In the isosceles triangle $ABC$ with $AB=AC$, the center of $\odot O$ is the midpoint of the side $BC$, and $AB,AC$ are tangent to the circle at points $E,F$ respectively. Point $G$ is on $\odot O$ with $\angle AGE = 90^{\circ}$. A tangent line of $\odot O$ passes through $G$, and meets $AC$ at $K$. Prove that line $BK$ bisects $EF$.

On the table was a pile of $135$ chocolate chips. Phil ate $\tfrac49$ of the chips, Eric ate $\tfrac4{15}$ of the chips, and Beverly ate the rest of the chips. How many chips did Beverly eat?

A cat can see $1$ mile in any direction. The cat walks around the $13$ mile perimeter of a triangle. Over the course of its walk, it sees every point inside of this triangle. What is the largest possible area, in square miles, of the total region it sees? [i]2020 CCA Math Bonanza Team Round #6[/i]

Let $ ABCD$ be a trapezoid with $ AB\parallel CD,AB>CD$ and $ \angle{A} \plus{} \angle{B} \equal{} 90^\circ$. Prove that the distance between the midpoints of the bases is equal to the semidifference of the bases.

As you know, the moon revolves around the earth. We assume that the Earth and the Moon are points, and the Moon rotates around the Earth in a circular orbit with a period of one revolution per month. The flying saucer is in the plane of the lunar orbit. It can be jumped through the Moon and the Earth - from the old place (point $A$), it instantly appears in the new (at point $A '$) so that either the Moon or the Earth is in the middle of segment $AA'$. Between the jumps, the flying saucer hangs motionless in outer space. 1) Determine the minimum number of jumps a flying saucer will need to jump from any point inside the lunar orbit to any other point inside the lunar orbit. 2) Prove that a flying saucer, using an unlimited number of jumps, can jump from any point inside the lunar orbit to any other point inside the lunar orbit for any period of time, for example, in a second.

Given the focus $F$ and the directrix $D$ of a parabola $P$ and a line $L$, describe a euclidean construction for the point or points of intersection of $P$ and $L.$ Be sure to identify the case for which there are no points of intersection.

In a parallelogram $ABCD$ of surface area $60$ cm$^2$ , a line is drawn by $D$ that intersects $BC$ at $P$ and the extension of $AB$ at $Q$. If the area of the quadrilateral $ABPD$ is $46$ cm$^2$ , find the area of triangle $CPQ$.

Prove that when three circles share the same chord $AB$, every line through $A$ different from $AB$ determines the same ratio $X Y : Y Z$, where $X$ is an arbitrary point different from $B$ on the first circle while $Y$ and $Z$ are the points where AX intersects the other two circles (labeled so that $Y$ is between $X$ and $Z$).

Consider a finite set of vectors in space $\{a_1, a_2, ... , a_n\}$ and the set $E$ of all vectors of the form $x=\sum_{i=1}^{n}{\lambda _i a_i}$, where $\lambda _i \in \mathbb{R}^{+}\cup \{0\}$. Let $F$ be the set consisting of all the vectors in $E$ and vectors parallel to a given plane $P$. Prove that there exists a set of vectors $\{b_1, b_2, ... , b_p\}$ such that $F$ is the set of all vectors $y$ of the form $y=\sum_{i=1}^{p}{\mu _i b_i}$, where $\mu _i \in \mathbb{R}^{+}\cup \{0\}$.

Let $ABCD$ be a cyclic quadrilateral with its diagonals intersecting at $E$. Let $M$ be the midpoint of $AB$. Suppose that $ME$ is perpendicular to $CD$. Show that either $AC$ is perpendicular to $BD$, or $AB$ is parallel to $CD$.

Let $a_1$ and $a_2$ be postive real numbers. Let $a_{n+2}=1+\frac{a_{n+1}}{a_{n}}$ Prove that $|a_{2012}-2|<10^{-200}$

A cube is cut into 99 smaller cubes, exactly 98 of which are unit cubes. Find the volume of the original cube. (V Proizvolov)

Jane can walk any distance in half the time it takes Hector to walk the same distance. They set off in opposite directions around the outside of the 18-block area as shown. When they meet for the first time, they will be closest to [asy] for(int i = -2; i <= 2; ++i) { draw((i,0)--(i,3),dashed); } draw((-3,1)--(3,1),dashed); draw((-3,2)--(3,2),dashed); draw((-3,0)--(-3,3)--(3,3)--(3,0)--cycle); dot((-3,0)); label("$A$",(-3,0),SW); dot((-3,3)); label("$B$",(-3,3),NW); dot((0,3)); label("$C$",(0,3),N); dot((3,3)); label("$D$",(3,3),NE); dot((3,0)); label("$E$",(3,0),SE); dot((0,0)); label("start",(0,0),S); label("$\longrightarrow$",(0,-0.75),E); label("$\longleftarrow$",(0,-0.75),W); label("$\textbf{Jane}$",(0,-1.25),W); label("$\textbf{Hector}$",(0,-1.25),E); [/asy] $\text{(A)}\ A \qquad \text{(B)}\ B \qquad \text{(C)}\ C \qquad \text{(D)}\ D \qquad \text{(E)}\ E$

Given a collection of sets $X = \{A_1, A_2, ..., A_n\}$. A set $\{a_1, a_2, ..., a_n\}$ is called a single representation of $X$ if $a_i \in A_i$ for all i. Let $|S| = mn$, $S = A_1\cup A_2 \cup ... \cup A_n = B_1 \cup B_2 \cup ... \cup B_n$ with $|A_i| = |B_i| = m$ for all $i$. Prove that $S = C_1 \cup C_2 \cup ... \cup C_n$ where for every $i, C_i $ is a single represenation for $\{A_j\}_{j=1}^n $and $\{B_j\}_{j=1}^n$.

In several cells of a square grid there live hedgehogs. Every hedgehog multiplies the number of hedgehogs in its row by the number of hedgehogs in its column. Is it possible that all the hedgehogs get distinct numbers?

In triangle $[ABC]$, the bisector in $C$ and the altitude passing through $B$ intersect at point $D$. Point $E$ is the symmetric of point $D$ wrt $BC$ and lies on the circle circumscribed to the triangle $[ABC]$. Prove that the triangle is $[ABC]$ isosceles.

Let $a, b, c$ be positive numbers with the sum $1$. Prove the inequality \[\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c} \geq \frac{2}{1+a}+\frac{2}{1+b}+\frac{2}{1+c}.\]

Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 3$. Prove that $$\frac{x + 3}{y + z} + \frac{y + 3}{z + x} + \frac{z + 3}{x + y} + 3 \ge 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^2}{(x + y + z)^3}.$$ Proposed by [i]Petar Filipovski, Macedonia[/i]

Every one of the six trucks of construction company drove for $8$ hours and they all together spent $720$ litres of oil. How many litres should $9$ trucks spend, if every one of them drives for $6$ hours?

A collection of $2n$ letters contains $2$ each of $n$ different letters. The collection is partitioned into $n$ pairs, each pair containing $2$ letters, which may be the same or different. Denote the number of distinct partitions by $u_n$. (Partitions differing in the order of the pairs in the partition or in the order of the two letters in the pairs are not considered distinct.) Prove that $u_{n+1}=(n+1)u_n - \frac{n(n-1)}{2} u_{n-2}.$ [i]Similar Problem :[/i] A pack of $2n$ cards contains $n$ pairs of $2$ identical cards. It is shuffled and $2$ cards are dealt to each of $n$ different players. Let $p_n$ be the probability that every one of the $n$ players is dealt two identical cards. Prove that $\frac{1}{p_{n+1}}=\frac{n+1}{p_n} + \frac{n(n-1)}{2p_{n-2}}.$

I'd really appreciate help on this. (a) Given a set $X$ of points in the plane, let $f_{X}(n)$ be the largest possible area of a polygon with at most $n$ vertices, all of which are points of $X$. Prove that if $m, n$ are integers with $m \geq n > 2$ then $f_{X}(m) + f_{X}(n) \geq f_{X}(m + 1) + f_{X}(n - 1)$. (b) Let $P_0$ be a $1 \times 2$ rectangle (including its interior) and inductively define the polygon $P_i$ to be the result of folding $P_{i-1}$ over some line that cuts $P_{i-1}$ into two connected parts. The diameter of a polygon $P_i$ is the maximum distance between two points of $P_i$. Determine the smallest possible diameter of $P_{2013}$.

Define $\{p_n\}_{n=0}^\infty\subset\mathbb N$ and $\{q_n\}_{n=0}^\infty\subset\mathbb N$ to be sequences of natural numbers as follows: [list] [*]$p_0=q_0=1$; [*]For all $n\in\mathbb N$, $q_n$ is the smallest natural number such that there exists a natural number $p_n$ with $\gcd(p_n,q_n)=1$ satisfying \[\dfrac{p_{n-1}}{q_{n-1}} < \dfrac{p_n}{q_n} < \sqrt 2.\] [/list] Find $q_3$.

Prove that the edges of a finite simple planar graph (with no loops, multiple edges) may be oriented in such a way that at most three fourths of the total number of dges of any cycle share the same orientation. Moreover, show that this is the best global bound possible. Comment: The actual problem in the TST asked to prove that the edges can be $2$-colored so that the same conclusion holds. Under this circumstances, the problem is wrong and a counterexample was found in the contest by Marius Tiba.

Let $n \geq 5$ be a positive integer. There are $n$ stars with values $1$ to $n$, respectively. Anya and Becky play a game. Before the game starts, Anya places the $n$ stars in a row in whatever order she wishes. Then, starting from Becky, each player takes the left-most or right-most star in the row. After all the stars have been taken, the player with the highest total value of stars wins; if their total values are the same, then the game ends in a draw. Find all $n$ such that Becky has a winning strategy. [i] Proposed by Ho-Chien Chen[/i]

p1. A telephone number with $7$ digits is called a [i]Beautiful Number [/i]if the digits are which appears in the first three numbers (the three must be different) repeats on the next three digits or the last three digits. For example some beautiful numbers: $7133719$, $7131735$, $7130713$, $1739317$, $5433354$. If the numbers are taken from $0, 1, 2, 3, 4, 5, 6, 7, 8$ or $9$, but the number the first cannot be $0$, how many Beautiful Numbers can there be obtained? p2. Find the number of natural numbers $n$ such that $n^3 + 100$ is divisible by $n +10$ p3. A function $f$ is defined as in the following table. [img]https://cdn.artofproblemsolving.com/attachments/5/5/620d18d312c1709b00be74543b390bfb5a8edc.png[/img] Based on the definition of the function $f$ above, then a sequence is defined on the general formula for the terms is as follows: $U_1=2$ and $U_{n+1}=f(U_n)$ , for $n = 1, 2, 3, ...$ p4. In a triangle $ABC$, point $D$ lies on side $AB$ and point $E$ lies on side $AC$. Prove for the ratio of areas: $\frac{ADE }{ABC}=\frac{AD\times AE}{AB\times AC}$ p5. In a chess tournament, a player only plays once with another player. A player scores $1$ if he wins, $0$ if he loses, and $\frac12$ if it's a draw. After the competition ended, it was discovered that $\frac12$ of the total value that earned by each player is obtained from playing with 10 different players who got the lowest total points. Especially for those in rank bottom ten, $\frac12$ of the total score one gets is obtained from playing with $9$ other players. How many players are there in the competition?