Olympiad problems

2024 AMC 12/AHSME, 4

Tags: factorial

What is the least value of $n$ such that $n!$ is a multiple of $2024$? $ \textbf{(A) }11 \qquad \textbf{(B) }21 \qquad \textbf{(C) }22 \qquad \textbf{(D) }23 \qquad \textbf{(E) }253 \qquad $

2023 Serbia Team Selection Test, P5

Tags: factorial , nt , weird , modular arithmetic , number theory

For positive integers $a$ and $b$, define \[a!_b=\prod_{1\le i\le a\atop i \equiv a \mod b} i\] Let $p$ be a prime and $n>3$ a positive integer. Show that there exist at least 2 different positive integers $t$ such that $1<t<p^n$ and $t!_p\equiv 1\pmod {p^n}$.

2023 Bulgarian Spring Mathematical Competition, 12.3

Tags: factorial , number theory

Given is a polynomial $f$ of degree $m$ with integer coefficients and positive leading coefficient. A positive integer $n$ is $\textit {good for f(x)}$ if there exists a positive integer $k_n$, such that $n!+1=f(n)^{k_n}$. Prove that there exist only finitely many integers good for $f$.

2007-2008 SDML (Middle School), 6

Tags: factorial

Find the smallest positive integer $k$ such that $k!$ ends in at least $43$ zeroes.

2012 AMC 12/AHSME, 11

Tags: probability , counting , distinguishability , factorial

Alex, Mel, and Chelsea play a game that has $6$ rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is $\frac{1}{2}$, and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round? $ \textbf{(A)}\ \frac{5}{72}\qquad\textbf{(B)}\ \frac{5}{36}\qquad\textbf{(C)}\ \frac{1}{6}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ 1 $

1946 Moscow Mathematical Olympiad, 114

Tags: double factorial , factorial

Prove that for any positive integer $n$ the following identity holds $\frac{(2n)!}{n!}= 2^n \cdot (2n - 1)!!$

2008 Putnam, B2

Tags: integration , limit , calculus , factorial , logarithm

Let $ F_0\equal{}\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n\plus{}1}(x)\equal{}\int_0^xF_n(t)\,dt.$ Evaluate $ \displaystyle\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$

2006 AIME Problems, 4

Tags: factorial , number theory , prime factorization

Let $N$ be the number of consecutive 0's at the right end of the decimal representation of the product $1!\times2!\times3!\times4!\cdots99!\times100!.$ Find the remainder when $N$ is divided by 1000.

2021 Irish Math Olympiad, 1

Tags: number theory , factorial

Let $N = 15! = 15\cdot 14\cdot 13 ... 3\cdot 2\cdot 1$. Prove that $N$ can be written as a product of nine different integers all between $16$ and $30$ inclusive.

2020 LIMIT Category 1, 19

Tags: limit , algebra , factorial

Let $a=2019^{1009}, b=2019!$ and $c=1010^{2019}$, then which of the following is true? (A)$c<b<a$ (B)$a<b<c$ (C)$b<a<c$ (D)$b<c<a$

2009 Princeton University Math Competition, 2

Tags: factorial

Suppose you are given that for some positive integer $n$, $1! + 2! + \ldots + n!$ is a perfect square. Find the sum of all possible values of $n$.

2013 NIMO Problems, 3

Tags: factorial , floor function

Let $a_1, a_2, \dots, a_{1000}$ be positive integers whose sum is $S$. If $a_n!$ divides $n$ for each $n = 1, 2, \dots, 1000$, compute the maximum possible value of $S$. [i]Proposed by Michael Ren[/i]

2014 NIMO Problems, 5

Tags: floor function , function , factorial

Find the largest integer $n$ for which $2^n$ divides \[ \binom 21 \binom 42 \binom 63 \dots \binom {128}{64}. \][i]Proposed by Evan Chen[/i]

2004 Federal Competition For Advanced Students, Part 1, 3

Tags: factorial

For natural numbers $a, b$, define $Z(a,b)=\frac{(3a)!\cdot (4b)!}{a!^4 \cdot b!^3}$. [b](a)[/b] Prove that $Z(a, b)$ is an integer for $a \leq b$. [b](b)[/b] Prove that for each natural number $b$ there are infinitely many natural numbers a such that $Z(a, b)$ is not an integer.[/list]

2021 Miklós Schweitzer, 2

Tags: factorial , diophantine equation , number theory

Prove that the equation \[ 2^x + 5^y - 31^z = n! \] has only a finite number of non-negative integer solutions $x,y,z,n$.

2002 Germany Team Selection Test, 3

Tags: number theory , decimal representation , digit , factorial

Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.

2023-IMOC, N2

Tags: factorial , number theory

Find all pairs of positive integers $(a, b)$ such that $a^b+b^a=a!+b^2+ab+1$.

2019 AMC 10, 14

Tags: factorial

The base-ten representation for $19!$ is $121,6T5,100,40M,832,H00$, where $T$, $M$, and $H$ denote digits that are not given. What is $T+M+H$? $\textbf{(A) }3 \qquad\textbf{(B) }8 \qquad\textbf{(C) }12 \qquad\textbf{(D) }14 \qquad\textbf{(E) } 17 $

1972 IMO Longlists, 15

Tags: binomial coefficient , factorial , number theory , recurrence relation

Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

2015 CCA Math Bonanza, L4.1

Tags: factorial

How many divisors of $12!$ are perfect squares? [i]2015 CCA Math Bonanza Lightning Round #4.1[/i]

2010 National Olympiad First Round, 4

Tags: factorial

How many positive integers less than $2010$ are there such that the sum of factorials of its digits is equal to itself? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

2022 Junior Macedonian Mathematical Olympiad, P1

Tags: number theory , diophantine equation , factorial

Determine all positive integers $a$, $b$ and $c$ which satisfy the equation $$a^2+b^2+1=c!.$$ [i]Proposed by Nikola Velov[/i]

2013 India Regional Mathematical Olympiad, 6

Tags: number theory , polynomial , root , factorial , greatest common divisor

Let $P(x)=x^3+ax^2+b$ and $Q(x)=x^3+bx+a$, where $a$ and $b$ are nonzero real numbers. Suppose that the roots of the equation $P(x)=0$ are the reciprocals of the roots of the equation $Q(x)=0$. Prove that $a$ and $b$ are integers. Find the greatest common divisor of $P(2013!+1)$ and $Q(2013!+1)$.

2011 Saudi Arabia BMO TST, 4

Tags: algebra , factorial

Consider a non-zero real number $a$ such that $\{a\} + \left\{\frac{1}{a}\right\}=1$, where $\{x\}$ denotes the fractional part of $x$. Prove that for any positive integer $n$, $\{a^n\} + \left\{\frac{1}{a^n}\right\}= 1$.

1996 AMC 12/AHSME, 3

Tags: factorial

$\frac{(3!)!}{3!} =$ $\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 6\qquad \text{(D)}\ 40\qquad \text{(E)}\ 120$