suppose that $\mathcal F\subseteq \bigcup_{j=k+1}^{n}X^{(j)}$ and $|X|=n$. we know that $\mathcal F$ is a sperner family and it's also $H_k$. prove that: $\sum_{B\in \mathcal F}\frac{1}{\dbinom{n-1}{|B|-1}}\le 1$ (15 points)

Evaluate $\int^2_0\frac{x(16-x^2)}{16-x^2+\sqrt{(4-x)(4+x)(12+x^2)}}dx$.

Find the remainder when \[\prod_{i=0}^{100}(1-i^2+i^4)\] is divided by $101$. [i]Victor Wang[/i]

Given $f:\mathbb{R}\to \mathbb{R}$ an arbitrary function and $g:\mathbb{R}\to \mathbb{R}$ a function of the second degree, with the property: for any real numbers m and n equation $f\left( x \right)=mx+n$ has solutions if and only if the equation $g\left( x \right)=mx+n$ has solutions Show that the functions $f$ and $g$ are equal.

Let $ABCD$ be a non-cyclic convex quadrilateral. The feet of perpendiculars from $A$ to $BC,BD,CD$ are $P,Q,R$ respectively, where $P,Q$ lie on segments $BC,BD$ and $R$ lies on $CD$ extended. The feet of perpendiculars from $D$ to $AC,BC,AB$ are $X,Y,Z$ respectively, where $X,Y$ lie on segments $AC,BC$ and $Z$ lies on $BA$ extended. Let the orthocenter of $\triangle ABD$ be $H$. Prove that the common chord of circumcircles of $\triangle PQR$ and $\triangle XYZ$ bisects $BH$.

Let $r$ be a positive constant. If 2 curves $C_1: y=\frac{2x^2}{x^2+1},\ C_2: y=\sqrt{r^2-x^2}$ have each tangent line at their point of intersection and at which their tangent lines are perpendicular each other, then find the area of the figure bounded by $C_1,\ C_2$.

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]

Given a set $S_n = \{1, 2, 3, \ldots, n\}$, we define a [i]preference list[/i] to be an ordered subset of $S_n$. Let $P_n$ be the number of preference lists of $S_n$. Show that for positive integers $n > m$, $P_n - P_m$ is divisible by $n - m$. [i]Note: the empty set and $S_n$ are subsets of $S_n$.[/i]

Let $k$ be a fixed positive integer, $k \le 10$. Given a list of ten numbers, the allowed operation is: choose $k$ numbers from the list, and add $1$ to each of them. Thus, a new list of ten numbers is obtained. If you initially have the list $1,2,3,4,5,6,7,8,9,10$, determine the values of $k$ for which it is possible, through a sequence of allowed operations, to obtain a list that has the ten equal numbers. In each case indicate the sequence.

A reaction in which a carboxylic acid reacts with an alcohol to form an organic compound and water is called $ \textbf{(A) } \text{esterification} \qquad\textbf{(B) } \text{hydrolysis}\qquad\textbf{(C) } \text{neutralization} \qquad\textbf{(D) } \text{saponification}\qquad$

Let $ABC$ be such a triangle that $\sphericalangle BAC = 45^{\circ}$ and $ \sphericalangle ACB > 90^{\circ}.$ Show that $BC + (\sqrt{2} - 1)\cdot CA < AB.$

Around a circular table sit $12$ people, and on the table there are $28$ vases. Two people can see each other, if and only if there is no vase lined with them. Prove that there are at least two people who can be seen.

The smallest possible volume of a cylinder that will fit nine spheres of radius 1 can be expressed as $x\pi$ for some value of $x$. Compute $x$. [i]2022 CCA Math Bonanza Team Round #3[/i]

The incircle of scalene triangle $\triangle{ABC}$ is tangent to $\overline{BC}, \overline{AC},$ and $\overline{AB}$ at points $D, E,$ and $F,$ respectively. The line $EF$ intersects line $BC$ at $P$ and line $AD$ at $Q.$ The circumcircle of $\triangle{AEF}$ intersects line $AP$ again at point $R \neq A.$ If $QE=3, QF=4, $ and $QR=8,$ find the area of triangle $\triangle{AEF}.$

Show that there are infinitely many polynomials P with real coefficients such that if x, y, and z are real numbers such that $x^2+y^2+z^2+2xyz=1$, then $$P\left(x\right)^2+P\left(y\right)^2+P\left(z\right)^2+2P\left(x\right)P\left(y\right)P\left(z\right) = 1$$

The 16 fields of a $4 \times 4$ checker board can be arranged in 18 lines as follows: the four lines, the four columns, the five diagonals from north west to south east and the five diagonals from north east to south west. These diagonals consists of 2,3 or 4 edge-adjacent fields of same colour; the corner fields of the chess board alone do not form a diagonal. Now, we put a token in 10 of the 16 fields. Each of the 18 lines contains an even number of tokens contains a point. What is the highest possible point number when can be achieved by optimal placing of the 10 tokens. Explain your answer.

Find all linear homogeneous differential equations with continuous coefficients (on the whole real line) such that for any solution $ f(t)$ and any real number $ c,f(t\plus{}c)$ is also a solution.

Real positive numbers $a, b, c$ satisfy $abc=1$. Prove the inequality $$\frac{a^2+b^2}{a^4+b^4}+\frac{b^2+c^2}{b^4+c^4}+\frac{c^2+a^2}{c^4+a^4}\leq a+b+c.$$

What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is [i]palindromic[/i] if the sequence of decimal digits are the same when read backwards.)

Find the sum of the squares of all real numbers satisfying the equation \[x^{256}-256^{32}=0.\] $\textbf{(A) }8\qquad\textbf{(B) }128\qquad\textbf{(C) }512\qquad\textbf{(D) }65,536\qquad\textbf{(E) }2(256^{32})$

\[\int_{-10}^{10}|4-|3-|2-|1-|x|||||\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1 ?$ [asy]unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white);[/asy] $\textbf{(A) } 4 \pi \sqrt{3} \qquad\textbf{(B) } 7 \pi \qquad\textbf{(C) } \pi(3\sqrt{3} +2) \qquad\textbf{(D) } 10 \pi (\sqrt{3} - 1) \qquad\textbf{(E) } \pi(\sqrt{3} + 6)$

The sides $AD$ and $BC$ of a convex quadrilateral $ABCD$ are extended to meet at $E$. Let $H$ and $G$ be the midpoints of $BD$ and $AC$, respectively. Find the ratio of the area of the triangle $EHG$ to that of the quadrilateral $ABCD$.

Ella the Witch was mixing a magic elixir which consisted of three components: $140$ ml of reindeer moss tea, $160$ ml of fly agaric extract, and $50$ ml of moonshine. She took an empty $350$ ml bottle, poured $140$ ml of reindeer moss tea into it and started adding fly agaric extract when she was disturbed by its black cat Mehsto. So she mistakenly poured too much fly agaric extract into the bottle and noticed her fault only later when the bottle Riled before all $50$ ml of moonshine was added. Ella made quick calculations, carefully shaked up the contents of the bottle, poured out some part of liquid and added some amount of mixture of reindeer moss tea and fly agaric extract taken in a certain proportion until the bottle was full again and the elixir had exactly the right compositsion. Which was the proportion of reindeer moss tea and fly agaric extract in the mixture that Ella added into the bottle?

A $300\times 300$ board is arbitrarily filled with $2\times 1$ dominoes with no overflow, underflow, or overlap. (Tokens can be placed vertically or horizontally.) Decide if it is possible to paint the tiles with three different colors, so that the following conditions are met: $\bullet$ Each token is painted in one and only one of the colors. $\bullet$ The same number of tiles are painted in each color. $\bullet$ No piece is a neighbor of more than two pieces of the same color. Note: Two dominoes are [i]neighbors [/i]if they share an edge.