Let $\mathbb{N}$ and $\mathbb{N}^*$ be the sets containing the natural numbers/positive integers respectively. We define a binary relation on $\mathbb{N}$ by $a\acute{\in}b$ iff the $a$-th bit in the binary representation of $b$ is $1$. We define a binary relation on $\mathbb{N}^*$ by $a\tilde{\in}b$ iff $b$ is a multiple of the $a$-th prime number $p_a$. i) Prove that there is no bijection $f:\mathbb{N}\to \mathbb{N}^*$ such that $a\acute{\in}b\Leftrightarrow f(a)\tilde{\in}f(b)$. ii) Prove that there is a bijection $g:\mathbb{N}\to \mathbb{N}^*$ such that $(a\acute{\in}b \vee b\acute{\in}a)\Leftrightarrow (g(a)\tilde{\in}g(b) \vee g(b)\tilde{\in}g(a))$.

Let $n$ be an arbitrary positive integer. (a) For every positive integers $a$ and $b$, show that $gcd(n^a + 1, n^b + 1) \le n^{gcd(a,b)} + 1$. (b) Show that there exist infinitely many composite pairs ($a, b)$, such that each of them is not a multiply of the other number and equality holds in (a).

The number $a_n$ is formed by writing in succession, without spaces, the numbers $1, 2, ..., n$ (for example, $a_{11} = 1234567891011$). Find the smallest number t such that $11 | a_t$.

Find all positive integers $a,b,c$ such that $$\begin{cases} a^2 = 4(b+c) \\ a^3 -2b^3 -4c^3 =\frac12 abc \end {cases}$$

Let $k$ be a positive integer, and let $s(n)$ denote the sum of the digits of $n$. Show that among the positive integers with $k$ digits, there are as many numbers $n$ satisfying $s(n) < s(2n)$ as there are numbers $n$ satisfying $s(n) > s(2n)$.

For every positive integer $n$, the symbol $a_n/b_n$ is the simplest form of the fraction $1+1/2+...+1/n$. Prove that for every pair of positive integers $(M, N)$ we can always find a positive integer $m$ where $(a_n, N) = 1$ for all $n = m, m + 1, ...,m + M$.

For positive integers $n$, let $\tau (n)$ denote the number of positive integer divisors of $n$, including $1$ and $n$. For example, $\tau (1)=1$ and $\tau(6) =4$. Define $S(n)$ by \[S(n)=\tau(1)+ \tau(2) + ... + \tau(n).\] Let $a$ denote the number of positive integers $n \leq 2005$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \leq 2005$ with $S(n)$ even. Find $|a-b|$.

Find all pairs of integers $(c, d)$, both greater than 1, such that the following holds: For any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p > c(2c+1)$, there exists a set $S$ of at most $\big(\tfrac{2c-1}{2c+1}\big)p$ integers, such that \[\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}\] contains a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$).

Define $a_0 = 2$ and $a_{n+1} = a^2_n + a_n -1$ for $n \ge 0$. Prove that $a_n$ is coprime to $2n + 1$ for all $n \in N$.

Find the smallest natural number $ k$ for which there exists natural numbers $ m$ and $ n$ such that $ 1324 \plus{} 279m \plus{} 5^n$ is $ k$-th power of some natural number.

Find all integer solutions of the equation \[\frac {x^{7} \minus{} 1}{x \minus{} 1} \equal{} y^{5} \minus{} 1.\]

The number $A$ is a product of $n$ distinct natural numbers. Prove that $A$ has at least $\frac{n(n-1)}{2}+1$ distinct divisors (including 1 and $A$).

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Compute $2^2 + 0 \cdot 0 + 2^2 + 3^3$. [b]p2.[/b] How many total letters (not necessarily distinct) are there in the names Jerry, Justin, Jackie, Jason, and Jeffrey? [b]p3.[/b] What is the remainder when $20232023$ is divided by $50$? [b]p4.[/b] Let $ABCD$ be a square. The fraction of the area of $ABCD$ that is the area of the intersection of triangles $ABD$ and $ABC$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ relatively prime positive integers. Find $a + b$. [b]p5.[/b] Raymond is playing basketball. He makes a total of $15$ shots, all of which are either worth $2$ or $3$ points. Given he scored a total of $40$ points, how many $2$-point shots did he make? [b]p6.[/b] If a fair coin is flipped $4$ times, the probability that it lands on heads more often than tails is $\frac{a}{b}$ , where $a$ and $b$ relatively prime positive integers. Find $a + b$. [b]p7.[/b] What is the sum of the perfect square divisors of $640$? [b]p8.[/b] A regular hexagon and an equilateral triangle have the same perimeter. The ratio of the area between the hexagon and equilateral triangle can be expressed in the form $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p9.[/b] If a cylinder has volume $1024\pi$, radius of $r$ and height $h$, how many ordered pairs of integers $(r, h)$ are possible? [b]p10.[/b] Pump $A$ can fill up a balloon in $3$ hours, while pump $B$ can fill up a balloon in $5$ hours. Pump $A$ starts filling up a balloon at $12:00$ PM, and pump $B$ is added alongside pump $A$ at a later time. If the balloon is completely filled at $2:00$ PM, how many minutes after $12:00$ PM was Pump $B$ added? [b]p11.[/b] For some positive integer $k$, the product $81 \cdot k$ has $20$ factors. Find the smallest possible value of $k$. [b]p12.[/b] Two people wish to sit in a row of fifty chairs. How many ways can they sit in the chairs if they do not want to sit directly next to each other and they do not want to sit with exactly one empty chair between them? [b]p13.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $2$ and $M$ be the midpoint of $BC$. Let $P$ be a point in the same plane such that $2PM = BC$. The minimum value of $AP$ can be expressed as $\sqrt{a}-b$, where $a$ and $b$ are positive integers such that $a$ is not divisible by any perfect square aside from $1$. Find $a + b$. [b]p14.[/b] What are the $2022$nd to $2024$th digits after the decimal point in the decimal expansion of $\frac{1}{27}$ , expressed as a $3$ digit number in that order (i.e the $2022$nd digit is the hundreds digit, $2023$rd digit is the tens digit, and $2024$th digit is the ones digit)? [b]p15.[/b] After combining like terms, how many terms are in the expansion of $(xyz+xy+yz+xz+x+y+z)^{20}$? [b]p16.[/b] Let $ABCD$ be a trapezoid with $AB \parallel CD$ where $AB > CD$, $\angle B = 90^o$, and $BC = 12$. A line $k$ is dropped from $A$, perpendicular to line $CD$, and another line $\ell$ is dropped from $C$, perpendicular to line $AD$. $k$ and $\ell$ intersect at $X$. If $\vartriangle AXC$ is an equilateral triangle, the area of $ABCD$ can be written as $m\sqrt{n}$, where $m$ and $n$ are positive integers such that $n$ is not divisible by any perfect square aside from $1$. Find $m + n$. [b]p17.[/b] If real numbers $x$ and $y$ satisfy $2x^2 + y^2 = 8x$, maximize the expression $x^2 + y^2 + 4x$. [b]p18.[/b] Let $f(x)$ be a monic quadratic polynomial with nonzero real coefficients. Given that the minimum value of $f(x)$ is one of the roots of $f(x)$, and that $f(2022) = 1$, there are two possible values of $f(2023)$. Find the larger of these two values. [b]p19.[/b] I am thinking of a positive integer. After realizing that it is four more than a multiple of $3$, four less than a multiple of $4$, four more than a multiple of 5, and four less than a multiple of $7$, I forgot my number. What is the smallest possible value of my number? [b]p20.[/b] How many ways can Aston, Bryan, Cindy, Daniel, and Evan occupy a row of $14$ chairs such that none of them are sitting next to each other? [b]p21.[/b] Let $x$ be a positive real number. The minimum value of $\frac{1}{x^2} +\sqrt{x}$ can be expressed in the form \frac{a}{b^{(c/d)}} , where $a$, $b$, $c$, $d$ are all positive integers, $a$ and $b$ are relatively prime, $c$ and $d$ are relatively prime, and $b$ is not divisible by any perfect square. Find $a + b + c + d$. [b]p22.[/b] For all $x > 0$, the function $f(x)$ is defined as $\lfloor x \rfloor \cdot (x + \{x\})$. There are $24$ possible $x$ such that $f(x)$ is an integer between $2000$ and $2023$, inclusive. If the sum of these $24$ numbers equals $N$, then find $\lfloor N \rfloor$. Note: Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, called the floor function. Also, $\{x\}$ is defined as $x - \lfloor x \rfloor$, called the fractional part function. [b]p23.[/b] Let $ABCD$ be a rectangle with $AD = 1$. Let $P$ be a point on diagonal $\overline{AC}$, and let $\omega$ and $\xi$ be the circumcircles of $\vartriangle APB$ and $\vartriangle CPD$, respectively. Line $\overleftrightarrow{AD}$ is extended, intersecting $\omega$ at $X$, and $\xi$ at $Y$ . If $AX = 5$ and $DY = 2$, find $[ABCD]^2$. Note: $[ABCD]$ denotes the area of the polygon $ABCD$. [b]p24.[/b] Alice writes all of the three-digit numbers on a blackboard (it’s a pretty big blackboard). Let $X_a$ be the set of three-digit numbers containing a somewhere in its representation, where a is a string of digits. (For example, $X_{12}$ would include $12$, $121$, $312$, etc.) If Bob then picks a value of $a$ at random so $0 \le a \le 999$, the expected number of elements in $X_a$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find$ m + n$. [b]p25.[/b] Let $f(x) = x^5 + 2x^4 - 2x^3 + 4x^2 + 5x + 6$ and $g(x) = x^4 - x^3 + x^2 - x + 1$. If $a$, $b$, $c$, $d$ are the roots of $g(x)$, then find $f(a) + f(b) + f(c) + f(d)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Dima has written number $ 1/80!,\,1/81!,\,\dots,1/99!$ on $ 20$ infinite pieces of papers as decimal fractions (the following is written on the last piece: $ \frac {1}{99!} \equal{} 0{,}{00\dots 00}10715\dots$, 155 0-s before 1). Sasha wants to cut a fragment of $ N$ consecutive digits from one of pieces without the comma. For which maximal $ N$ he may do it so that Dima may not guess, from which piece Sasha has cut his fragment? [i]A. Golovanov[/i]

Let $n = 2^k$. Prove that we can select $n$ integers from any $2n-1$ integers such that their sum is divisible by $n$.

Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?

A quadruple $(a; b; c; d)$ of positive integers with $a \leq b \leq c \leq d$ is called good if we can colour each integer red, blue, green or purple, in such a way that $i$ of each $a$ consecutive integers at least one is coloured red; $ii$ of each $b$ consecutive integers at least one is coloured blue; $iii$ of each $c$ consecutive integers at least one is coloured green; $iiii$ of each $d$ consecutive integers at least one is coloured purple. Determine all good quadruples with $a = 2.$

Let $r$ be the remainder when $2017^{2025!}-1$ is divided by $2025!.$ Compute $\tfrac{r}{2025!}.$ (Note that $2017$ is prime.)

Assume that $a$ and $b$ are integers. Prove that the equation $a^2 +b^2 +x^2 = y^2$ has an integer solution $x,y$ if and only if the product $ab$ is even.

Find the remainder of $\binom{3^n}{2^n}$ modulo $3^{n+1}$. [i]Proposed by V. Rastorguev[/i]

Solve in prime numbers the equation $p(p+1)+q(q+1)=r(r+1)$.

Let $p$ and $q$ be prime numbers. Show that $p^2+q^2+2020$ is composite.

Let $p$ be a prime number. We construct a directed graph of $p$ vertices, labeled with integers from $0$ to $p-1$. There is an edge from vertex $x$ to vertex $y$ if and only if $x^2+1\equiv y \pmod{p}$. Let $f(p)$ denotes the length of the longest directed cycle in this graph. Prove that $f(p)$ can attain arbitrarily large values.

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

Find all positive integers $a$ such that there exists a set $X$ of $6$ integers satisfying the following conditions: for every $k=1,2,\ldots ,36$ there exist $x,y\in X$ such that $ax+y-k$ is divisible by $37$.