Let $a_1$, $a_2$, $a_3$, $\ldots$ be a sequence of positive integers such that $a_1=2021$ and $$\sqrt{a_{n+1}-a_n}=\lfloor \sqrt{a_n} \rfloor. $$ Show that there are infinitely many odd numbers and infinitely many even numbers in this sequence. [i] Proposed by Li4, Tsung-Chen Chen, and Ming Hsiao.[/i]

To prepare for an Olympiad $20$ students went to a coach. The coach gave them $20$ problems and it turned out that (a) each of the students solved two problems and (b) each problem was solved by twostudents. Prove that it is possible to organize the coaching so that each student would discuss one of the problems that (s)he had solved, and so that all problems would be discussed.

A sequence of positive integers $a_1,a_2,\ldots $ is such that for each $m$ and $n$ the following holds: if $m$ is a divisor of $n$ and $m<n$, then $a_m$ is a divisor of $a_n$ and $a_m<a_n$. Find the least possible value of $a_{2000}$.

Let $A, B \in M_6 (\mathbb{Z} )$ such that $A \equiv I \equiv B \text{ mod }3$ and $A^3 B^3 A^3 = B^3$. Prove that $A = I$. Here $M_6 (\mathbb{Z} )$ indicates the $6$ by $6$ matrices with integer entries, $I$ is the identity matrix, and $X \equiv Y \text{ mod }3$ means all entries of $X-Y$ are divisible by $3$.

Exists a triangle whose three altitudes have lengths $1, 2$ and $3$ respectively?

If $\left(\log_3 x\right)\left(\log_x 2x\right)\left( \log_{2x} y\right)=\log_{x}x^2$, then $y$ equals: $ \textbf{(A)}\ \frac92\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 81 $

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

Find all functions $f: \mathbb{R}\to\mathbb{R}$ that satidfy : $$2f(x+y+xy)= a f(x)+ bf(y)+f(xy)$$ for any $x,y \in\mathbb{R}$ όπου $a,b\in\mathbb{R}$ with $a^2-a\ne b^2-b$

$P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches. Then $PB$, in inches, equals: $\textbf{(A) }2\sqrt{3}\qquad\textbf{(B) }3\sqrt{2}\qquad\textbf{(C) }3\sqrt{3}\qquad\textbf{(D) }4\sqrt{2}\qquad \textbf{(E) }2$ [asy] draw((0,0)--(6.5,0)--(6.5,4.5)--(0,4.5)--cycle); draw((2.5,1.5)--(0,0)); draw((2.5,1.5)--(0,4.5)); draw((2.5,1.5)--(6.5,4.5)); draw((2.5,1.5)--(6.5,0),linetype("8 8")); label("$A$",(0,0),dir(-135)); label("$B$",(6.5,0),dir(-45)); label("$C$",(6.5,4.5),dir(45)); label("$D$",(0,4.5),dir(135)); label("$P$",(2.5,1.5),dir(-90)); label("$3$",(1.25,0.75),dir(120)); label("$4$",(1.25,3),dir(35)); label("$5$",(4.5,3),dir(120)); //Credit to bobthesmartypants for the diagram [/asy]

For real $a > 1$ find$$\lim \limits_{n \to \infty}\sqrt[n]{\prod \limits_{k=2}^n \left(a-a^{\frac{1}{k}}\right)}$$

1. What is the probability that a randomly chosen word of this sentence has exactly four letters?

Consider the polynomials $P\left(x\right)=16x^4+40x^3+41x^2+20x+16$ and $Q\left(x\right)=4x^2+5x+2$. If $a$ is a real number, what is the smallest possible value of $\frac{P\left(a\right)}{Q\left(a\right)}$? [i]2016 CCA Math Bonanza Team #6[/i]

I ponder some numbers in bed, All products of three primes I've said, Apply $\phi$ they're still fun: now Elev'n cubed plus one. What numbers could be in my head?

Two numbers $1+\sqrt[3]{2}+\sqrt[3]{4}$ and $1+2\sqrt[3]{2}+3\sqrt[3]{4}$ are given. In one move you can do one of the following operations: 1. Replace one of the numbers $a$ with either $a-\sqrt[3]{2}$ or $-2a$ 2. Replace both numbers $a$ and $b$ with $a-b$ and $a+b$ (you can choose the order of $a$ and $b$ yourself) Prove that the obtained numbers are always non-zero

Suppose $ABC$ is a triangle with $M$ the midpoint of $BC$. Suppose that $AM$ intersects the incircle at $K,L$. We draw parallel line from $K$ and $L$ to $BC$ and name their second intersection point with incircle $X$ and $Y$. Suppose that $AX$ and $AY$ intersect $BC$ at $P$ and $Q$. Prove that $BP=CQ$.

Prove that in any set consisting of $117$ pairwise distinct three-digit numbers, you can choose $4$ pairwise disjoint subsets in which the sums of numbers are equal.

Each field of the table $(mn + 1) \times (mn + 1)$ contains a real number from the interval $[0, 1]$. The sum the numbers in each square section of the table with dimensions $n x n$ is equal to $n$. Determine how big it can be sum of all numbers in the table.

A palindrome is a number that is the same when its digits are reversed. Find all numbers that have at least one multiple that is a palindrome.

Each vertex of a convex hexagon is the center of a circle whose radius is equal to the shorter side of the hexagon that contains the vertex. Show that if the intersection of all six circles (including their boundaries) is not empty, then the hexagon is regular.

Let $p$ be a prime and let $f(x) = ax^2 + bx + c$ be a quadratic polynomial with integer coefficients such that $0 < a, b, c \le p$. Suppose $f(x)$ is divisible by $p$ whenever $x$ is a positive integer. Find all possible values of $a + b + c$.

Let $ABC$ be a triangle inscribed in circle $C$. Circles $C_1, C_2, C_3$ are tangent internally with circle $C$ in $A_1, B_1, C_1$ and tangent to sides $[BC], [CA], [AB]$ in points $A_2, B_2, C_2$ respectively, so that $A, A_1$ are on one side of $BC$ and so on. Lines $A_1A_2, B_1B_2$ and $C_1C_2$ intersect the circle $C$ for second time at points $A’,B’$ and $C’$, respectively. If $ M = BB’ \cap CC’$, prove that $m (\angle MAA’) = 90^\circ$ .

For all positive reals $a$ and $b$, show that $$\frac{a^2+b^2}{2a^5b^5} + \frac{81a^2b^2}{4} + 9ab > 18.$$

Let $A(n)$ denote the number of sequences $a_1\ge a_2\ge\cdots{}\ge a_k$ of positive integers for which $a_1+\cdots{}+a_k = n$ and each $a_i +1$ is a power of two $(i = 1,2,\cdots{},k)$. Let $B(n)$ denote the number of sequences $b_1\ge b_2\ge \cdots{}\ge b_m$ of positive integers for which $b_1+\cdots{}+b_m =n$ and each inequality $b_j\ge 2b_{j+1}$ holds $(j=1,2,\cdots{}, m-1)$. Prove that $A(n) = B(n)$ for every positive integer $n$. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

For positive integers $n,$ let $f(n)$ equal the number of subsets of the first $13$ positive integers whose members sum to $n.$ Compute \[ \sum_{n=46}^{86} f(n). \]

A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle? $ \textbf{(A) }2\sqrt{3}\qquad \textbf{(B) }4\qquad \textbf{(C) }3\sqrt{2}\qquad \textbf{(D) }2\sqrt{5}\qquad \textbf{(E) }5\qquad $