Let $n$, $k$ be positive integers such that $n$ is not divisible by $3$ and $k\ge n$. Prove that there exists a positive integer m which is divisible by $n$ and the sum of its digits in the decimal representation is $k$.

Find all natural integers $n$ such that $(n^3 + 39n - 2)n! + 17\cdot 21^n + 5$ is a square.

For how many pairs of positive integers $(x,y)$ is $x+3y=100$?

For positive integers $k,m$, where $1\le k\le 5$, define the function $f(m,k)$ as \[f(m,k)=\sum_{i=1}^{5}\left[m\sqrt{\frac{k+1}{i+1}}\right]\] where $[x]$ denotes the greatest integer not exceeding $x$. Prove that for any positive integer $n$, there exist positive integers $k,m$, where $1\le k\le 5$, such that $f(m,k)=n$.

What is the smallest non-square positive integer that is the product of four prime numbers (not necessarily distinct)?

[b]p1.[/b] Evaluate $(2-0)^2 \cdot 3+ \frac{20}{2+3}$ . [b]p2.[/b] Let $x = 11 \cdot 99$ and $y = 9 \cdot 101$. Find the sumof the digits of $x \cdot y$. [b]p3.[/b] A rectangle is cut into two pieces. The ratio between the areas of the two pieces is$ 3 : 1$ and the positive difference between those areas is $20$. What’s the area of the rectangle? [b]p4.[/b] Edgeworth is scared of elevators. He is currently on floor $50$ of a building, and he wants to go down to floor $1$. Edgeworth can go down at most $4$ floors each time he uses the elevator. What’s the minimum number of times he needs to use the elevator to get to floor $1$? [b]p5.[/b] There are $20$ people at a party. Fifteen of those people are normal and $5$ are crazy. A normal person will shake hands once with every other normal person, while a crazy person will shake hands twice with every other crazy person. How many total handshakes occur at the party? [b]p6.[/b] Wam and Sang are chewing gum. Gum comes in packages, each package consisting of $14$ sticks of gum. Wam eats $6$ packs and $9$ individual sticks of gum. Sang wants to eat twice as much gum as Wam. How many packs of gum must Sang buy? [b]p7.[/b] At Lakeside Health School (LHS), $40\%$ of students are male and $60\%$ of the students are female. If half of the students at the school take biology, and the same number ofmale and female students take biology, to the nearest percent, what percent of female students take biology? [b]p8.[/b] Evin is bringing diluted raspberry iced tea to the annual LexingtonMath Team party. He has a cup with $10$ mL of iced tea and a $2000$ mL cup of water with $10\%$ raspberry iced tea. If he fills up the cup with $20$ more mL of $10\%$ raspberry iced tea water, what percent of the solution will be iced tea? [b]p9.[/b] Tree $1$ starts at height $220$ m and grows continuously at $3$ m per year. Tree $2$ starts at height $20$ m and grows at $5$ m during the first year, $7$ m per during the second year, $9$ m during the third year, and in general $(3+2n)$ m in the nth year. After which year is Tree $2$ taller than Tree $1$? [b]p10.[/b] Leo and Chris are playing a game in which Chris flips a coin. The coin lands on heads with probability $\frac{499}{999}$ , tails with probability $\frac{499}{999}$ , and it lands on its side with probability $\frac{1}{999}$ . For each flip of the coin, Leo agrees to give Chris $4$ dollars if it lands on heads, nothing if it lands on tails, and $2$ dollars if it lands on its side. What’s the expected value of the number of dollars Chris gets after flipping the coin $17$ times? [b]p11.[/b] Ephram has a pile of balls, which he tries to divide into piles. If he divides the balls into piles of $7$, there are $5$ balls that don’t get divided into any pile. If he divides the balls into piles of $11$, there are $9$ balls that aren’t in any pile. If he divides the balls into piles of $13$, there are $11$ balls that aren’t in any pile. What is the minimumnumber of balls Ephram has? [b]p12.[/b] Let $\vartriangle ABC$ be a triangle such that $AB = 3$, $BC = 4$, and $C A = 5$. Let $F$ be the midpoint of $AB$. Let $E$ be the point on $AC$ such that $EF \parallel BC$. Let CF and $BE$ intersect at $D$. Find $AD$. [b]p13.[/b] Compute the sum of all even positive integers $n \le 1000$ such that: $$lcm(1,2, 3, ..., (n -1)) \ne lcm(1,2, 3,, ...,n)$$. [b]p14.[/b] Find the sum of all palindromes with $6$ digits in binary, including those written with leading zeroes. [b]p15.[/b] What is the side length of the smallest square that can entirely contain $3$ non-overlapping unit circles? [b]p16.[/b] Find the sum of the digits in the base $7$ representation of $6250000$. Express your answer in base $10$. [b]p17.[/b] A number $n$ is called sus if $n^4$ is one more than a multiple of $59$. Compute the largest sus number less than $2023$. [b]p18.[/b] Michael chooses real numbers $a$ and $b$ independently and randomly from $(0, 1)$. Given that $a$ and $b$ differ by at most $\frac14$, what is the probability $a$ and $b$ are both greater than $\frac12$ ? [b]p19.[/b] In quadrilateral $ABCD$, $AB = 7$ and $DA = 5$, $BC =CD$, $\angle BAD = 135^o$ and $\angle BCD = 45^o$. Find the area of $ABCD$. [b]p20.[/b] Find the value of $$\sum_{i |210} \sum_{j |i} \left \lfloor \frac{i +1}{j} \right \rfloor$$ [b]p21.[/b] Let $a_n$ be the number of words of length $n$ with letters $\{A,B,C,D\}$ that contain an odd number of $A$s. Evaluate $a_6$. [b]p22.[/b] Detective Hooa is investigating a case where a criminal stole someone’s pizza. There are $69$ people involved in the case, among whom one is the criminal and another is a witness of the crime. Every day, Hooa is allowed to invite any of the people involved in the case to his rather large house for questioning. If on some given day, the witness is present and the criminal is not, the witness will reveal who the criminal is. What is the minimum number of days of questioning required such that Hooa is guaranteed to learn who the criminal is? [b]p23.[/b] Find $$\sum^{\infty}_{n=2} \frac{2n +10}{n^3 +4n^2 +n -6}.$$ [b]p24.[/b] Let $\vartriangle ABC$ be a triangle with circumcircle $\omega$ such that $AB = 1$, $\angle B = 75^o$, and $BC =\sqrt2$. Let lines $\ell_1$ and $\ell_2$ be tangent to $\omega$ at $A$ and $C$ respectively. Let $D$ be the intersection of $\ell_1$ and $\ell_2$. Find $\angle ABD$ (in degrees). [b]p25.[/b] Find the sum of the prime factors of $14^6 +27$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n>1$ be a natural number and $x_k{}$ be the residue of $n^2$ modulo $\lfloor n^2/k\rfloor+1$ for all natural $k{}$. Compute the sum \[\bigg\lfloor\frac{x_2}{1}\bigg\rfloor+\bigg\lfloor\frac{x_3}{2}\bigg\rfloor+\cdots+\left\lfloor\frac{x_n}{n-1}\right\rfloor.\]

Determine all pairs of positive integers $(a,b)$ such that \[ \dfrac{a^2}{2ab^2-b^3+1} \] is a positive integer.

Let $ab$ be a two digit number. A positive integer $n$ is a [i]relative[/i] of $ab$ if: [list] [*] The units digit of $n$ is $b$. [*] The remaining digits of $n$ are nonzero and add up to $a$.[/list] Find all two digit numbers which divide all of their relatives.

What is remainder when $2014^{2015}$ is divided by $121$? $ \textbf{(A)}\ 45 \qquad\textbf{(B)}\ 34 \qquad\textbf{(C)}\ 23 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 1 $

Prove: [b](a)[/b] There are infinitely many positive intengers $n$, satisfying: $$\gcd(n,[\sqrt2n])=1.$$ [b](b)[/b] There are infinitely many positive intengers $n$, satisfying: $$\gcd(n,[\sqrt2n])>1.$$

The sum $$\sum_{k=3}^{\infty} \frac{1}{k(k^4-5k^2+4)^2}$$ is equal to $\frac{m^2}{2n^2}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

For every positive integer $n$, $s(n)$ denotes the sum of the digits in the decimal representation of $n$. Prove that for every integer $n \ge 5$, we have $$S(1)S(3)...S(2n-1) \ge S(2)S(4)...S(2n)$$

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

Find the largest positive integer $n$ satisfying the following: [center] "There are precisely $53$ integers in the list of integers $1, 2, \ldots, n$ that are either perfect squares, perfect cubes or both."[/center]

Let $ a,b$ be real nonzero numbers, such that number $ \lfloor an \plus{} b \rfloor$ is an even integer for every $ n \in \mathbb{N}$. Prove that $ a$ is an even integer.

The number $60$ is written on a blackboard. In every move, Andris wipes the numbers on the board one by one, and writes all its divisors in its place (including itself). After $10$ such moves, how many times will $1$ appear on the board?

Given two cubes $R$ and $S$ with integer sides of lengths $r$ and $s$ units respectively . If the difference between volumes of the two cubes is equal to the difference in their surface areas , then prove that $r=s$.

Let $n>3$ be a positive integer satisfying $2^n+1=3p$, where $p$ is a prime. Let $s_0=\frac{2^{n-2}+1}{3}$ and $s_i=s_{i-1}^2-2$ for $i>0$. Show that $p \mid 2s_{n-2}-3$.

a) Find all multiplicative functions $f: \mathbb Z_{p}^{*}\longrightarrow\mathbb Z_{p}^{*}$ (i.e. that $\forall x,y\in\mathbb Z_{p}^{*}$, $f(xy)=f(x)f(y)$.) b) How many bijective multiplicative does exist on $\mathbb Z_{p}^{*}$ c) Let $A$ be set of all multiplicative functions on $\mathbb Z_{p}^{*}$, and $VB$ be set of all bijective multiplicative functions on $\mathbb Z_{p}^{*}$. For each $x\in \mathbb Z_{p}^{*}$, calculate the following sums :\[\sum_{f\in A}f(x),\ \ \sum_{f\in B}f(x)\]

Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$. Prove that for no $n$: $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square. Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$. Prove that $S_1 \cdot S_2$ is divisible by $150$.

Find a necessary and sufficient condition of $a,b,n\in\mathbb{N^*}$ such that for $S=\{a+bt\mid t=0,1,2,\cdots,n-1\}$, there exists a one-to-one mapping $f: S\to S$ such that for all $x\in S$, $\gcd(x,f(x))=1$.

Determine that all $ k \in \mathbb{Z}$ such that $ \forall n$ the numbers $ 4n\plus{}1$ and $ kn\plus{}1$ have no common divisor.

Find all positive integers $n$ such that there exists a prime number $p$, such that $p^n-(p-1)^n$ is a power of $3$. Note. A power of $3$ is a number of the form $3^a$ where $a$ is a positive integer.

For positive integers $ n$, $ f_n \equal{} \lfloor2^n\sqrt {2008}\rfloor \plus{} \lfloor2^n\sqrt {2009}\rfloor$. Prove there are infinitely many odd numbers and infinitely many even numbers in the sequence $ f_1,f_2,\ldots$.