Consider the sequence $\{a_k\}_{k \geq 1}$ defined by $a_1 = 1$, $a_2 = \frac{1}{2}$ and \[ a_{k + 2} = a_k + \frac{1}{2}a_{k + 1} + \frac{1}{4a_ka_{k + 1}}\ \textrm{for}\ k \geq 1. \] Prove that \[ \frac{1}{a_1a_3} + \frac{1}{a_2a_4} + \frac{1}{a_3a_5} + \cdots + \frac{1}{a_{98}a_{100}} < 4. \]

Let $k, n$ be positive integers, and let $\alpha_1, \alpha_2, \ldots, \alpha_n$ all be $k$-th roots of unity, satisfying: \[ \alpha_1^j + \alpha_2^j + \cdots + \alpha_n^j = 0 \quad \text{for any } j (0 < j < k). \] Prove that among $\alpha_1, \alpha_2, \ldots, \alpha_n$, each $k$-th root of unity appears the same number of times.

Let $a_1\geq a_2\geq \cdots \geq a_n >0 .$ Prove that$$ \left( \frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)^2\geq \sum_{k=1}^{n} \frac{k(2k-1)}{a^2_1+a^2_2+\cdots+a^2_k}$$

Let $\alpha,\beta,\gamma$ be the angles of a triangle opposite to its sides with lengths $a,b,c$ respectively. Prove the inequality \[a\left(\frac{1}{\beta}+\frac{1}{\gamma}\right)+b\left(\frac{1}{\gamma}+\frac{1}{\alpha}\right)+c\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)\ge2\left(\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\right)\]

Let be $a$ be positive real number. Prove that there are no real numbers $b$ and $c$, with $b < c$, so that for any distinct numbers $x, y \in (b, c)$ we have $|\frac{x+y} {x-y}| \leq a$.

Nine quadratics, $x^2+a_1x+b_1, x^2+a_2x+b_2,...,x^2+a_9x+b_9$ are written on the board. The sequences $a_1, a_2,...,a_9$ and $b_1, b_2,...,b_9$ are arithmetic. The sum of all nine quadratics has at least one real root. What is the the greatest possible number of original quadratics that can have no real roots?

Let $r$ be a positive integer. Find the smallest positive integer $m$ satisfying the condition: For all sets $A_1, A_2, \dots, A_r$ with $A_i \cap A_j = \emptyset$, for all $i \neq j$, and $\bigcup_{k = 1}^{r} A_k = \{ 1, 2, \dots, m \}$, there exists $a, b \in A_k$ for some $k$ such that $1 \leq \frac{b}{a} \leq 1 + \frac{1}{2022}$.

The sequence $(a_n)_{n \in\mathbb{N}}$ is defined by $a_1 = 8, a_2 = 18, a_{n+2} = a_{n+1}a_{n}$. Find all terms which are perfect squares.

The sequence $ \{x_{n}\}_{n \ge 1}$ is defined by \[ x_{1} \equal{} 2, x_{n \plus{} 1} \equal{} \frac {2 \plus{} x_{n}}{1 \minus{} 2x_{n}}\;\; (n \in \mathbb{N}). \] Prove that a) $ x_{n}\not \equal{} 0$ for all $ n \in \mathbb{N}$, b) $ \{x_{n}\}_{n \ge 1}$ is not periodic.

Mr Green measures his rectangular garden by walking two of the sides and finds that it is 15 steps by 20 steps. Each or Mr Green's steps is two feet long. Mr Green expect half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr Green expect from his garden? $ \textbf{(A) }600\qquad\textbf{(B) }800\qquad\textbf{(C) }1000\qquad\textbf{(D) }1200\qquad\textbf{(E) }1400 $

Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations: [list=] [*]$L,$ a rotation of $90^{\circ}$ counterclockwise around the origin; [*]$R,$ a rotation of $90^{\circ}$ clockwise around the origin; [*]$H,$ a reflection across the $x$-axis; and [*]$V,$ a reflection across the $y$-axis. [/list] Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.) $\textbf{(A)}\ 2^{37} \qquad\textbf{(B)}\ 3\cdot 2^{36} \qquad\textbf{(C)}\ 2^{38} \qquad\textbf{(D)}\ 3\cdot 2^{37} \qquad\textbf{(E)}\ 2^{39}$

A point $(X, Y, Z)$ is chosen uniformly at random from the ball of radius $4$ centered at the origin (i.e., the set $\{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \leq 4^2\}$). Compute the probability that the inequalities $X^2 \leq 1$ and $X^2 + Y^2 + Z^2 \geq 1$ simultaneously hold.

Determine all points on the straight line which joins $(4, 11)$ to $(16, 1)$ and whose coordinates are positive integers.

A regular hexagon of area $H$ is inscribed in a convex polygon of area $P$. Show that $P \leq \frac{3}{2}H$. When does the equality occur?

Let $a_0,a_1,\ldots$ be a sequence of positive integers such that $a_0=1$, and for all positive integers $n$, $a_n$ is the smallest composite number relatively prime to all of $a_0,a_1,\ldots,a_{n-1}$. Compute $a_{10}$. [i]2020 CCA Math Bonanza Lightning Round #4.2[/i]

Magicman and his helper want to do some magic trick. They have special card desk. Back of all cards is common color and face is one of $2017$ colors. Magic trick: magicman go away from scene. Then viewers should put on the table $n>1$ cards in the row face up. Helper looks at these cards, then he turn all cards face down, except one, without changing order in row. Then magicman returns on the scene, looks at cards, then show on the one card, that lays face down and names it face color. What is minimal $n$ such that magicman and his helper can has strategy to make magic trick successfully?

Let $I$ be the incenter of triangle $ABC$, and let $A'$, $B'$, and $C'$ be the circumcenters of triangles $IBC$, $ICA$, and $IAB$, respectively. Prove that the circumcircles of triangles $ABC$ and $A'B'C'$ are concentric.

Let O be the circumcenter of triangle ABC. Let H be the orthocenter of triangle ABC. the perpendicular bisector of AB meet AC,BC at D,E. the circumcircle of triangle DEH meet AC,BC,OH again at F,G,L. CH meet FG at T,and ABCT is concyclic. Prove that LHBC is concyclic. graph: https://cdn.luogu.com.cn/upload/image_hosting/w6z6mvm4.png

Let $a,b,c$ be positive real numbers for which $a+b+c=3$. Prove the inequality \[\frac{a^3+2}{b+2}+\frac{b^3+2}{c+2}+\frac{c^3+2}{a+2}\ge3\]

$5$ airway companies operate in a country consisting of $36$ cities. Between any pair of cities exactly one company operates two way flights. If some air company operates between cities $A, B$ and $B, C$ we say that the triple $A, B, C$ is [i]properly-connected[/i]. Determine the largest possible value of $k$ such that no matter how these flights are arranged there are at least $k$ properly-connected triples.

Let $ABC$ be a right angled triangle with $\angle B=90^{\circ}$. Let $I$ be the incentre of triangle $ABC$. Suppose $AI$ is extended to meet $BC$ at $F$ . The perpendicular on $AI$ at $I$ is extended to meet $AC$ at $E$ . Prove that $IE = IF$.

Let $ABC$ be a right triangle with $\angle A=90^\circ.$ Let $A'$ be the midpoint of $BC,$ $M$ be the midpoint of the height $AD$ and $P$ be the intersection of $BM$ and $AA'.$ Prove that $\tan\angle PCB=\sin C\cdot\cos C.$ [i]Daniel Văcărețu[/i]

Given a convex quadrilateral $ABCD$.Let $\ell_A,\ell_B,\ell_C,\ell_D$ be exterior angle bisectors of quadrilateral $ABCD$. Let $\ell_A \cap \ell_B=K,\ell_B \cap \ell_C=L,\ell_C \cap \ell_D=M,\ell_D \cap \ell_A=N$.Prove that if circumcircles of triangles $ABK$ and $CDM$ be externally tangent to each other then circumcircles of the triangles $BCL$ and $DAN$ are externally tangent to each other.(L.Emelyanov)

Let $x$ be the smallest positive integer for which $2x$ is the square of an integer, $3x$ is the third power of an integer, and $5x$ is the fifth power of an integer. Find the prime factorization of $x$. (St. Wagner, Stellenbosch University)

The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]