Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.

Let n be the natural number ($n\ge 2$). All natural numbers from $1$ up to $n$ ,inclusive, are written on the board in some order: $a_1$, $a_2$ , $...$ , $a_n$. Determine all natural numbers $n$ ($n\ge 2$), for which the product $$P = (1 + a_1) \cdot (2 + a_2) \cdot ... \cdot (n + a_n)$$ is an even number, whatever the arrangement of the numbers written on the board.

Find a countable family of natural solutions to $ \frac{1}{a} +\frac{1}{b} +\frac{1}{ab}=\frac{1}{c} . $

Determine the fractions of a fraction of the form $\frac{1}{ab}$ where $a,b$ are prime natural numbers such that $0 < a < b \le 200$ and $a + b > 200$

[b]p1.[/b] Frankie the frog likes to hop. On his first hop, he hops $1$ meter. On each successive hop, he hops twice as far as he did on the previous hop. For example, on his second hop, he hops $2$ meters, and on his third hop, he hops $4$ meters. How many meters, in total, has he travelled after $6$ hops? [b]p2.[/b] Anton flips $5$ fair coins. The probability that he gets an odd number of heads can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p3.[/b] April discovers that the quadratic polynomial $x^2 + 5x + 3$ has distinct roots $a$ and $b$. She also discovers that the quadratic polynomial $x^2 + 7x + 4$ has distinct roots $c$ and $d$. Compute $$ac + bc + bd + ad + a + b.$$ [b]p4.[/b] A rectangular picture frame that has a $2$ inch border can exactly fit a $10$ by $7$ inch photo. What is the total area of the frame's border around the photo, in square inches? [b]p5.[/b] Compute the median of the positive divisors of $9999$. [b]p6.[/b] Kaity only eats bread, pizza, and salad for her meals. However, she will refuse to have salad if she had pizza for the meal right before. Given that she eats $3$ meals a day (not necessarily distinct), in how many ways can we arrange her meals for the day? [b]p7.[/b] A triangle has side lengths $3$, $4$, and $x$, and another triangle has side lengths $3$, $4$, and $2x$. Assuming both triangles have positive area, compute the number of possible integer values for $x$. [b]p8.[/b] In the diagram below, the largest circle has radius $30$ and the other two white circles each have a radius of $15$. Compute the radius of the shaded circle. [img]https://cdn.artofproblemsolving.com/attachments/c/1/9eaf1064b2445edb15782278fc9c6efd1440b0.png[/img] [b]p9.[/b] What is the remainder when $2022$ is divided by $9$? [b]p10.[/b] For how many positive integers $x$ less than $2022$ is $x^3 - x^2 + x - 1$ prime? [b]p11.[/b] A sphere and cylinder have the same volume, and both have radius $10$. The height of the cylinder can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p12.[/b] Amanda, Brianna, Chad, and Derrick are playing a game where they pass around a red flag. Two players "interact" whenever one passes the flag to the other. How many different ways can the flag be passed among the players such that (1) each pair of players interacts exactly once, and (2) Amanda both starts and ends the game with the flag? [b]p13.[/b] Compute the value of $$\dfrac{12}{1 + \dfrac{12}{1+ \dfrac{12}{1+...}}}$$ [b]p14.[/b] Compute the sum of all positive integers $a$ such that $a^2 - 505$ is a perfect square. [b]p15.[/b] Alissa, Billy, Charles, Donovan, Eli, Faith, and Gerry each ask Sara a question. Sara must answer exactly $5$ of them, and must choose an order in which to answer the questions. Furthermore, Sara must answer Alissa and Billy's questions. In how many ways can Sara complete this task? [b]p16.[/b] The integers $-x$, $x^2 - 1$, and $x3$ form a non-decreasing arithmetic sequence (in that order). Compute the sum of all possible values of $x^3$. [b]p17.[/b] Moor and his $3$ other friends are trying to split burgers equally, but they will have $2$ left over. If they find another friend to split the burgers with, everyone can get an equal amount. What is the fewest number of burgers that Moor and his friends could have started with? [b]p18.[/b] Consider regular dodecagon $ABCDEFGHIJKL$ below. The ratio of the area of rectangle $AFGL$ to the area of the dodecagon can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/8/3/c38c10a9b2f445faae397d8a7bc4c8d3ed0290.png[/img] [b]p19.[/b] Compute the remainder when $3^{4^{5^6}}$ is divided by $4$. [b]p20.[/b] Fred is located at the middle of a $9$ by $11$ lattice (diagram below). At every second, he randomly moves to a neighboring point (left, right, up, or down), each with probability $1/4$. The probability that he is back at the middle after exactly $4$ seconds can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/7/c/f8e092e60f568ab7b28964d23b2ee02cdba7ad.png[/img] PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a prime number $p{}$ and integers $x{}$ and $y$, find the remainder of the sum $x^0y^{p-1}+x^1y^{p-2}+\ldots+x^{p-2}y^1+x^{p-1}y^0$ upon division by $p{}$.

Find all natural numbers $n$, such that $3$ divides the number $n\cdot 2^n+1$.

Find all pairs $(x,y)$ of positive integers such that $x(x+y)=y^2+1.$

Prove that for each positive integer $k$ there exists a number base $b$ along with $k$ triples of Fibonacci numbers $(F_u,F_v,F_w)$ such that when they are written in base $b$, their concatenation is also a Fibonacci number written in base $b$. (Fibonacci numbers are defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$.) [i]Proposed by Serbia[/i]

Find, and write out explicitly, a permutation $\{p(1),p(2),\ldots,p(20)\}$ of $\{1,2,\ldots,20\}$ such that $k+p(k)$ is a power of $2$ for $k=1,2,\ldots,20$, and prove that only one such permutation exists.

Prove that the equation \[ x^{2008}\plus{} 2008!\equal{} 21^{y}\] doesn't have solutions in integers.

A positive integer $ m$ is called [i]good[/i] if there is a positive integer $ n$ such that $ m$ is the quotient of $ n$ by the number of positive integer divisors of $ n$ (including $ 1$ and $ n$ itself). Prove that $ 1, 2, \ldots, 17$ are good numbers and that $ 18$ is not a good number.

Show that $ n\equiv 0\pmod 9 $ if $ 2^n\equiv -1\pmod n, $ where $ n $ is a natural number greater than $ 3. $

Two different $3$ digit numbers are picked and then for every of them is calculated sum of all $5$ numbers which are getting when digits of picked number change place (etc. if one of the number is $707$, the sum is $2401=770+77+77+770+707$). Do the given results must be different?

Prove that every natural number which is not an integer power of $2$ is the sum of two or more consecutive natural numbers.

Given positive integer $n$. Prove that for any integers $a_1,a_2,\cdots,a_n,$ at least $\lceil \tfrac{n(n-6)}{19} \rceil$ numbers from the set $\{ 1,2, \cdots, \tfrac{n(n-1)}{2} \}$ cannot be represented as $a_i-a_j (1 \le i, j \le n)$.

Let $k$ a positive integer number such that $\frac{k(k+1)}{3}$ is a perfect square. Show that $\frac{k}{3}$ and $k+1$ are both perfect squares.

[b]p1.[/b] What is $(2 + 4 + ... + 20) - (1 + 3 + ...+ 19)$? [b]p2.[/b] Two ants start on opposite vertices of a dodecagon ($12$-gon). Each second, they randomly move to an adjacent vertex. What is the probability they meet after four moves? [b]p3.[/b] How many distinct $8$-letter strings can be made using $8$ of the $9$ letters from the words $FORK$ and $KNIFE$ (e.g., $FORKNIFE$)? [b]p4.[/b] Let $ABC$ be an equilateral triangle with side length $8$ and let $D$ be a point on segment $BC$ such that $BD = 2$. Given that $E$ is the midpoint of $AD$, what is the value of $CE^2 - BE^2$? [b][color=#f00](mistyped p4)[/color][/b] Let $ABC$ be an equilateral triangle with side length $8$ and let $D$ be a point on segment $BC$ such that $BD = 2$. Given that $E$ is the midpoint of $AD$, what is the value of $CE^2 + BE^2$? [b]p5.[/b] You have two fair six-sided dice, one labeled $1$ to $6$, and for the other one, each face is labeled $1$, $2$, $3$, or $4$ (not necessarily all numbers are used). Let $p$ be the probability that when the two dice are rolled, the number on the special die is smaller than the number on the normal die. Given that $p = 1/2$, how many distinct combinations of $1$, $2$, $3$, $4$ can appear on the special die? The arrangement of the numbers on the die does not matter. [b]p6.[/b] Let $\omega_1$ and $\omega_2$ be two circles with centers $A$ and $B$ and radii $3$ and $13$, respectively. Suppose $AB = 10$ and that $C$ is the midpoint of $AB$. Let $\ell$ be a line that passes through $C$ and is tangent to $\omega_1$ at $P$. Given that $\ell$ intersects $\omega_2$ at $X$ and $Y$ such that $XP < Y P$, what is $XP$? [b]p7.[/b] Let $f(x)$ be a cubic polynomial. Given that $f(1) = 13$, $f(4) = 19$, $f(7) = 7$, and $f(10) = 13$, find $f(13)$. [b]p8.[/b] For all integers $0 \le n \le 202$ not divisible by seven, define $f(n) = \{\sqrt{7n}\}$. For what value $n$ does $f(n)$ take its minimum value? (Note: $\{x\} = x - \lfloor x \rfloor$, where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) [b]p9.[/b] Let $ABC$ be a triangle with $AB = 14$ and $AC = 25$. Let the incenter of $ABC$ be $I$. Let line $AI$ intersect the circumcircle of $BIC$ at $D$ (different from $I$). Given that line $DC$ is tangent to the circumcircle of $ABC$, find the area of triangle $BCD$. [b]p10.[/b] Evaluate the infinite sum $$\frac{4^2 + 3}{1 \cdot 3 \cdot 5 \cdot 7} +\frac{6^2 + 3}{3 \cdot 5 \cdot 7 \cdot 9}+\frac{8^2 + 3}{5 \cdot 7 \cdot 9 \cdot 11}+ ...$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A finite set $S$ of positive integers has the property that, for each $s \in S,$ and each positive integer divisor $d$ of $s$, there exists a unique element $t \in S$ satisfying $\text{gcd}(s, t) = d$. (The elements $s$ and $t$ could be equal.) Given this information, find all possible values for the number of elements of $S$.

The number $2020$ has three different prime factors. What is their sum?

For every positive integer $x$, let $k(x)$ denote the number of composite numbers that do not exceed $x$. Find all positive integers $n$ for which $(k (n))! $ lcm $(1, 2,..., n)> (n - 1) !$ .

Find all positive integer pairs $(m, n)$ such that $m- n$ is a positive prime number and $mn$ is a perfect square. Justify your answer.

Positive integer $m$ is such that the sum of decimal digits of $8^m$ equals 8. Can the last digit of $8^m$ be equal 6? (Author: V. Senderov) (compare with http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=431860)

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Let $a$, $b$, $c$, $d$, $e$, $f$ and $g$ be seven distinct positive integers not bigger than $7$. Find all primes which can be expressed as $abcd+efg$