The volume of a rectangular solid each of whose side, front, and bottom faces are $12\text{ in}^2$, $8\text{ in}^2$, and $6\text{ in}^2$ respectively is: $\textbf{(A)}\ 576\text{ in}^3 \qquad \textbf{(B)}\ 24\text{ in}^3 \qquad \textbf{(C)}\ 9\text{ in}^3 \qquad \textbf{(D)}\ 104\text{ in}^3 \qquad \textbf{(E)}\ \text{None of these}$

For the natural numbers $x$ and $y$, $2x^2 + x = 3y^2 + y$ . Prove that then $x-y$, $2x + 2y + 1$ and $3x + 3y + 1$ are perfect squares.

You are given a number, and round it to the nearest thousandth, round this result to nearest hundredth, and round this result to the nearest tenth. If the final result is $.7$, what is the smallest number you could have been given? As is customary, $5$’s are always rounded up. Give the answer as a decimal.

given $p_1,p_2,...$ be a sequence of integer and $p_1=2$, for positive integer $n$, $p_{n+1}$ is the least prime factor of $np_1^{1!}p_2^{2!}...p_n^{n!}+1 $ prove that all primes appear in the sequence (Proposed by Beatmania)

Let $A=(a_{ij})$ be the $n\times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j+j^i$ by $3$ for $i,j=1,2,\ldots,n$. Find the greatest $n$ for which $\det A\ne0$.

Prove that the equation $4x^3-3x+1=2y^2$ has at least $31$ solutions in positive integers $x$ and $y$ with $x\leq 2005$.

Call a year [i]ultra-even[/i] if all of its digits are even. Thus $2000,2002,2004,2006$, and $2008$ are all [i]ultra-even[/i] years. They are all $2$ years apart, which is the shortest possible gap. $2009$ is not an [i]ultra-even[/i] year because of the $9$, and $2010$ is not an ultra-even year because of the $1$. (a) In the years between the years $1$ and $10000$, what is the longest possible gap between two [i]ultra-even[/i] years? Give an example of two ultra-even years that far apart with no [i]ultra-even[/i] years between them. Justify your answer. (b) What is the second-shortest possible gap (that is, the shortest gap longer than $2$ years) between two [i]ultra-even[/i] years? Again, give an example, and justify your answer.

Proce that number $$\underbrace{11...11}_{2n \,\, digits}-\underbrace{22 ... 22}_{n \,\, digits}$$ is, for every natural $n$, a perfect square.

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Find all integers $n$ such that $\frac{4n}{n^2 +3 }$is an integer.

Jerry likes to play with numbers. One day, he wrote all the integers from $1$ to $2024$ on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them with either their sum or their product. (For example, Jerry's first step might have been to erase $1, 2, 3$, and $5$, and then write either $11$, their sum, or $30$, their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the board were odd. What is the maximum possible number of integers on the board at that time? $ \textbf{(A) }1010 \qquad \textbf{(B) }1011 \qquad \textbf{(C) }1012 \qquad \textbf{(D) }1013 \qquad \textbf{(E) }1014 \qquad $

A natural number is called [i]chaotigal [/i] if it and its successor both have the sum of their digits divisible by $2021$. How many digits are in the smallest chaotigal number?

Find all positive integers $(a,b,c)$ such that $$ab-c,\quad bc-a,\quad ca-b$$ are all powers of $2$. [i]Proposed by Serbia[/i]

Prove that for no integer $ n$ is $ n^7 \plus{} 7$ a perfect square.

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

The $x$ and $y$ figures satisfy a condition: for every $n\ge1$ the number $$xx...x6yy...y4$$ ($n$ times $x$ and $n$ times $y$) is a perfect square. Find all possible $x$ and $y$.

Let $n \ge 2$ be a positive integer. Suppose $a_1, a_2, \dots, a_n$ are distinct integers. For $k = 1, 2, \dots, n$, let \[ s_k := \prod_{\substack{i \not= k, \\ 1 \le i \le n}} |a_k - a_i|, \] i.e. $s_k$ is the product of all terms of the form $|a_k - a_i|$, where $i \in \{ 1, 2, \dots, n \}$ and $i \not= k$. Find the largest positive integer $M$ such that $M$ divides the least common multiple of $s_1, s_2, \dots, s_n$ for any choices of $a_1, a_2, \dots, a_n$.

Let $ a$ and $ k$ be positive integers. Let $ a_i$ be the sequence defined by $ a_1 \equal{} a$ and $ a_{n \plus{} 1} \equal{} a_n \plus{} k\pi(a_n)$ where $ \pi(x)$ is the product of the digits of $ x$ (written in base ten) Prove that we can choose $ a$ and $ k$ such that the infinite sequence $ a_i$ contains exactly $ 100$ distinct terms

Given a finite set $ P$ of prime numbers, prove that there exists a positive integer $ x$ such that it can be written in the form $ a^p \plus{} b^p$ ($ a,b$ are positive integers), for each $ p\in P$, and cannot be written in that form for each $ p$ not in $ P$.

Prove that any rational $r \in (0, 1)$ can be written uniquely in the form $$r=\frac{a_1}{1!}+\frac{a_2}{2!}+\frac{a_3}{3!}+\cdots+\frac{a_k}{k!}$$ where $a_i\text{’s}$ are nonnegative integers with $a_i\leq i-1$ for all $i$.

If \[\frac{1} {x} + \frac{1} {2x^2} +\frac{1} {4x^3}+\frac{1}{8x^4}+\frac{1}{16x^5}+\cdots=\frac{1} {64}, \] and $x$ can be expressed in the form $\frac{m}{n},$ where $m,n$ are relatively prime positive integers, find $m+n$. [i]Author: Ray Li[/i]

Start with a finite sequence $ a_1,a_2,\dots,a_n$ of positive integers. If possible, choose two indices $ j < k$ such that $ a_j$ does not divide $ a_k$ and replace $ a_j$ and $ a_k$ by $ \gcd(a_j,a_k)$ and $ \text{lcm}\,(a_j,a_k),$ respectively. Prove that if this process is repeated, it must eventually stop and the final sequence does not depend on the choices made. (Note: $ \gcd$ means greatest common divisor and lcm means least common multiple.)

The diagram shows a large circular dart board with four smaller shaded circles each internally tangent to the larger circle. Two of the internal circles have half the radius of the large circle, and are, therefore, tangent to each other. The other two smaller circles are tangent to these circles. If a dart is thrown so that it sticks to a point randomly chosen on the dart board, then the probability that the dart sticks to a point in the shaded area is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] size(150); defaultpen(linewidth(0.8)); filldraw(circle((0,0.5),.5),gray); filldraw(circle((0,-0.5),.5),gray); filldraw(circle((2/3,0),1/3),gray); filldraw(circle((-2/3,0),1/3),gray); draw(unitcircle); [/asy]

Find all nonzero subsets $A$ of the set $\left\{2,3,4,5,\cdots\right\}$ such that $\forall n\in A$, we have that $n^{2}+4$ and $\left\lfloor{\sqrt{n}\right\rfloor}+1$ are both in $A$.

Given a finite sequence of integers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ for $n\geq 2.$ Show that there exists a subsequence $a_{k_{1}},$ $a_{k_{2}},$ $...,$ $a_{k_{m}},$ where $1\leq k_{1}\leq k_{2}\leq...\leq k_{m}\leq n,$ such that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}$ is divisible by $n.$ [b]Note by Darij:[/b] Of course, the $1\leq k_{1}\leq k_{2}\leq ...\leq k_{m}\leq n$ should be understood as $1\leq k_{1}<k_{2}<...<k_{m}\leq n;$ else, we could take $m=n$ and $k_{1}=k_{2}=...=k_{m},$ so that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}=n^{2}a_{k_{1}}^{2}$ will surely be divisible by $n.$