This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 310

2008 Putnam, B2
Tags: calculus , integration , limit , 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}.$

2018 CCA Math Bonanza, I1
Tags: factorial
What is the tens digit of the sum \[\left(1!\right)^2+\left(2!\right)^2+\left(3!\right)^2+\ldots+\left(2018!\right)^2?\] [i]2018 CCA Math Bonanza Individual Round #1[/i]

2018 Rio de Janeiro Mathematical Olympiad, 1
Tags: factorial , rio
A natural number is a [i]factorion[/i] if it is the sum of the factorials of each of its decimal digits. For example, $145$ is a factorion because $145 = 1! + 4! + 5!$. Find every 3-digit number which is a factorion.

2014 Singapore Senior Math Olympiad, 16
Tags: factorial
Evaluate the sum $\frac{3!+4!}{2(1!+2!)}+\frac{4!+5!}{3(2!+3!)}+\cdots+\frac{12!+13!}{11(10!+11!)}$

2009 Math Prize For Girls Problems, 18
Tags: factorial
The value of $ 21!$ is $ 51{,}090{,}942{,}171{,}abc{,}440{,}000$, where $ a$, $ b$, and $ c$ are digits. What is the value of $ 100a \plus{} 10b \plus{} c$?

2016 IFYM, Sozopol, 5
Tags: number theory , factorial
Find all pairs of integers $(x,y)$ for which $x^z+z^x=(x+z)!$.

2023 Singapore Senior Math Olympiad, 4
Tags: number theory , factorial
Find all positive integers $m, n$ satisfying $n!+2^{n-1}=2^m$.

2022/2023 Tournament of Towns, P1
Tags: number theory , factorial
Find the maximum integer $m$ such that $m! \cdot 2022!$ is a factorial of an integer.

2001 IMO Shortlist, 1
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 $k$.

PEN A Problems, 47
Tags: floor function , number theory , factorial , prime number , divisibility theory
Let $n$ be a positive integer with $n>1$. Prove that \[\frac{1}{2}+\cdots+\frac{1}{n}\] is not an integer.

2020 Estonia Team Selection Test, 1
Tags: number theory , factorial , least common multiple
For every positive integer $x$, let $k(x)$ denote the number of composite numbers that do not exceed $x$. Find all positive integers $n$ for which $(k (n))! $ lcm $(1, 2,..., n)> (n - 1) !$ .

2023 Bulgarian Spring Mathematical Competition, 12.3
Tags: number theory , factorial
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 $f$.

2002 Germany Team Selection Test, 3
Tags: decimal representation , number theory , factorial , digit
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 $k$.

2013 India Regional Mathematical Olympiad, 6
Tags: factorial , polynomial , root , greatest common divisor , number theory
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 $Q(2013!+1)$.

2006 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: inequalities , factorial , induction , number theory
(a) For each integer $k\ge 3$, find a positive integer $n$ that can be represented as the sum of exactly $k$ mutually distinct positive divisors of $n$. (b) Suppose that $n$ can be expressed as the sum of exactly $k$ mutually distinct positive divisors of $n$ for some $k\ge 3$. Let $p$ be the smallest prime divisor of $n$. Show that \[\frac1p+\frac1{p+1}+\cdots+\frac{1}{p+k-1}\ge1.\]

2019 Ramnicean Hope, 1
Tags: real analysis , factorial , trigonometric limit , limit
Calculate $ \lim_{n\to\infty }\left(\lim_{x\to 0} \left( -\frac{n}{x}+1+\frac{1}{x}\sum_{r=2}^{n+1}\sqrt[r!]{1+\sin rx}\right)\right) . $ [i]Constantin Rusu[/i]

2019 International Zhautykov OIympiad, 1
Tags: combinatorics , factorial
Prove that there exist at least $100!$ ways to write $100!$ as sum of elements of set {$1!,2!,3!...99!$} (each number in sum can be two or more times)

2022 Grosman Mathematical Olympiad, P1
Tags: number theory , factorial
For each positive integer $n$ denote: \[n!=1\cdot 2\cdot 3\dots n\] Find all positive integers $n$ for which $1!+2!+3!+\cdots+n!$ is a perfect square.

1968 AMC 12/AHSME, 21
Tags: factorial , modular arithmetic
If $S=1!+2!+3!+ \cdots +99!$, then the units' digit in the value of $S$ is: $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 0$

2012 NIMO Summer Contest, 3
Tags: factorial
Let \[ S = \sum_{i = 1}^{2012} i!. \] The tens and units digits of $S$ (in decimal notation) are $a$ and $b$, respectively. Compute $10a + b$. [i]Proposed by Lewis Chen[/i]

2022 Bulgarian Autumn Math Competition, Problem 9.3
Tags: number theory , diophantine equation , factorial
Find all the pairs of natural numbers $(a, b),$ such that \[a!+1=(a+1)^{(2^b)}\]

2023 Grand Duchy of Lithuania, 4
Tags: number theory , divide , factorial
Note that $k\ge 1$ for an odd natural number $$k! ! = k \cdot (k - 2) \cdot ... \cdot 1.$$ Prove that $2^n$ divides $(2^n -1)!! -1$ for all $n \ge 3$.

1961 Putnam, B5
Tags: factorial , inequalities
Let $k$ be a positive integer, and $n$ a positive integer greater than $2$. Define $$f_{1}(n)=n,\;\; f_{2}(n)=n^{f_{1}(n)},\;\ldots\;, f_{j+1}(n)=n^{f_{j}(n)}.$$ Prove either part of the inequality $$f_{k}(n) < n!! \cdots ! < f_{k+1}(n),$$ where the middle term has $k$ factorial symbols.

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 $

2024 AMC 10, 5
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 $