The localities $P_1, P_2, \dots, P_{1983}$ are served by ten international airlines $A_1,A_2, \dots , A_{10}$. It is noticed that there is direct service (without stops) between any two of these localities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines can offer a round trip with an odd number of landings.

Circles $c_1$ and $c_2$ are tangent internally to circle $c$ at points $A$ and $B$ , respectively, as seen in the figure. The inner tangent common to $c_1$ and $c_2$ touches these circles in $P$ and $Q$ , respectively. Show that the $AP$ and $BQ$ lines intersect the circle $c$ at diametrically opposite points. [img]https://cdn.artofproblemsolving.com/attachments/0/a/9490a4d7ba2038e490a858b14ba21d07377c5d.gif[/img]

Prove that there exists infinetly many natural number $n$ such that at least one of the numbers $2^{2^n}+1$ and $2018^{2^n}+1$ is not a prime.

An [i]$n$-city[/i] is an $n \times n$ grid of positive integers such that every entry greater than 1 is the sum of an entry in the same row and an entry in the same column. Shown below is an example $3$-city. $$\begin{pmatrix} 1 & 1 & 2 \\ 2 & 3 & 1 \\ 6 & 4 & 1 \end{pmatrix}$$ (a) Construct a $5$-city that includes some entry that is at least $150$. (It is acceptable simply to write the $5$-city. You do not need to explain how you found it.) (b) Show that for all $n \ge 1$, the largest entry in an $n$-city is at most $3^{\binom{n}{2}}$.

The lengths, $a$ and $b$, of two sides of a triangle are known. (a) What length should the third side be, in order for the largest angle of the triangle to be of the least possible value? (b) What length should the third side be in order for the smallest angle of the triangle to be of the greatest possible value?

How many strings of ones and zeroes of length 10 are there such that there is an even number of ones, and no zero follows another zero?

Using each of the first eight primes exactly once and several algebraic operations, obtain the result $2010$.

Let $x_1,...,x_n$ be positive real numbers, satisfying $x_1+\dots+x_n=n$. Prove that $\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\leq\frac{4}{x_1\cdot x_2\cdot\dots\cdot x_n}+n-4$.

Let $ABC$ be and acute triangle. Construct a triangle $PQR$ such that $ AB = 2PQ $, $ BC = 2QR $, $ CA = 2 RP $, and the lines $ PQ, QR,$ and $RP$ pass through the points $ A, B , $ and $ C $, respectively. (All six points $ A, B, C, P, Q, $ and $ R $ are distinct.)

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?