Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0), (2,0), (2,1),$ and $(0,1)$. $R$ can be divided into two unit squares, as shown; the resulting figure has seven edges. [asy] size(3cm); draw((0,0)--(2,0)--(2,1)--(0,1)--cycle); draw((1,0)--(1,1)); [/asy] How many subsets of these seven edges form a connected figure?

[b]p1.[/b] Alex and Sam have a friend Pat, who is younger than they are. Alex, Sam and Pat all share a birthday. When Pat was born, Alex’s age times Sam’s age was $42$. Now Pat’s age is $33$ and Alex’s age is a prime number. How old is Sam now? Show your work and justify your answer. (All ages are whole numbers.) [b]p2.[/b] Let $ABCD$ be a square with side length $2$. The four sides of $ABCD$ are diameters of four semicircles, each of which lies inside the square. The set of all points which lie on or inside two of these semicircles is a four petaled flower. Find (with proof) the area of this flower. [img]https://cdn.artofproblemsolving.com/attachments/5/5/bc724b9f74c3470434c322020997a533986d33.png[/img] [b]p3.[/b] A prime number is called [i]strongly prime[/i] if every integer obtained by permuting its digits is also prime. For example $113$ is strongly prime, since $113$, $131$, and $311$ are all prime numbers. Prove that there is no strongly prime number such that each of the digits $1, 3, 7$, and $9$ appears at least once in its decimal representation. [b]p4.[/b] Suppose $n$ is a positive integer. Let an be the number of permutations of $1, 2, . . . , n$, where $i$ is not in the $i$-th position, for all $i$ with $1 \le i \le n$. For example $a_3 = 2$, where the two permutations that are counted are $231$, and $312$. Let bn be the number of permutations of $1, 2, . . . , n$, where no $i$ is followed by $i + 1$, for all $i$ with $1 \le i \le n - 1$. For example $b_3 = 3$, where the three permutations that are counted are $132$, $213$, and $321$. For every $n \ge 1$, find (with proof) a simple formula for $\frac{a_{n+1}}{b_n}$. Your formula should not involve summations. Use your formula to evaluate $\frac{a_{2020}}{b_{2019}}$. [b]p5.[/b] Let $n \ge 2$ be an integer and $a_1, a_2, ... , a_n$ be positive real numbers such that $a_1 + a_2 +... + a_n = 1$. Prove that $$\sum^n_{k=1}\frac{a_k}{1 + a_{k+1} - a_{k-1}}\ge 1.$$ (Here $a_0 = a_n$ and $a_{n+1} = a_1$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the sum of the coefficients of the polynomial $(63x-61)^4$.

Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $ \$1$ more than a pink pill, and Al’s pills cost a total of $ \$546$ for the two weeks. How much does one green pill cost? $ \textbf{(A)}\ \$7 \qquad \textbf{(B)}\ \$14 \qquad \textbf{(C)}\ \$19 \qquad \textbf{(D)}\ \$20 \qquad \textbf{(E)}\ \$39$

The incircle of triangle $ABC$ touches its sides $BC$, $AC$, $AB$ at the points $A_{1}$, $B_{1}$, $C_{1}$ respectively. A segment $AA_{1}$ intersects the incircle at the point $Q\ne A_{1}$. A line $\ell$ through $A$ is parallel to $BC$. Lines $A_{1}C_{1}$ and $A_{1}B_{1}$ intersect $\ell$ at the points $P$ and $R$ respectively. Prove that $\angle PQR=\angle B_{1}QC_{1}$. [i]A. Polyansky[/i]

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

Find all natural numbers $ n $ for which there exists two natural numbers $ a,b $ such that $$ n=S(a)=S(b)=S(a+b) , $$ where $ S(k) $ denotes the sum of the digits of $ k $ in base $ 10, $ for any natural number $ k. $ [i]Vasile Zidaru[/i] and [i]Mircea Lascu[/i]

Suzie flips a fair coin 6 times. The probability that Suzie flips 3 heads in a row but not 4 heads in a row is given by $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

The buttons $\{\times, +, \div\}$ on a calculator have their functions swapped. A button instead performs one of the other two functions; no two buttons have the same function. The calculator claims that $2 + 3 \div 4 = 10$ and $4 \times 2 \div 3 = 5$. What does $4 + 3 \times 2 \div 1$ equal on this calculator? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }7\qquad\textbf{(D) }9\qquad\textbf{(E) }10$

If $F(\frac{1-x}{1+x})=x$, then $\text{(A)}F(-2-x)=-2-F(x)\qquad\text{(B)}F(-x)=F(\frac{1+x}{1-x})$ $\text{(C)}F(\frac{1}{x})=F(x)\qquad\text{(D)}F(F(-x))=-x$

Numbers $a, b, c$ are the length of the medians of some triangle. If $ab + bc + ac = 1$ prove that a) $a^2b + b^2c + c^2a > \frac13$ b) $a^2b + b^2c + c^2a > \frac12$ (I. Bliznets)

Determine all positive integers $n>2$, such that $n = a^3 + b^3$, where $a$ is the smallest positive divisor of $n$ greater than $1$ and $b$ is an arbitrary positive divisor of $n$.

A sequence of positive integer numbers $a_1,a_2,\ldots$ for $i \geq 3$ satisfies $$a_{i+1}=a_i+gcd(a_{i-1},a_{i-2})$$ Prove that there exist two positive integer numbers $N, M$, such that $a_{n+1}-a_n=M$ for all $n \geq N$

Define $f\left(n\right)=\textrm{LCM}\left(1,2,\ldots,n\right)$. Determine the smallest positive integer $a$ such that $f\left(a\right)=f\left(a+2\right)$. [i]2017 CCA Math Bonanza Lightning Round #2.4[/i]

Let $P$ be an arbitrary point inside the triangle $ABC$. Let $A_1, B_1$ and $C_1$ denote the intersection points of the straight lines $AP, BP$ and $CP$, respectively, with the sides $BC, CA$ and $AB$. We order the areas of the triangles $AB_1C_1,A_1BC_1,A_1B_1C$. Denote the smaller by $S_1$, the middle by $S_2$, and the larger by $S_3$. Prove that $\sqrt{S_1 S_2} \le S \le \sqrt{S_2 S_3}$ ,where $S$ is the area of the triangle $A_1B_1S_1$.

Given a positive integer whose base-$10$ representation is $\overline{d_k\ldots d_0}$ for some integer $k \geq 0$, where $d_k \neq 0$, a move consists of selecting some integers $0 \leq i \leq j \leq k$, such that the digits $d_j,\ldots,d_i$ are not all $0$, erasing them from $n$, and replacing them with a divisor of $\overline{d_j\ldots d_i}$ (this divisor need not have the same number of digits as $\overline{d_j\ldots d_i}$). Prove that for all sufficiently large even integers $n$, we may apply some sequence of moves to $n$ to transform it into $2024$. [i]Allen Wang[/i]

There is a chessboard with $m$ columns and $n$ rows. In each blanks, an integer is given. If a rectangle $R$ (in this chessboard) has an integer $h$ satisfying the following two conditions, we call $R$ as a 'shelf'. (i) All integers contained in $R$ are bigger than $h$. (ii) All integers in blanks, which are not contained in $R$ but meet with $R$ at a vertex or a side, are not bigger than $h$. Assume that all integers are given to make shelves as much as possible. Find the number of shelves.

Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?

Prove that for any non-zero real numbers $a, b, c,$ \[\frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac 35.\]

a) Given a circle with two inscribed triangles $T_1$ and $T_2$. The vertices of $T_1$ are the midpoints of the arcs with the ends in the vertices of $T_2$. Consider a hexagon -- the intersection of $T_1$ and $T_2$. Prove that its main diagonals are parallel to $T_1$ sides and are intersecting in one point. b) The segment, that connects the midpoints of the arcs $AB$ and $AC$ of the circle circumscribed around the $ABC$ triangle, intersects $[AB]$ and $[AC]$ sides in $D$ and $K$ points. Prove that the points $A,D,K$ and $O$ -- the centre of the circle -- are the vertices of a diamond.

Let $n > 1$ be an integer. A set $S \subseteq \{ 0, 1, 2, \cdots , 4n - 1 \}$ is called ’sparse’ if for any $k \in \{ 0, 1, 2, \cdots , n - 1 \}$ the following two conditions are satisfied: $(a)$ The set $S \cap \{4k - 2, 4k - 1, 4k, 4k + 1, 4k + 2 \}$ has at most two elements; $(b)$ The set $S \cap \{ 4k +1, 4k +2, 4k +3 \}$ has at most one element. Prove that there are exactly $8 \cdot 7^{n-1}$ sparse subsets.

In $\Delta ABC$ with $AC=10$ and $BC=15$ the points $G$ and $I$ are its centroid and the center of its inscribed circle respectively. Find $AB$, if $\angle GIC=90^\circ$.

Given $n$ real numbers $a_1$, $a_2$ $\ldots$ $a_n$. ($n\geq 1$). Prove that there exists real numbers $b_1$, $b_2$ $\ldots$ $b_n$ satisfying: (a) For any $1 \leq i \leq n$, $a_i - b_i$ is a positive integer. (b)$\sum_{1 \leq i < j \leq n} (b_i - b_j)^2 \leq \frac{n^2-1}{12}$

$16$ progamers are playing in another single elimination tournament. Each round, each of the remaining progamers plays against another and the loser is eliminated. Additionally, each time a progamer wins, he will have a ceremony to celebrate. A player's rst ceremony is ten seconds long, and afterward each ceremony is ten seconds longer than the last. What is the total length in seconds of all the ceremonies over the entire tournament?

How namy subsets with $3$ elements of set $S=\{1,2,3,...,19,20\}$ exist, such that their product is divisible by $4$.