Let $k$ be a positive integer congruent to $1$ modulo $4$ which is not a perfect square and let $a=\frac{1+\sqrt{k}}{2}$. Show that $\{\left \lfloor{a^2n}\right \rfloor-\left \lfloor{a\left \lfloor{an}\right \rfloor}\right \rfloor : n \in \mathbb{N}_{>0}\}=\{1 , 2 , \ldots ,\left \lfloor{a}\right \rfloor\}$.

Find all triples $(a,b,c)$ of non-negative integers satisfying $a\ge b\ge c$ and \[1\cdot a^3+9\cdot b^2+9\cdot c+7=1997 \]

Define real number $y$ as the fractional part of real number $x$ such that $0\leq y<1$ and $x-y$ is integer. Denote this by $<x>$. For real number $a$, define an infinite sequence $\{a_n\}\ (n=1,\ 2,\ 3,\ \cdots)$ inductively as follows. (i) $a_1=<a>$ (ii) If $a\n\neq 0$, then $a_{n+1}=\left<\frac{1}{a_n}\right>$, if $a_n=0$, then $a_{n+1}=0$. (1) For $a=\sqrt{2}$, find $a_n$. (2) For any natural number $n$, find real number $a\geq \frac 13$ such that $a_n=a$. (3) Let $a$ be a rational number. When we express $a=\frac{p}{q}$ with integer $p$, natural number $q$, prove that $a_n=0$ for any natural number $n\geq q$. [i]2011 Tokyo University entrance exam/Science, Problem 2[/i]

Find all positive integers $n$ for which there do not exist $n$ consecutive composite positive integers less than $n!$.

Assume that the set of all positive integers is decomposed into $ r$ (disjoint) subsets $ A_1 \cup A_2 \cup \ldots \cup A_r \equal{} \mathbb{N}.$ Prove that one of them, say $ A_i,$ has the following property: There exists a positive $ m$ such that for any $ k$ one can find numbers $ a_1, a_2, \ldots, a_k$ in $ A_i$ with $ 0 < a_{j \plus{} 1} \minus{} a_j \leq m,$ $ (1 \leq j \leq k \minus{} 1)$.

Let $k>1$ be an integer. The sequence $a_1,a_2, \cdots$ is defined as: $a_1=1, a_2=k$ and for all $n>1$ we have: $a_{n+1}-(k+1)a_n+a_{n-1}=0$ Find all positive integers $n$ such that $a_n$ is a power of $k$. [i]Proposed by Amirhossein Pooya[/i]

Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.

Let $n$ be a natural number. Two numbers are called "unsociable" if their greatest common divisor is $1$. The numbers $\{1,2,...,2n\}$ are partitioned into $n$ pairs. What is the minimum number of "unsociable" pairs that are formed?

Let $A_1,A_2,\cdots , A_n$ be arithmetic progressions of integers, each of $k$ terms, such that any two of these arithmetic progressions have at least two common elements. Suppose $b$ of these arithmetic progressions have common difference $d_1$ and the remaining arithmetic progressions have common difference $d_2$ where $0<b<n$. Prove that \[b \le 2\left(k-\frac{d_2}{gcd(d_1,d_2)}\right)-1.\]

Find all prime numbers $ p,q,r$, such that $ \frac{p}{q}\minus{}\frac{4}{r\plus{}1}\equal{}1$

How many positive integers $n$ are there such that $n$ is an exact divisors of at least one of the numbers $10^{40}$ and $20^{30}$?

Show that, if $m$ and $n$ are non-zero integers of like parity, and $n^2 -1$ is divisible by $m^2 - n^2 + 1$, then $m^2 - n^2 + 1$ is the square of an integer. Amer. Math. Monthly

For a positive integer $n$, $a_1, a_2, \cdots a_k$ are all positive integers without repetition that are not greater than $n$ and relatively prime to $n$. If $k>8$, prove the following. $$\sum_{i=1}^k |a_i-\frac{n}{2}|<\frac{n(k-4)}{2}$$

Given a circle $C:x^2+y^2=r^2$ ($r$ is an odd number). $P(u,v)\in C$, satisfying: $u=p^m, v=q^n$($p,q$ are prime numbers, $m,n$ are integers, $u>v$). Define $A,B,C,D,M,N:A(r,0),B(-r,0),C(0,-r),D(0,r),M(u,0),N(0,v)$. Prove that $|AM|=1,|BM|=9,|CN|=8,|DN|=2$.

Define the sequence $a_1, a_2, a_3, ...$ by $a_1 = 1$, $a_n = a_{n-1} + a_{[n/2]}$. Does the sequence contain infinitely many multiples of $7$?

How many positive integers less that $200$ are relatively prime to either $15$ or $24$?

[b]p1.[/b] What is $6\times 7 + 4 \times 7 + 6\times 3 + 4\times 3$? [b]p2.[/b] How many integers $n$ have exactly $\sqrt{n}$ factors? [b]p3.[/b] A triangle has distinct angles $3x+10$, $2x+20$, and $x+30$. What is the value of $x$? [b]p4.[/b] If $4$ people of the Math Club are randomly chosen to be captains, and Henry is one of the $30$ people eligible to be chosen, what is the probability that he is not chosen to be captain? [b]p5.[/b] $a, b, c, d$ is an arithmetic sequence with difference $x$ such that $a, c, d$ is a geometric sequence. If $b$ is $12$, what is $x$? (Note: the difference of an aritmetic sequence can be positive or negative, but not $0$) [b]p6.[/b] What is the smallest positive integer that contains only $0$s and $5$s that is a multiple of $24$. [b]p7.[/b] If $ABC$ is a triangle with side lengths $13$, $14$, and $15$, what is the area of the triangle made by connecting the points at the midpoints of its sides? [b]p8.[/b] How many ways are there to order the numbers $1,2,3,4,5,6,7,8$ such that $1$ and $8$ are not adjacent? [b]p9.[/b] Find all ordered triples of nonnegative integers $(x, y, z)$ such that $x + y + z = xyz$. [b]p10.[/b] Noah inscribes equilateral triangle $ABC$ with area $\sqrt3$ in a cricle. If $BR$ is a diameter of the circle, then what is the arc length of Noah's $ARC$? [b]p11.[/b] Today, $4/12/14$, is a palindromic date, because the number without slashes $41214$ is a palindrome. What is the last palindromic date before the year $3000$? [b]p12.[/b] Every other vertex of a regular hexagon is connected to form an equilateral triangle. What is the ratio of the area of the triangle to that of the hexagon? [b]p13.[/b] How many ways are there to pick four cards from a deck, none of which are the same suit or number as another, if order is not important? [b]p14.[/b] Find all functions $f$ from $R \to R$ such that $f(x + y) + f(x - y) = x^2 + y^2$. [b]p15.[/b] What are the last four digits of $1(1!) + 2(2!) + 3(3!) + ... + 2013(2013!)$/ [b]p16.[/b] In how many distinct ways can a regular octagon be divided up into $6$ non-overlapping triangles? [b]p17.[/b] Find the sum of the solutions to the equation $\frac{1}{x-3} + \frac{1}{x-5} + \frac{1}{x-7} + \frac{1}{x-9} = 2014$ . [b]p18.[/b] How many integers $n$ have the property that $(n+1)(n+2)(n+3)(n+4)$ is a perfect square of an integer? [b]p19.[/b] A quadrilateral is inscribed in a unit circle, and another one is circumscribed. What is the minimum possible area in between the two quadrilaterals? [b]p20.[/b] In blindfolded solitary tic-tac-toe, a player starts with a blank $3$-by-$3$ tic-tac-toe board. On each turn, he randomly places an "$X$" in one of the open spaces on the board. The game ends when the player gets $3$ $X$s in a row, in a column, or in a diagonal as per normal tic-tac-toe rules. (Note that only $X$s are used, not $O$s). What fraction of games will run the maximum $7$ amount of moves? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all non-negative integer numbers $n$ for which there exists integers $a$ and $b$ such that $n^2=a+b$ and $n^3=a^2+b^2.$

If you multiply all positive integer factors of $24$, you get $24^x$. Find $x$.

Let $S$ be the set of integers that can be written in the form $50m + 3n$ where $m$ and $n$ are non-negative integers. For example $3, 50, 53$ are all in $S$. Find the sum of all positive integers not in $S$.

A [i]super-integer[/i] triangle is defined to be a triangle whose lengths of all sides and at least one height are positive integers. We will deem certain positive integer numbers to be [i]good[/i] with the condition that if the lengths of two sides of a super-integer triangle are two (not necessarily different) good numbers, then the length of the remaining side is also a good number. Let $5$ be a good number. Prove that all integers larger than $2$ are good numbers.

Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that \[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \] and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$. [i]Proposed by Victor Wang[/i]

Let $f : R^+ \to R^+$ be a function such that for all $x, y \in R^+$, $f(x)f(y) = f(xy) + f\left( \frac{x}{y}\right)$, where $R^+$ represents the positive real numbers. Given that $f(2) = 3$, compute the last two digits of $f(2^{2^{2020}})$. .

Find all pairs of positive integers $(a,b)$ such that there exists a finite set $S$ satisfying that any positive integer can be written in the form $$n = x^a + y^b + s$$where $x,y$ are nonnegative integers and $s \in S$ [i]CSJL[/i]

Find all functions $f:\mathbb{N}\to \mathbb{N}$ so that for every natural numbers $m,n$ :$f(n)+2mn+f(m)$ is a perfect square.