Found problems: 526
Let $x,y$ be positive integers. If for each positive integer $n$ we have that $$(ny)^2+1\mid x^{\varphi(n)}-1.$$
Prove that $x=1$.
[i](Silouanos Brazitikos, Greece)[/i]
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.
[i]Proposed by James Rickards, Canada[/i]
Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$
[i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]
Let $m,n$ be positive integers. Find the minimum positive integer $N$ which satisfies the following condition. If there exists a set $S$ of integers that contains a complete residue system module $m$ such that $| S | = N$, then there exists a nonempty set $A \subseteq S$ so that $n\mid {\sum\limits_{x \in A} x }$.
For a finite non empty set of primes $P$, let $m(P)$ denote the largest possible number of consecutive positive integers, each of which is divisible by at least one member of $P$.
(i) Show that $|P|\le m(P)$, with equality if and only if $\min(P)>|P|$.
(ii) Show that $m(P)<(|P|+1)(2^{|P|}-1)$.
(The number $|P|$ is the size of set $P$)
[i]Dan Schwarz, Romania[/i]
Find the smallest positive integer solution to the equation $2^{2^k}\equiv k\pmod{29}$.
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
[i]Proposed by Hong Kong[/i]
For any natural number $n > 1$ write the finite decimal expansion of $\frac{1}{n}$ (for example we write $\frac{1}{2}=0.4\overline{9}$ as its infinite decimal expansion not $0.5)$. Determine the length of non-periodic part of the (infinite) decimal expansion of $\frac{1}{n}$.
Let $a\in\mathbb N$ and $m=a^2+a+1$. Find the number of $0\leq x\leq m$ that:\[x^3\equiv1(\mbox{mod}\ m)\]
Does there exist an infinite set $\displaystyle{M}$ consisting of positive integers such that for any $\displaystyle{a,b \in M}$ with $\displaystyle{a < b}$ the sum $\displaystyle{a + b}$ is square-free?
[b]Note.[/b] A positive integer is called square-free if no perfect square greater than $\displaystyle{1}$ divides it.
[i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]
Does there exist such infinite set of positive integers $S$ that satisfies the condition below?
*for all $a,b$ in $S$, there exists an odd integer $k$ that $a$ divides $b^k+1$.
Find all positive integers $a$ such that for any positive integer $n\ge 5$ we have $2^n-n^2\mid a^n-n^a$.
For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?
Let $\mathbb N$ denote the set of positive integers. A function $f\colon\mathbb N\to\mathbb N$ has the property that for all positive integers $m$ and $n$, exactly one of the $f(n)$ numbers
\[f(m+1),f(m+2),\ldots,f(m+f(n))\]
is divisible by $n$. Prove that $f(n)=n$ for infinitely many positive integers $n$.
Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that
$$f(x + f(y)) = f(x) + f(y)$$
for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.
Let $f(x)$ be a polynomial with integer coefficients, for which there exist $a,b\in \mathbb{Z}$ ($a\neq b$), such that $f(a)$ and $f(b)$ are coprime. Prove that there exist infinitely many values for $x$, such that each $f(x)$ is coprime with any other.
Finitely many cards are placed in two stacks, with more cards in the left stack than the right. Each card has one or more distinct names written on it, although different cards may share some names. For each name, we define a “shuffle” by moving every card that has this name written on it to the opposite stack. Prove that it is always possible to end up with more cards in the right stack by picking several distinct names, and doing in turn the shuffle corresponding to each name.
A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$.
[i]Proposed by Jaroslaw Wroblewski, Poland[/i]
Compute
$$
2022^{\left(2022^{\cdot ^ {\cdot ^{\cdot ^{\left(2022^{2022}\right)}}}}\right)} \pmod{111}
$$
where there are $2022$ $2022$s. (Give the answer as an integer from $0$ to $110$).
[i]Proposed by David Tang[/i]
Let $a,b,c,d$ be positive integers such that $ad \neq bc$ and $gcd(a,b,c,d)=1$. Let $S$ be the set of values attained by $\gcd(an+b,cn+d)$ as $n$ runs through the positive integers. Show that $S$ is the set of all positive divisors of some positive integer.
For a positive integer $k,$ call an integer a $pure$ $k-th$ $power$ if it can be represented as $m^k$ for some integer $m.$ Show that for every positive integer $n,$ there exists $n$ distinct positive integers such that their sum is a pure $2009-$th power and their product is a pure $2010-$th power.
[u]Set 1[/u]
[b]p1.[/b] Find $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11$.
[b]p2.[/b] Find $1 \cdot 11 + 2 \cdot 10 + 3 \cdot 9 + 4 \cdot 8 + 5 \cdot 7 + 6 \cdot 6$.
[b]p3.[/b] Let $\frac{1}{1\cdot 2} +\frac{1}{2\cdot 3} +\frac{1}{3\cdot 4} +\frac{1}{4\cdot 5} +\frac{1}{5\cdot 6} +\frac{1}{6\cdot 7} +\frac{1}{7\cdot 8} +\frac{1}{8\cdot 9} +\frac{1}{9\cdot 10} +\frac{1}{10\cdot 11} =\frac{m}{n}$ , where $m$ and $n$ are positive integers that share no prime divisors. Find $m + n$.
[u]Set 2[/u]
[b]p4.[/b] Define $0! = 1$ and let $n! = n \cdot (n - 1)!$ for all positive integers $n$. Find the value of $(2! + 0!)(1! + 8!)$.
[b]p5.[/b] Rachel’s favorite number is a positive integer $n$. She gives Justin three clues about it:
$\bullet$ $n$ is prime.
$\bullet$ $n^2 - 5n + 6 \ne 0$.
$\bullet$ $n$ is a divisor of $252$.
What is Rachel’s favorite number?
[b]p6.[/b] Shen eats eleven blueberries on Monday. Each day after that, he eats five more blueberries than the day before. For example, Shen eats sixteen blueberries on Tuesday. How many blueberries has Shen eaten in total before he eats on the subsequent Monday?
[u]Set 3[/u]
[b]p7.[/b] Triangle $ABC$ satisfies $AB = 7$, $BC = 12$, and $CA = 13$. If the area of $ABC$ can be expressed in the form $m\sqrt{n}$, where $n$ is not divisible by the square of a prime, then determine $m + n$.
[b]p8.[/b] Sebastian is playing the game Split! on a coordinate plane. He begins the game with one token at $(0, 0)$. For each move, he is allowed to select a token on any point $(x, y)$ and take it off the plane, replacing it with two tokens, one at $(x + 1, y)$, and one at $(x, y + 1)$. At the end of the game, for a token on $(a, b)$, it is assigned a score $\frac{1}{2^{a+b}}$ . These scores are summed for his total score. Determine the highest total score Sebastian can get in $100$ moves.
[b]p9.[/b] Find the number of positive integers $n$ satisfying the following two properties:
$\bullet$ $n$ has either four or five digits, where leading zeros are not permitted,
$\bullet$ The sum of the digits of $n$ is a multiple of $3$.
[u]Set 4[/u]
[b]p10.[/b] [i]A unit square rotated $45^o$ about a vertex,
Sweeps the area for Farmer Khiem’s pen.
If $n$ is the space the pigs can roam,
Determine the floor of $100n$.[/i]
If $n$ is the area a unit square sweeps out when rotated 4$5$ degrees about a vertex, determine $\lfloor 100n \rfloor$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
[img]https://cdn.artofproblemsolving.com/attachments/b/1/129efd0dbd56dc0b4fb742ac80eaf2447e106d.png[/img]
[b]p11.[/b][i] Michael is planting four trees,
In a grid, three rows of three,
If two trees are close,
Then both are bulldozed,
So how many ways can it be?[/i]
In a three by three grid of squares, determine the number of ways to select four squares such that no two share a side.
[b]p12.[/b] [i]Three sixty-seven
Are the last three digits of
$n$ cubed. What is $n$?[/i]
If the last three digits of $n^3$ are $367$ for a positive integer $n$ less than $1000$, determine $n$.
[u]Set 5[/u]
[b]p13.[/b] Determine $\sqrt[4]{97 + 56\sqrt{3}} + \sqrt[4]{97 - 56\sqrt{3}}$.
[b]p14. [/b]Triangle $\vartriangle ABC$ is inscribed in a circle $\omega$ of radius $12$ so that $\angle B = 68^o$ and $\angle C = 64^o$ . The perpendicular from $A$ to $BC$ intersects $\omega$ at $D$, and the angle bisector of $\angle B$ intersects $\omega$ at $E$. What is the value of $DE^2$?
[b]p15.[/b] Determine the sum of all positive integers $n$ such that $4n^4 + 1$ is prime.
[u]Set 6[/u]
[b]p16.[/b] Suppose that $p, q, r$ are primes such that $pqr = 11(p + q + r)$ such that $p\ge q \ge r$. Determine the sum of all possible values of $p$.
[b]p17.[/b] Let the operation $\oplus$ satisfy $a \oplus b =\frac{1}{1/a+1/b}$ . Suppose $$N = (...((2 \oplus 2) \oplus 2) \oplus ... 2),$$ where there are $2018$ instances of $\oplus$ . If $N$ can be expressed in the form $m/n$, where $m$ and $n$ are relatively prime positive integers, then determine $m + n$.
[b]p18.[/b] What is the remainder when $\frac{2018^{1001} - 1}{2017}$ is divided by $2017$?
PS. You had better use hide for answers. Last sets have been posted [url=https://artofproblemsolving.com/community/c4h2777307p24369763]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $n$ be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer. Compute the remainder when $3^n$ is divided by $1000$.
$(\text{a})$ Do there exist 14 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 11$?
$(\text{b})$ Do there exist 21 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 13$?