A trapezoid has bases with lengths equal to $5$ and $15$ and legs with lengths equal to $13$ and $13.$ Determine the area of the trapezoid.

In a pile you have 100 stones. A partition of the pile in $ k$ piles is [i]good[/i] if: 1) the small piles have different numbers of stones; 2) for any partition of one of the small piles in 2 smaller piles, among the $ k \plus{} 1$ piles you get 2 with the same number of stones (any pile has at least 1 stone). Find the maximum and minimal values of $ k$ for which this is possible.

Demonstrate that the condition necessary so that, in triangle ${ABC}$, the median from ${B}$ is divided into three equal parts by the inscribed circumference of a circle is: ${A/5 = B/10 = C/13}$.

If $57a + 88b + 125c \geq 1148$, where $a,b,c > 0$, what is the minimum value of \[ a^3 + b^3 + c^3 + 5a^2 + 5b^2 + 5c^2? \]

Let $k \ge 0$ and $a, b, c$ be three positive real numbers such that $$\frac{a}{b}+\frac{b}{c}+ \frac{c}{a}= (k + 1)^2 + \frac{2}{k+ 1}.$$ Prove that $$a^2 + b^2 + c^2 \le (k^2 + 1)(ab + bc + ca).$$

The schoolboy wrote a homework essay on the topic “How I spent my summer.” Two of his comrades from a neighboring school decided not to bother themselves with work and rewrote his essay. But while rewriting they made several mistakes - each their own. Before submitting their work, both students gave their essays to four other friends to rewrite (each gave them to two acquaintances). These four schoolchildren do the same, and so on. With each rewrite, all previous mistakes are saved and, possibly, new ones are made. It is known that on some day each new essay contained at least $10$ errors. Prove that there was a day when at least $11$ new mistakes were made in total.

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

[u]Round 5[/u] [b]5.1.[/b] In a triangle $\vartriangle ABC$ with $AB = 48$, let the angle bisectors of $\angle BAC$ and $\angle BCA$ meet at $I$. Given $\frac{[ABI]}{[BCI]}=\frac{24}{7}$ and $\frac{[ACI]}{[ABI]}=\frac{25}{24}$ , find the area of $\vartriangle ABC$. [b]5.2.[/b] At a dinner party, $9$ people are to be seated at a round table. If person $A$ cannot be seated next to person $B$ and person $C$ cannot be next to person $D$, how many ways can the $9$ people be seated? Rotations of the table are indistinguishable. [b]5.3.[/b] Let $f(x)$ be a monic cubic polynomial such that $f(1) = f(7) = f(10) = a$ and $f(2) = f(5) = f(11) = b$. Find $|a - b|$. [u]Round 6[/u] [b]6.1.[/b] If $N$ has $16$ positive integer divisors and the sum of all divisors of $N$ that are multiples of $3$ is $39$ times the sum of divisors of $N$ that are not multiples of $3$, what is the smallest value of $N$? [b]6.2.[/b] In the two parabolas $y = x^2/16$ and $x = y^2/16$, the single line tangent to both parabolas intersects the parabolas at $A$ and $B$. If the parabolas intersect each other at $C$ which is not the origin, find the area of $\vartriangle ABC$. [b]6.3.[/b] Five distinguishable noncollinear points are drawn. How many ways are there to draw segments connecting the points, such that there are exactly two disjoint groups of connected points? Note that a single point can be considered a connected group of points. [u]Round 7[/u] [b]7.1.[/b] Let $a, b$ be positive integers, and $1 = d_1 < d_2 < d_3 < ... < d_n = a$ be the divisors of $a$, and $1 = e_1 < e_2 <e_3 < ... < e_m = b$ be the divisors of b. Given $gcd(a, b) = d_2 = e_6$, find the smallest possible value of $a + b$. [b]7.2.[/b] Let $\vartriangle ABC$ be a triangle such that $AB = 2$ and $AC = 3$. Let X be the point on $BC$ such that $m \angle BAX =\frac13 m\angle BAC$. Given that $AX = 1$, the sum of all possible values of $CX^2$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Find $a + b$. [b]7.3.[/b] Bob has a playlist of $6$ different songs in some order, and he listens to his playlist repeatedly. Every time he finishes listening to the third song in the playlist, he randomly shuffles his playlist and listens to the playlist starting with the new first song. The expected number of times Bob shuffles his songs before he listens each one of his $6$ songs at least once can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Find $a+b$. [u]Round 8[/u] [b]8.1.[/b] $\underline{A}, \underline{B}, \underline{C}, \underline{D}, \underline{E}, \underline{F}, \underline{G}, \underline{H}, \underline{I}$, and $\underline{J}$ represent distinct digits ($0$ to $9$) in the equation $\underline{FBGA} - \underline{ABAC} = \underline{DCE}$ (where $\underline{ABAC}$ and $\underline{F BGA}$ are four-digit numbers, and $\underline{DCE }$ is a three-digit number). If $\underline{A} < \underline{B} < \underline{C} < \underline{D}$ and $\underline{ABCDEF GHIJ}$ is minimized, find $\underline{ABCD} + \underline{EF G} + \underline{HI} + \underline{J}$. [b]8.2.[/b] $\underline{A}, \underline{B}, \underline{C}, \underline{D}, \underline{E}$,,, and $\underline{F}$ represent distinct digits ($0$ to $9$) in the equations $\underline{ABC} \cdot \underline{C} = \underline{DEA}, \underline{ABC} \cdot \underline{D} = \underline{BAF E}$, and $ \underline{DEA} + \underline{BAF E}0 = \underline{BF ACA}$ (where $\underline{ABC}$ and $\underline{DEA}$ are three-digit numbers, $\underline{BAF E}$ is a four-digit number, and $\underline{BF ACA}$ is a five-digit number). Find $\underline{ABC} + \underline{DE} + \underline{F}$. [b]8.3.[/b] $\underline{A}, \underline{B}, \underline{C}, \underline{D}, \underline{E}, \underline{F}, \underline{G}$, and $\underline{H}$ represent distinct digits ($0$ to $9$) in the equations $\underline{ABC } \cdot \underline{D} = \underline{AF GE}$, $\underline{ABC } \cdot \underline{C} = \underline{GHC}$, $\underline{GHC} + \underline{HF F} = \underline{AEHC}$, and $\underline{AF GE}0 + \underline{AEHC} = \underline{AEABC}$ (where $\underline{ABC}$, $\underline{GHC}$ and $\underline{HF F}$ are three-digit numbers, $\underline{AF GE}$ is a four-digit number, and $\underline{AEABC}$ is a five-digit number). Find $\underline{ABCD} + \underline{EF GH}$. [u]Round 9[/u] Estimate the arithmetic mean of all answers to this question. Only integer answers between $0$ to $100, 000$ will count for credit and count toward the average. Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05|I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3129699p28347299]here [/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Several square-shaped papers are situated on a table such that every side of the paper is positioned parallel to the sides of the table. Each paper has a colour, and there are $n$ different coloured papers. It is known that for every $n$ papers with distinct colors, we can always find an overlapping pair of papers. Prove that, using $2n- 2$ nails, it is possible to hammer all the squares of a certain colour to the table.

$AM$ is the median of triangle $ABC$. A perpendicular $CC_1$ is drawn from point $C$ on the bisector of angle $\angle CMA$, a perpendicular $BB_1$ is drawn from point $B$ on the bisector of angle $\angle BMA$. Prove that line $AM$ intersects segment $B_1C_1$ at its midpoint.

A merganser mates every 7th day, a scaup mates every 11th day, and a gadwall mates every 13th day. A merganser, scaup, and gadwall all mate on Day 0. On Days N, N+1, and N+2 the merganser, scaup, and gadwall mate in some order with no two birds mating on the same day. Determine the smallest possible value of N. [i]2022 CCA Math Bonanza Lightning Round 3.4[/i]

Seven identical-looking coins are given, of which four are real and three are counterfeit. The three counterfeit coins have equal weight, and the four real coins have equal weight. It is known that a counterfeit coin is lighter than a real one. In one weighing, one can select two sets of coins and check which set has a smaller total weight, or if they are of equal weight. How many weightings are needed to identify one counterfeit coin?

Let $ \mathbb{A} \subset \mathbb{N} $ such that all elements in $ \mathbb{A} $ can be representable in the form of $ x^2+2y^2 $ , $ x,y \in \mathbb{N} $, and $ x>y $. Let $ \mathbb{B} \subset \mathbb{N} $ such that all elements in $ \mathbb{B} $ can be representable in the form of $\displaystyle \frac{a^3+b^3+c^3}{a+b+c} $ , $ a,b,c \in \mathbb{N} $, and $ a,b,c $ are distinct. a) Prove that $ \mathbb{A} \subset \mathbb{B} $. b) Prove that there exist infinitely many positive integers $n$ satisfy $ n \in \mathbb{B}$ and $ n \not \in \mathbb{A} $

Find all integers $n\geq 3$ for which the following statement is true: If $\mathcal{P}$ is a convex $n$-gon such that $n-1$ of its sides have equal length and $n-1$ of its angles have equal measure, then $\mathcal{P}$ is a regular polygon. (A [i]regular [/i]polygon is a polygon with all sides of equal length, and all angles of equal measure.) [i]Proposed by Ivan Borsenco and Zuming Feng[/i]

In the plane are given $1997$ points. Show that among the pairwise distances between these points, there are at least $32$ different values.

Unit square $ZINC$ is constructed in the interior of hexagon $CARBON$. What is the area of triangle $BIO$?

Let $n$ be a positive integer. Show that there exists a one-to-one function $\sigma : \{1,2,...,n\} \to \{1,2,...,n\}$ such that $$\sum_{k=1}^{n} \frac{k}{(k+\sigma(k))^2} < \frac{1}{2}.$$

An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After five changes, what fractional part of the original area of the black triangle remains black? [asy] unitsize(36); fill((0,0)--(2,0)--(1,sqrt(3))--cycle,gray); draw((0,0)--(2,0)--(1,sqrt(3))--cycle,linewidth(1)); fill((4,0)--(6,0)--(5,sqrt(3))--cycle,gray); fill((5,0)--(9/2,sqrt(3)/2)--(11/2,sqrt(3)/2)--cycle,white); draw((5,sqrt(3))--(4,0)--(5,0)--(9/2,sqrt(3)/2)--(11/2,sqrt(3)/2)--(5,0)--(6,0)--cycle,linewidth(1)); fill((8,0)--(10,0)--(9,sqrt(3))--cycle,gray); fill((9,0)--(17/2,sqrt(3)/2)--(19/2,sqrt(3)/2)--cycle,white); fill((17/2,0)--(33/4,sqrt(3)/4)--(35/4,sqrt(3)/4)--cycle,white); fill((9,sqrt(3)/2)--(35/4,3*sqrt(3)/4)--(37/4,3*sqrt(3)/4)--cycle,white); fill((19/2,0)--(37/4,sqrt(3)/4)--(39/4,sqrt(3)/4)--cycle,white); draw((9,sqrt(3))--(35/4,3*sqrt(3)/4)--(37/4,3*sqrt(3)/4)--(9,sqrt(3)/2)--(35/4,3*sqrt(3)/4)--(33/4,sqrt(3)/4)--(35/4,sqrt(3)/4)--(17/2,0)--(33/4,sqrt(3)/4)--(8,0)--(9,0)--(17/2,sqrt(3)/2)--(19/2,sqrt(3)/2)--(9,0)--(19/2,0)--(37/4,sqrt(3)/4)--(39/4,sqrt(3)/4)--(19/2,0)--(10,0)--cycle,linewidth(1)); label("Change 1",(3,3*sqrt(3)/4),N); label("$\Longrightarrow $",(3,5*sqrt(3)/8),S); label("Change 2",(7,3*sqrt(3)/4),N); label("$\Longrightarrow $",(7,5*sqrt(3)/8),S); [/asy] $\text{(A)}\ \frac{1}{1024} \qquad \text{(B)}\ \frac{15}{64} \qquad \text{(C)}\ \frac{243}{1024} \qquad \text{(D)}\ \frac{1}{4} \qquad \text{(E)}\ \frac{81}{256}$

Let $\Delta ABC$ be an acute triangle with incenter $I$ and orthocenter $H$. $AI$ meets the circumcircle of $\Delta ABC$ again at $M$. Suppose the length $IM$ is exactly the circumradius of $\Delta ABC$. Show that $AH\geq AI$.

For n points \[ P_1;P_2;...;P_n \] in that order on a straight line. We colored each point by 1 in 5 white, red, green, blue, and purple. A coloring is called acceptable if two consecutive points \[ P_i;P_{i+1} (i=1;2;...n-1) \] is the same color or 2 points with at least one of 2 points are colored white. How many ways acceptable color?

Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.

We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without one cell in corner a $P$-rectangle. We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without two cells in opposite (under center of rectangle) corners a $S$-rectangle. Using some squares of size $2 \times 2$, some $P$-rectangles and some $S$-rectangles, one form one rectangle of size $1993 \times 2000$ (figures don’t overlap each other). Let $s$ denote the sum of numbers of squares and $S$-rectangles used in such tiling. Find the maximal value of $s$.

Suppose you have identical coins distributed in several piles with one or more coins in each pile. An action consists of taking two piles, which have an even total of coins among them, and redistribute their coins in two piles so that they end up with the same number of coins. A distribution is [i]levelable[/i] if it is possible, by means of 0 or more operations, to end up with all the piles having the same number of coins. Determine all positive integers $n$ such that, for all positive integers $k$, any distribution of $nk$ coins in $n$ piles is levelable.

Numbers $1992,1993, ... ,2000$ are written in a $3 \times 3$ table to form a magic square (i.e. the sums of numbers in rows, columns and big diagonals are all equal). Prove that the number in the center is $1996$. Which numbers are placed in the corners?

Let set $S$ be the smallest set of positive integers satisfying the following properties: [list] [*] $2$ is in set $S$. [*] If $n^2$ is in set $S$, then $n$ is also in set $S$. [*] If $n$ is in set $S$, then $(n+5)^2$ is also in set $S$. [/list] Determine which positive integers are not in set $S$.