}\ 30 } $

We may say concerning the solution of \[ |x|^2 \plus{} |x| \minus{} 6 \equal{} 0 \] that: $ \textbf{(A)}\ \text{there is only one root}\qquad \textbf{(B)}\ \text{the sum of the roots is }{\plus{}1}\qquad \textbf{(C)}\ \text{the sum of the roots is }{0}\qquad \\ \textbf{(D)}\ \text{the product of the roots is }{\plus{}4}\qquad \textbf{(E)}\ \text{the product of the roots is }{\minus{}6}$

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[2, \; 2^{2}, \; 2^{2^{2}}, \; 2^{2^{2^{2}}}, \cdots \pmod{n}\] is eventually constant.

Points $A$ and $B$ lie on circle $\omega$. Point $P$ lies on the extension of segment $AB$ past $B$. Line $\ell$ passes through $P$ and is tangent to $\omega$. The tangents to $\omega$ at points $A$ and $B$ intersect $\ell$ at points $D$ and $C$ respectively. Given that $AB=7$, $BC=2$, and $AD=3$, compute $BP$.

The marked price of a book was $ 30\%$ less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay? $ \textbf{(A)}\ 25\% \qquad \textbf{(B)}\ 30\% \qquad \textbf{(C)}\ 35\% \qquad \textbf{(D)}\ 60\% \qquad \textbf{(E)}\ 65\%$

From a squared sheet of paper of size $29 \times 29, 99$ pieces, each a $2\times 2$ square, are cut off (all cutting is along the lines bounding the squares). Prove that at least one more piece of size $2\times 2$ may be cut from the remaining part of the sheet. (S Fomin, Leningrad)

Prove that if $1 < a < b < c$, then $$\log_a(\log_a b) + \log_b(\log_b c) + \log_c(\log_c a) > 0.$$

For each positive integer $n,$ let $f(n)$ be either the unique integer $r \in \{0,1, \ldots, n-1\}$ such that $n$ divides $15r-1,$ or $0$ if such $r$ does not exist. Compute $$f(16)+f(17)+f(18)+\cdots+f(300).$$

Prove for positive $a,b,c$ that $$ (a^2+\frac{b^2}{c^2})(b^2+\frac{c^2}{a^2})(c^2+\frac{a^2}{b^2}) \geq abc (a+\frac{1}{a})(b+\frac{1}{b})(c+\frac{1}{c})$$

Answer the following questions. (1) By setting $ x\plus{}\sqrt{x^2\minus{}1}\equal{}t$, find the indefinite integral $ \int \sqrt{x^2\minus{}1}\ dx$. (2) Given two points $ P(p,\ q)\ (p>1,\ q>0)$ and $ A(1,\ 0)$ on the curve $ x^2\minus{}y^2\equal{}1$. Find the area $ S$ of the figure bounded by two lines $ OA,\ OP$ and the curve in terms of $ p$. (3) Let $ S\equal{}\frac{\theta}{2}$. Express $ p,\ q$ in terms of $ \theta$.

In a regular hexagon $ABCDEF$ of side length $8$ and center $K$, points $W$ and $U$ are chosen on $\overline{AB}$ and $\overline{CD}$ respectively such that $\overline{KW} = 7$ and $\angle WKU = 120^{\circ}$. Find the area of pentagon $WBCUK$. [i]Proposed by Bradley Guo[/i]

Prove that a bounded figure cannot have more than one center of symmetry.

Two triangles $ABC$ and $A'B'C'$ are given. The lines $AB$ and $A'B'$ meet at $C_1$ and the lines parallel to them and passing through $C$ and $C'$ meet at $C_2$. The points $A_1,A_2$, $B_1,B_2$ are defined similarly. Prove that $A_1A_2,B_1B_2,C_1C_1$ are either parallel or concurrent.

CCA's B building has 6 rooms on the second floor, labeled B201 to B206, as well as 8 rooms on the first floor, labeled B101 to B108. Annie is currently in room B205. Each minute, she chooses to stay or change floors with equal probability, and chooses a classroom on that floor to go to at random (she can stay in the classroom that she's already in). B104, B108, and B203 are the only rooms that have teachers who will scold her for randomly walking around during class time. The probability that she is first scolded in room B203 can be expressed as $\frac{p}{q}$. Compute $p+q$. [i]2022 CCA Math Bonanza Team Round #2[/i]

The squares in a $9\times 9$ grid are numbered from $11$ to $99$, where the first digit is the row and the second the column. Each square is colored black or white. Squares $44$ and $49$ are black. Every black square shares an edge with at most one other black square, and each white square shares an edge with at most one other white square. What color is square $99$?

Suppose that $P(z), Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z)=R(z)$. What is the minimum possible value of $N$? $\textbf{(A)}\: 0\qquad\textbf{(B)} \: 1\qquad\textbf{(C)} \: 2\qquad\textbf{(D)} \: 3\qquad\textbf{(E)} \: 5$