Do there exist two positive powers of $5$ such that the number obtained by writing one after the other is also a power of $5$?

Let $s(a)$ denote the sum of digits of a given positive integer a. The sequence $a_1, a_2,..., a_n, ...$ of positive integers is such that $a_{n+1} = a_n+s(a_n)$ for each positive integer $n$. Find the greatest possible n for which it is possible to have $a_n = 2008$.

Find the sum of all such natural numbers from $1$ to $500$ that are not divisible by $5$ or $7$.

Find all integers that can be written in the form $\frac{(x+y+z)^2}{xyz}$, where $x,y,z$ are positive integers.

Let $n$ be a positive integer with the following property: $2^n-1$ divides a number of the form $m^2+81$, where $m$ is a positive integer. Find all possible $n$.

Determine all positive integers $ n$, for which $ 2^{n\minus{}1}n\plus{}1$ is a perfect square.

find all positive integer n such that there exists positive integers (a,b) such that (a^n + b^n)/n! is a positive integer smaller than 101

Let $f:Z \to N -\{0\}$ such that: $f(x + y)f(x-y) = (f(x)f(y))^2$ and $f(1)\ne 1$. Provethat $\log_{f(1)}f(z)$ is a perfect square for every integer $z$.

[b]p1.[/b] Chris goes to Matt's Hamburger Shop to buy a hamburger. Each hamburger must contain exactly one bread, one lettuce, one cheese, one protein, and at least one condiment. There are two kinds of bread, two kinds of lettuce, three kinds of cheese, three kinds of protein, and six different condiments: ketchup, mayo, mustard, dill pickles, jalape~nos, and Matt's Magical Sunshine Sauce. How many different hamburgers can Chris make? [b]p2.[/b] The degree measures of the interior angles in convex pentagon $NICKY$ are all integers and form an increasing arithmetic sequence in some order. What is the smallest possible degree measure of the pentagon's smallest angle? [b]p3.[/b] Daniel thinks of a two-digit positive integer $x$. He swaps its two digits and gets a number $y$ that is less than $x$. If $5$ divides $x-y$ and $7$ divides $x+y$, find all possible two-digit numbers Daniel could have in mind. [b]p4.[/b] At the Lio Orympics, a target in archery consists of ten concentric circles. The radii of the circles are $1$, $2$, $3$, $...$, $9$, and $10$ respectively. Hitting the innermost circle scores the archer $10$ points, the next ring is worth $9$ points, the next ring is worth 8 points, all the way to the outermost ring, which is worth $1$ point. If a beginner archer has an equal probability of hitting any point on the target and never misses the target, what is the probability that his total score after making two shots is even? [b]p5.[/b] Let $F(x) = x^2 + 2x - 35$ and $G(x) = x^2 + 10x + 14$. Find all distinct real roots of $F(G(x)) = 0$. [b]p6.[/b] One day while driving, Ivan noticed a curious property on his car's digital clock. The sum of the digits of the current hour equaled the sum of the digits of the current minute. (Ivan's car clock shows $24$-hour time; that is, the hour ranges from $0$ to $23$, and the minute ranges from $0$ to $59$.) For how many possible times of the day could Ivan have observed this property? [b]p7.[/b] Qi Qi has a set $Q$ of all lattice points in the coordinate plane whose $x$- and $y$-coordinates are between $1$ and $7$ inclusive. She wishes to color $7$ points of the set blue and the rest white so that each row or column contains exactly $1$ blue point and no blue point lies on or below the line $x + y = 5$. In how many ways can she color the points? [b]p8.[/b] A piece of paper is in the shape of an equilateral triangle $ABC$ with side length $12$. Points $A_B$ and $B_A$ lie on segment $AB$, such that $AA_B = 3$, and $BB_A = 3$. Define points $B_C$ and $C_B$ on segment $BC$ and points $C_A$ and $A_C$ on segment $CA$ similarly. Point $A_1$ is the intersection of $A_CB_C$ and $A_BC_B$. Define $B_1$ and $C_1$ similarly. The three rhombi - $AA_BA_1A_C$,$BB_CB_1B_A$, $CC_AC_1C_B$ - are cut from triangle $ABC$, and the paper is folded along segments $A_1B_1$, $B_1C_1$, $C_1A_1$, to form a tray without a top. What is the volume of this tray? [b]p9.[/b] Define $\{x\}$ as the fractional part of $x$. Let $S$ be the set of points $(x, y)$ in the Cartesian coordinate plane such that $x + \{x\} \le y$, $x \ge 0$, and $y \le 100$. Find the area of $S$. [b]p10.[/b] Nicky likes dolls. He has $10$ toy chairs in a row, and he wants to put some indistinguishable dolls on some of these chairs. (A chair can hold only one doll.) He doesn't want his dolls to get lonely, so he wants each doll sitting on a chair to be adjacent to at least one other doll. How many ways are there for him to put any number (possibly none) of dolls on the chairs? Two ways are considered distinct if and only if there is a chair that has a doll in one way but does not have one in the other. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f : Z^+ \to Z^+$ satisfying $f (1) = 2, f (2) \ne 4$, and max $\{f (m) + f (n), m + n\} |$ min $\{2m + 2n, f (m + n) + 1\}$ for all $m, n \in Z^+$.

The sequence $\left(a_n \right)$ is defined by $a_1=1, \ a_2=2$ and $$a_{n+2} = 2a_{n+1}-pa_n, \ \forall n \ge 1,$$ for some prime $p.$ Find all $p$ for which there exists $m$ such that $a_m=-3.$

Solve in positive integers: \[x^{y^2+1}+y^{x^2+1}=2^z\]

Let $n \ge 3$ be a positive integer. For every set $S$ with $n$ distinct positive integers, prove that there exists a bijection $f: \{1,2, \cdots n\} \rightarrow S$ which satisfies the following condition. For all $1 \le i < j < k \le n$, $f(j)^2 \neq f(i) \cdot f(k)$.

Suppose that the greatest common divisor of $n$ and $5040$ is equal to $120.$ Determine the sum of the four smallest possible positive integers $n.$

[b]p1.[/b] For positive real numbers $x, y$, the operation $\otimes$ is given by $x \otimes y =\sqrt{x^2 - y}$ and the operation $\oplus$ is given by $x \oplus y =\sqrt{x^2 + y}$. Compute $(((5\otimes 4)\oplus 3)\otimes2)\oplus 1$. [b]p2.[/b] Janabel is cutting up a pizza for a party. She knows there will either be $4$, $5$, or $6$ people at the party including herself, but can’t remember which. What is the least number of slices Janabel can cut her pizza to guarantee that everyone at the party will be able to eat an equal number of slices? [b]p3.[/b] If the numerator of a certain fraction is added to the numerator and the denominator, the result is $\frac{20}{19}$ . What is the fraction? [b]p4.[/b] Let trapezoid $ABCD$ be such that $AB \parallel CD$. Additionally, $AC = AD = 5$, $CD = 6$, and $AB = 3$. Find $BC$. [b]p5.[/b] AtMerrick’s Ice Cream Parlor, customers can order one of three flavors of ice cream and can have their ice cream in either a cup or a cone. Additionally, customers can choose any combination of the following three toppings: sprinkles, fudge, and cherries. How many ways are there to buy ice cream? [b]p6.[/b] Find the minimum possible value of the expression $|x+1|+|x-4|+|x-6|$. [b]p7.[/b] How many $3$ digit numbers have an even number of even digits? [b]p8.[/b] Given that the number $1a99b67$ is divisible by $7$, $9$, and $11$, what are $a$ and $b$? Express your answer as an ordered pair. [b]p9.[/b] Let $O$ be the center of a quarter circle with radius $1$ and arc $AB$ be the quarter of the circle’s circumference. Let $M$,$N$ be the midpoints of $AO$ and $BO$, respectively. Let $X$ be the intersection of $AN$ and $BM$. Find the area of the region enclosed by arc $AB$, $AX$,$BX$. [b]p10.[/b] Each square of a $5$-by-$1$ grid of squares is labeled with a digit between $0$ and $9$, inclusive, such that the sum of the numbers on any two adjacent squares is divisible by $3$. How many such labelings are possible if each digit can be used more than once? [b]p11.[/b] A two-digit number has the property that the difference between the number and the sum of its digits is divisible by the units digit. If the tens digit is $5$, how many different possible values of the units digit are there? [b]p12.[/b] There are $2019$ red balls and $2019$ white balls in a jar. One ball is drawn and replaced with a ball of the other color. The jar is then shaken and one ball is chosen. What is the probability that this ball is red? [b]p13.[/b] Let $ABCD$ be a square with side length $2$. Let $\ell$ denote the line perpendicular to diagonal $AC$ through point $C$, and let $E$ and $F$ be themidpoints of segments $BC$ and $CD$, respectively. Let lines $AE$ and $AF$ meet $\ell$ at points $X$ and $Y$ , respectively. Compute the area of $\vartriangle AXY$ . [b]p14.[/b] Express $\sqrt{21-6\sqrt6}+\sqrt{21+6\sqrt6}$ in simplest radical form. [b]p15.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length two. Let $D$ and $E$ be on $AB$ and $AC$ respectively such that $\angle ABE =\angle ACD = 15^o$. Find the length of $DE$. [b]p16.[/b] $2018$ ants walk on a line that is $1$ inch long. At integer time $t$ seconds, the ant with label $1 \le t \le 2018$ enters on the left side of the line and walks among the line at a speed of $\frac{1}{t}$ inches per second, until it reaches the right end and walks off. Determine the number of ants on the line when $t = 2019$ seconds. [b]p17.[/b] Determine the number of ordered tuples $(a_1,a_2,... ,a_5)$ of positive integers that satisfy $a_1 \le a_2 \le ... \le a_5 \le 5$. [b]p18.[/b] Find the sum of all positive integer values of $k$ for which the equation $$\gcd (n^2 -n -2019,n +1) = k$$ has a positive integer solution for $n$. [b]p19.[/b] Let $a_0 = 2$, $b_0 = 1$, and for $n \ge 0$, let $$a_{n+1} = 2a_n +b_n +1,$$ $$b_{n+1} = a_n +2b_n +1.$$ Find the remainder when $a_{2019}$ is divided by $100$. [b]p20.[/b] In $\vartriangle ABC$, let $AD$ be the angle bisector of $\angle BAC$ such that $D$ is on segment $BC$. Let $T$ be the intersection of ray $\overrightarrow{CB}$ and the line tangent to the circumcircle of $\vartriangle ABC$ at $A$. Given that $BD = 2$ and $TC = 10$, find the length of $AT$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$, with $i = 1,2, ..., 2015$.

Find all nonnegative integers $k, n$ which satisfy $2^{2k+1} + 9\cdot 2^k + 5 = n^2.$

Suppose that $a$ and $b$ are integers. Prove that there are integers $c$ and $d$ such that $a+b+c+d=0$ and $ac+bd=0$, if and only if $a-b$ divides $2ab$.

Show that there exist integers $m$ and $n$ such that \[\dfrac{m}{n}=\sqrt[3]{\sqrt{50}+7}-\sqrt[3]{\sqrt{50}-7}.\]

The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

Find all real numbers a such that all solutions to the quadratic equation $ x^2 \minus{} ax \plus{} a \equal{} 0$ are integers.

In the beginning, the number 1 is written on the board 9999 times. We are allowed to perform the following actions: [list] [*] Erase four numbers of the form $x,x,y,y$, and instead write the two numbers $x+y,x-y$. (The order or location of the erased numbers does not matter) [*] Erase the number 0 from the board, if it's there. [/list] Is it possible to reach a state where: [list=a] [*] Only one number remains on the board? [*] At most three numbers remain on the board? [/list]

Let $\omega$ be a nonreal $47$th root of unity. Suppose that $\mathcal{S}$ is the set of polynomials of degree at most $46$ and coefficients equal to either $0$ or $1$. Let $N$ be the number of polynomials $Q \in \mathcal{S}$ such that \[ \sum_{j = 0}^{46} \frac{Q(\omega^{2j}) - Q(\omega^{j})}{\omega^{4j} + \omega^{3j} + \omega^{2j} + \omega^j + 1} = 47. \] The prime factorization of $N$ is $p_1^{\alpha_1}p_2^{\alpha_2} \dots p_s^{\alpha_s}$ where $p_1, \ldots, p_s$ are distinct primes and $\alpha_1, \alpha_2, \ldots, \alpha_s$ are positive integers. Compute $\sum_{j = 1}^s p_j\alpha_j$.

$a_1,a_2,\dots,a_{14}$ are different positive integers. All 196 numbers of the form $a_k+a_l$ with $1\leq k,l\leq 14$ are written on a board. Is it possible that for any two-digit combination, there exists a number among all 196 that ends with that combination (i.e., there exist numbers ending with $00, 01, \dots, 99$)? (Author: P. Kozhevnikov)

Find the smallest possible natural number $n = \overline{a_m ...a_2a_1a_0} $ (in decimal system) such that the number $r = \overline{a_1a_0a_m ..._20} $ equals $2n$.