The average (arithmetic mean) of $10$ different positive whole numbers is $10$. The largest possible value of any of these numbers is $\text{(A)}\ 10 \qquad \text{(B)}\ 50 \qquad \text{(C)}\ 55 \qquad \text{(D)}\ 90 \qquad \text{(E)}\ 91$

Jonathan has a magical coin machine which takes coins in amounts of $7, 8$, and $9$. If he puts in $7$ coins, he gets $3$ coins back; if he puts in $8$, he gets $11$ back; and if he puts in $9$, he gets $4$ back. The coin machine does not allow two entries of the same amount to happen consecutively. Starting with $15$ coins, what is the minimum number of entries he can make to end up with $4$ coins?

Let $ f(x,y,z)$ be the polynomial with integer coefficients. Suppose that for all reals $ x,y,z$ the following equation holds: \[ f(x,y,z) \equal{} \minus{} f(x,z,y) \equal{} \minus{} f(y,x,z) \equal{} \minus{} f(z,y,x) \] Prove that if $ a,b,c\in\mathbb{Z}$ then $ f(a,b,c)$ takes an even value

Refer to the diagram below. Let $ABCD$ be a cyclic quadrilateral with center $O$. Let the internal angle bisectors of $\angle A$ and $\angle C$ intersect at $I$ and let those of $\angle B$ and $\angle D$ intersect at $J$. Now extend $AB$ and $CD$ to intersect $IJ$ and $P$ and $R$ respectively and let $IJ$ intersect $BC$ and $DA$ at $Q$ and $S$ respectively. Let the midpoints of $PR$ and $QS$ be $M$ and $N$ respectively. Given that $O$ does not lie on the line $IJ$, show that $OM$ and $ON$ are perpendicular.

If an angle $0^\circ<\theta<30^\circ$ satisfies $\sin\left(90^\circ-\theta\right)\sin\left(60^\circ-\theta\right)\sin\left(30^\circ-\theta\right)=\sin^3\left(\theta\right)$, compute $\sin\left(\theta\right)$. [i]2019 CCA Math Bonanza Lightning Round #4.4[/i]

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

$$\int_0^\frac{\pi}{2}\frac{1}{4-3\cos^2(x)}\,dx$$ [i]Proposed by Connor Gordon[/i]

Let $a,b,c,$ the lengths of the sides of a triangle. Prove that \[\sqrt{\frac{(3a+b)(3b+a)}{(2a+c)(2b+c)}} + \sqrt{\frac{(3b+c)(3c+b)}{(2b+a)(2c+a)}} + \sqrt{\frac{(3c+a)(3a+c)}{(2c+b)(2a+b)}} \geq 4.\]

Let $S$ be a randomly chosen $6$-element subset of the set $\{0,1,2,\ldots,n\}.$ Consider the polynomial $P(x)=\sum_{i\in S}x^i.$ Let $X_n$ be the probability that $P(x)$ is divisible by some nonconstant polynomial $Q(x)$ of degree at most $3$ with integer coefficients satisfying $Q(0) \neq 0.$ Find the limit of $X_n$ as $n$ goes to infinity.

In the parallelogram $ABCD$, a line through $C$ intersects the diagonal $BD$ at $E$ and $AB$ at $F$. If $F$ is the midpoint of $AB$ and the area of $\vartriangle BEC$ is $100$, find the area of the quadrilateral $AFED$.

Let $f(x)$ be a function such that $f(1)=1$ and for $x \geq 1$ $$f'(x)= \frac{1}{x^2 +f(x)^{2}}.$$ Prove that $$\lim_{x\to \infty} f(x)$$ exists and is less than $1+ \frac{\pi}{4}.$

In a triangle $\triangle ABC$, $D$ and $E$ are respectively on $AB$ and $AC$ such that $DE\parallel BC$. $P$ is the intersection of $BE$ and $CD$. $M$ is the second intersection of $(APD)$ and $(BCD)$ , $N$ is the second intersection of $(APE)$ and $(BCE)$. $w$ is the circle passing through $M$ and $N$ and tangent to $BC$. Prove that the lines tangent to $w$ at $M$ and $N$ intersect on $AP$.

Find all primes $1 < p < 100$ such that the equation $x^2-6y^2=p$ has an integer solution $(x, y)$.

For all integers $ {a}_{0},{a}_{1}, \cdot\cdot\cdot {a}_{100}$, find the maximum of ${a}_{5}-2{a}_{40}+3{a}_{60}-4{a}_{95} $ $\bigstar$ 1) ${a}_{0}={a}_{100}=0$ 2) for all $i=0,1,\cdot \cdot \cdot 99, $ $|{a}_{i+1}-{a}_{i}|\le1$ 3) $ {a}_{10}={a}_{90} $

For his birthday, Bert gets a box that holds 125 jellybeans when filled to capacity. A few weeks later, Carrie gets a larger box full of jellybeans. Her box is twice as high, twice as wide and twice as long as Bert's. Approximately, how many jellybeans did Carrie get? $ \text{(A)}\ 250\qquad\text{(B)}\ 500\qquad\text{(C)}\ 625\qquad\text{(D)}\ 750\qquad\text{(E)}\ 1000 $

In a right triangle with legs $a$, $b$ and hypotenuse $c$, draw semicircles with diameters on the sides of the triangle as indicated in the figure. The purple areas have values $X,Y$ . Calculate $X + Y$. [img]https://cdn.artofproblemsolving.com/attachments/1/a/5086dc7172516b0a986ef1af192c15eba4d6fc.png[/img]

Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

Let $\Bbb{Z}^+$ denote the set of positive integers. Find all functions $f:\Bbb{Z}^+\to \Bbb{Z}^+$ that satisfy the following condition: for each positive integer $n,$ there exists a positive integer $k$ such that $$\sum_{i=1}^k f_i(n)=kn,$$ where $f_1(n)=f(n)$ and $f_{i+1}(n)=f(f_i(n)),$ for $i\geq 1. $

in the acute triangle $\triangle ABC$. M is a point in the interior of the segment AC and N is a point on the extension of segment AC such that MN=AC. let D,E be the feet of perpendiculars from M,N onto lines BC,AB respectively prove that the orthocentre of $\triangle ABC$ lies on circumcircle of $\triangle BED$

Suppose that $x$ and $y$ are positive real numbers such that $x^2-xy+2y^2=8$. Find the maximum possible value of $x^2+xy+2y^2$.

Jorge writes a list with an even number of integers, not all equal to $0$ (there may be repeated numbers). Show that Martin can cross out a number from the list, of his choice, so that it is impossible for Jorge to separate the remaining numbers into two groups in such a way that the sum of all the numbers in one group is equal to the sum of all the others. numbers from the other group.

The convex pentagon $ABCDE$ has $\angle A=\angle B=120^{\circ}$, $EA=AB=BC=2$ and $CD=DE=4$. What is the area of $ABCDE$? [asy] draw((0,0)--(1,0)--(1.5,sqrt(3)/2)--(0.5,3sqrt(3)/2)--(-0.5,sqrt(3)/2)--cycle); dot((0,0)); dot((1,0)); dot((1.5,sqrt(3)/2)); dot((0.5,3sqrt(3)/2)); dot((-0.5,sqrt(3)/2)); label("A", (0,0), SW); label("B", (1,0), SE); label("C", (1.5,sqrt(3)/2), E); label("D", (0.5,3sqrt(3)/2), N); label("E", (-.5, sqrt(3)/2), W); [/asy] $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 7\sqrt{3} \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 9\sqrt{3} \qquad\textbf{(E)}\ 12\sqrt{5} $

Two players play a game as follows: there $n > 1$ rounds and $d \geq 1$ is fixed. In the first round A picks a positive integer $m_1$, then B picks a positive integer $n_1 \not = m_1$. In round $k$ (for $k = 2, \ldots , n$), A picks an integer $m_k$ such that $m_{k-1} < m_k \leq m_{k-1} + d$. Then B picks an integer $n_k$ such that $n_{k-1} < n_k \leq n_{k-1} + d$. A gets $\gcd(m_k,n_{k-1})$ points and B gets $\gcd(m_k,n_k)$ points. After $n$ rounds, A wins if he has at least as many points as B, otherwise he loses. For each $(n, d)$ which player has a winning strategy?

Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=12.$ A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P,$ which is 7 units from $\overline{BG}$ and 5 units from $\overline{BC}.$ The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube is given by $m\sqrt{n},$ where $m$ and $n$ are integers and $n$ is not divisible by the square of any prime. Find $m+n.$