Each diagonal of a quadrangle divides it into two isosceles triangles. Is it true that the quadrangle is a diamond?

In $\varDelta{ABC}$,the excircle of $A$ is tangent to segment $BC$,line $AB$ and $AC$ at $E,D,F$ respectively.$EZ$ is the diameter of the circle.$B_1$ and $C_1$ are on $DF$, and $BB_1\perp{BC}$,$CC_1\perp{BC}$.Line $ZB_1,ZC_1$ intersect $BC$ at $X,Y$ respectively.Line $EZ$ and line $DF$ intersect at $H$,$ZK$ is perpendicular to $FD$ at $K$.If $H$ is the orthocenter of $\varDelta{XYZ}$,prove that:$H,K,X,Y$ are concyclic.

Nancy shuffles a deck of $52$ cards and spreads the cards out in a circle face up, leaving one spot empty. Andy, who is in another room and does not see the cards, names a card. If this card is adjacent to the empty spot, Nancy moves the card to the empty spot, without telling Andy; otherwise nothing happens. Then Andy names another card and so on, as many times as he likes, until he says "stop." [list][b](a)[/b] Can Andy guarantee that after he says "stop," no card is in its initial spot? [b](b)[/b] Can Andy guarantee that after he says "stop," the Queen of Spades is not adjacent to the empty spot?[/list]

The area of a triangle is $S$ and the sum of the lengths of its sides is $L$. Prove that $36S \leq L^2\sqrt 3$ and give a necessary and sufficient condition for equality.

A number is [i]interesting [/i]if it is a $6$-digit integer that contains no zeros, its first $3$ digits are strictly increasing, and its last $3$ digits are non-increasing. What is the average of all interesting numbers?

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: [list=disc] [*]every term in the sequence is less than or equal to $2^{2023}$, and [*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\] [/list]

This graph depicts the torque output of a hypothetical gasoline engine as a function of rotation frequency. The engine is incapable of running outside of the graphed range. [asy] size(200); draw((0,0)--(10,0)--(10,7)--(0,7)--cycle); for (int i=1;i<10;++i) { draw((i,0)--(i,7),dashed+linewidth(0.5)); } for (int j=1;j<7;++j) { draw((0,j)--(10,j),dashed+linewidth(0.5)); } draw((0,0)--(0,-0.3)); draw((4,0)--(4,-0.3)); draw((8,0)--(8,-0.3)); draw((0,0)--(-0.3,0)); draw((0,2)--(-0.3,2)); draw((0,4)--(-0.3,4)); draw((0,6)--(-0.3,6)); label("0",(0,-0.5),S); label("1000",(4,-0.5),S); label("2000",(8,-0.5),S); label("0",(-0.5,0),W); label("10",(-0.5,2),W); label("20",(-0.5,4),W); label("30",(-0.5,6),W); label("I",(1,-1.5),S); label("II",(6,-1.5),S); label("III",(9,-1.5),S); label(scale(0.95)*"Engine Revolutions per Minute",(5,-3.5),N); label(scale(0.95)*rotate(90)*"Output Torque (Nm)",(-1.5,3),W); path A=(0.9,2.7)--(1.213, 2.713)-- (1.650, 2.853)-- (2.087, 3)-- (2.525, 3.183)-- (2.963, 3.471)-- (3.403, 3.888)-- (3.823, 4.346)-- (4.204, 4.808)-- (4.565, 5.277)-- (4.945, 5.719)-- (5.365, 6.101)-- (5.802, 6.298)-- (6.237, 6.275)-- (6.670, 6.007)-- (7.101, 5.600)-- (7.473, 5.229)-- (7.766, 4.808)-- (8.019, 4.374)-- (8.271, 3.894)-- (8.476, 3.445)-- (8.568, 2.874)-- (8.668, 2.325)-- (8.765, 1.897)-- (8.794, 1.479)--(8.9,1.2); draw(shift(0.1*right)*shift(0.2*down)*A,linewidth(3)); [/asy] At what engine RPM (revolutions per minute) does the engine produce maximum power? (A) $\text{I}$ (B) At some point between $\text{I}$ and $\text{II}$ (C) $\text{II}$ (D) At some point between $\text{II}$ and $\text{III}$ (E) $\text{III}$

For any positive integer $k$ consider the sequence $$a_n=\sqrt{k+\sqrt{k+\dots+\sqrt k}},$$ where there are $n$ square-root signs on the right-hand side. (a) Show that the sequence converges, for every fixed integer $k\ge 1$. (b) Find $k$ such that the limit is an integer. Furthermore, prove that if $k$ is odd, then the limit is irrational.

For every positive integer $n$, we define the function $p_n$ for $x\ge 1$ by $$p_n(x) = \frac12 \left(\left(x+\sqrt{x^2-1}\right)^n+\left(x-\sqrt{x^2-1}\right)^n\right).$$ Prove that $p_n(x) \ge 1$ and that $p_{mn}(x) = p_m(p_n(x))$.

The lock of a safe consists of 3 wheels, each of which may be set in 8 different ways positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations which must be tried if one is to guarantee being able to open the safe (assuming the "right combination" is not known)?

A point $X$ is marked on the base $BC$ of an isosceles $\triangle ABC$, and points $P$ and $Q$ are marked on the sides $AB$ and $AC$ so that $APXQ$ is a parallelogram. Prove that the point $Y$ symmetrical to $X$ with respect to line $PQ$ lies on the circumcircle of the $\triangle ABC$. [i]($5$ points)[/i]

Compute the probability that a random permutation of the letters in BERKELEY does not have the three E’s all on the same side of the Y.

The positive integers $x$ and $y$ are such that $x^{2022}+x+y^2$ is divisible by $xy$. a) Give an example of such integers $x$ and $y$, with $x>y$. b) Prove that $x$ is a perfect square.

We observe that number $10001=73\cdot137$ is not prime. Show that every member of infinite sequence $10001, 100010001, 1000100010001,...$ is not prime

Zadam Heng bets Saniel Dun that he can win in a free throw contest. Zadam shoots until he has made $5$ shots. He wins if this takes him $9$ or fewer attempts. The probability that Zadam makes any given attempt is $\frac{1}{2}$. What is the probability that Zadam Heng wins the bet? [i]2018 CCA Math Bonanza Individual Round #4[/i]