The sequence $ \{a_n \} $ is defined as follows: $ a_0 = 1 $ and $ {a_n} = \sum \limits_ {k = 1} ^ {[\sqrt n]} {{a_ {n - {k ^ 2 }}}} $ for $ n \ge 1. $ Prove that among $ a_1, a_2, \ldots, a_ {10 ^ 6} $ there are at least $500$ even numbers. (Here, $ [x] $ is the largest integer not exceeding $ x $.)

Is $\log_{10}{8}$ rational?

The area of a convex pentagon $ABCDE$ is $S$, and the circumradii of the triangles $ABC$, $BCD$, $CDE$, $DEA$, $EAB$ are $R_1$, $R_2$, $R_3$, $R_4$, $R_5$. Prove the inequality \[ R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2. \]

Consider two distinct positive integers $a$ and $b$ having integer arithmetic, geometric and harmonic means. Find the minimal value of $|a-b|$. [i]Mircea Fianu[/i]

If $\text{A}$ represents the central atom, in which molecule is the $\text{F-A-F}$ angle the smallest? $ \textbf{(A) } \ce{BF3} \qquad\textbf{(B) }\ce{CF4} \qquad\textbf{(C) }\ce{NF3} \qquad\textbf{(D) }\ce{OF2} \qquad $

Let $ABC$ be a triangle with area $K$. Points $A^*$, $B^*$, and $C^*$ are chosen on $AB$, $BC$, and $CA$ respectively such that $\triangle{A^*B^*C^*}$ has area $J$. Suppose that \[\frac{AA^*}{AB}=\frac{BB^*}{BC}=\frac{CC^*}{CA}=\frac{J}{K}=x\] for some $0<x<1$. What is $x$? [i]2019 CCA Math Bonanza Lightning Round #4.3[/i]

Let $D,E,F$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$ such that $BD=CE=AF$ and $\angle BDF=\angle CED=\angle AFE.$ Show that $\triangle ABC$ is equilateral.

Let $\{u_{n}\}_{n \ge 0}$ be a sequence of positive integers defined by \[u_{0}= 1, \;u_{n+1}= au_{n}+b,\] where $a, b \in \mathbb{N}$. Prove that for any choice of $a$ and $b$, the sequence $\{u_{n}\}_{n \ge 0}$ contains infinitely many composite numbers.

Ella lays out $16$ coins heads up in a $4\times 4$ grid as shown. [img]https://cdn.artofproblemsolving.com/attachments/3/3/a728be9c51b27f442109cc8613ef50d61182a0.png[/img] On a move, Ella can flip all the coins in any row, column, or diagonal (including small diagonals such as $H_1$ & $H_4$). If rotations are considered distinct, how many distinct grids of coins can she create in a finite number of moves?

$n$ pairs are formed from $n$ girls and $n$ boys at random. What is the probability of having at least one pair of girls? For which $n$ the probability is over $0,9?$

In chess tournament participates $n$ participants ($n >1$). In tournament each of participants plays with each other exactly $1$ game. For each game participant have $1$ point if he wins game, $0,5$ point if game is drow and $0$ points if he lose game. If after ending of tournament participant have at least $ 75 % $ of maximum possible points he called $winner$ $of$ $tournament$. Find maximum possible numbers of $winners$ $of$ $tournament$.

Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than $4$. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game? $\textbf{(A) } \text{Bela will always win.}$ $\textbf{(B) } \text{Jenn will always win.} $ $\textbf{(C) } \text{Bela will win if and only if }n \text{ is odd.}$ $\textbf{(D) } \text{Jenn will win if and only if }n \text{ is odd.} $ $\textbf{(E) } \text{Jenn will win if and only if }n > 8.$

A triangle with perimeter $7$ has integer sidelengths. What is the maximum possible area of such a triangle?

Let $a$ and $b$ be real numbers greater than 1 such that $ab=100$. The maximum possible value of $a^{(\log_{10}b)^2}$ can be written in the form $10^x$ for some real number $x$. Find $x$.

Convex pentagon $ABCDE$ has side lengths $AB=5$, $BC=CD=DE=6$, and $EA=7$. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$.

Let $a$ be a positive integer and let $\{a_n\}$ be defined by $a_0 = 0$ and \[a_{n+1 }= (a_n + 1)a + (a + 1)a_n + 2 \sqrt{a(a + 1)a_n(a_n + 1)} \qquad (n = 1, 2 ,\dots ).\] Show that for each positive integer $n$, $a_n$ is a positive integer.