Define $f_{0}(x)=e^x$ and $f_{n+1}(x)=x f_{n}'(x)$. Show that $\sum_{n=0}^{\infty} \frac{f_{n}(1)}{n!}=e^e$.

[b]Q.[/b] Prove that: $$\sum_{\text{cyc}}\tan (\tan A) - 2 \sum_{\text{cyc}} \tan \left(\cot \frac{A}{2}\right) \geqslant -3 \tan (\sqrt 3)$$where $A, B$ and $C$ are the angles of an acute-angled $\triangle ABC$. [i]Proposed by SA2018[/i]

The diagonals of a pentagon $ABCDE$ determine another pentagon $MNPQR$. If $MNPQR$ and $ABCDE$ are similar, must $ABCDE$ be regular?

Let $k$ be a fixed positive integer. The sequence $\{a_{n}\}_{n\ge1}$ is defined by \[a_{1}=k+1, a_{n+1}=a_{n}^{2}-ka_{n}+k.\] Show that if $m \neq n$, then the numbers $a_{m}$ and $a_{n}$ are relatively prime.

The digits from $1$ to $9$ are placed in the figure below with one digit in each square. The sum of three numbers placed in the same horizontal or vertical line is $13$. Show that the marked place says $4$. [img]https://cdn.artofproblemsolving.com/attachments/a/f/517b644caf59bbc57701662f21d57465855dc1.png[/img]

Determine whether there is a tetrahedron $ABCD$ with the longest edge of length 1 such that all its faces are similar right triangles with right angles at vertices $B,C.$ If so, determine which edge is the longest, which is the shortest and what is its length.

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 $a, b, c,$ \[\frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac 35.\]