Olympiad problems

2000 AIME Problems, 7

Given that \[ \frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!} \] find the greatest integer that is less than $\frac N{100}.$

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 $k$.

2022 Singapore MO Open, Q3

Tags: functional equation , number theory , factorial , function

Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$. [i]Proposed by DVDthe1st[/i]

2022 JHMT HS, 2

Tags: factorial , number theory

Find the number of ordered pairs of positive integers $(m,n)$, where $m,n\leq 10$, such that $m!+n!$ is a multiple of $10$.

1934 Eotvos Mathematical Competition, 1

Tags: number theory , integer , factorial

Let $n$ be a given positive integer and $$A =\frac{1 \cdot 3 \cdot 5 \cdot ... \cdot (2n- 1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot 2n}$$ Prove that at least one term of the sequence $A, 2A,4A,8A,...,2^kA, ... $ is an integer.

2000 Belarusian National Olympiad, 2

Tags: number theory , factorial

Find the number of pairs $(n, q)$, where $n$ is a positive integer and $q$ a non-integer rational number with $0 < q < 2000$, that satisfy $\{q^2\}=\left\{\frac{n!}{2000}\right\}$

1969 IMO Longlists, 64

Tags: inequality , factorial , algebra

$(USS 1)$ Prove that for a natural number $n > 2, (n!)! > n[(n - 1)!]^{n!}.$

2009 AMC 12/AHSME, 12

Tags: factorial , ratio , geometric sequence

The fifth and eighth terms of a geometric sequence of real numbers are $ 7!$ and $ 8!$ respectively. What is the first term? $ \textbf{(A)}\ 60\qquad \textbf{(B)}\ 75\qquad \textbf{(C)}\ 120\qquad \textbf{(D)}\ 225\qquad \textbf{(E)}\ 315$

1988 Iran MO (2nd round), 1

Tags: polynomial , induction , factorial , algebra

[b](a)[/b] Prove that for all positive integers $m,n$ we have \[\sum_{k=1}^n k(k+1)(k+2)\cdots (k+m-1)=\frac{n(n+1)(n+2) \cdots (n+m)}{m+1}\] [b](b)[/b] Let $P(x)$ be a polynomial with rational coefficients and degree $m.$ If $n$ tends to infinity, then prove that \[\frac{\sum_{k=1}^n P(k)}{n^{m+1}}\] Has a limit.

2018 Romania Team Selection Tests, 4

Tags: pell equation , sophie germain identity , factorial , number theory

Given an non-negative integer $k$, show that there are infinitely many positive integers $n$ such that the product of any $n$ consecutive integers is divisible by $(n+k)^2+1$.

2006 AMC 10, 11

Tags: factorial

What is the tens digit in the sum $ 7! \plus{} 8! \plus{} 9! \plus{} \cdots \plus{} 2006!$? $ \textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 9$

2017 Saudi Arabia BMO TST, 1

Tags: number theory , divide , divisible , factorial

Prove that there are infinitely many positive integer $n$ such that $n!$ is divisible by $n^3 -1$.

2023 Bangladesh Mathematical Olympiad, P1

Tags: factorial , number theory , diophantine equation

Find all possible non-negative integer solution ($x,$ $y$) of the following equation- $$x!+2^y=z!$$ Note: $x!=x\cdot(x-1)!$ and $0!=1$. For example, $5!=5\times4\times3\times2\times1=120$.

2024 Bangladesh Mathematical Olympiad, P9

Tags: number theory , factorial , divisibility , p-adic

Find all pairs of positive integers $(k, m)$ such that for any positive integer $n$, the product\[(n+m)(n+2m)\cdots(n+km)\]is divisible by $k!$.

2017 CMIMC Number Theory, 10

Tags: number theory , greatest common divisor , factorial

For each positive integer $n$, define \[g(n) = \gcd\left\{0! n!, 1! (n-1)!, 2 (n-2)!, \ldots, k!(n-k)!, \ldots, n! 0!\right\}.\] Find the sum of all $n \leq 25$ for which $g(n) = g(n+1)$.

2011 Saudi Arabia IMO TST, 1

Tags: number theory , remainder , factorial

Find all integers $n$, $n \ge 2$, such that the numbers $1!, 2 !,..., (n - 1)!$ give distinct remainders when divided by $n$.

1986 AMC 12/AHSME, 10

Tags: algorithm , factorial

The 120 permutations of the AHSME are arranged in dictionary order as if each were an ordinary five-letter word. The last letter of the 85th word in this list is: $ \textbf{(A)}\ \text{A} \qquad \textbf{(B)}\ \text{H} \qquad \textbf{(C)}\ \text{S} \qquad \textbf{(D)}\ \text{M} \qquad \textbf{(E)}\ \text{E} $

2008 Harvard-MIT Mathematics Tournament, 13

Tags: algebra , polynomial , factorial

Let $ P(x)$ be a polynomial with degree 2008 and leading coefficient 1 such that \[ P(0) \equal{} 2007, P(1) \equal{} 2006, P(2) \equal{} 2005, \dots, P(2007) \equal{} 0. \]Determine the value of $ P(2008)$. You may use factorials in your answer.

2019 AMC 12/AHSME, 24

Tags: factorial

For how many integers $n$ between $1$ and $50$, inclusive, is \[ \frac{(n^2-1)!}{(n!)^n} \]an integer? (Recall that $0! = 1$.) $\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$

2019 Taiwan TST Round 1, 2

Tags: factorial , number theory

Given a positive integer $ n $, let $ A, B $ be two co-prime positive integers such that $$ \frac{B}{A} = \left(\frac{n\left(n+1\right)}{2}\right)!\cdot\prod\limits_{k=1}^{n}{\frac{k!}{\left(2k\right)!}} $$ Prove that $ A $ is a power of $ 2 $.

2006 Germany Team Selection Test, 2

Tags: factorial , modular arithmetic , number theory , divisibility , exponential , chinese remainder theorem

Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]

2007 ITest, 6

Tags: factorial

Find the units digit of the sum \[(1!)^2+(2!)^2+(3!)^2+(4!)^2+\cdots+(2007!)^2.\] $\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }3$ $\textbf{(D) }5\hspace{14em}\textbf{(E) }7\hspace{14em}\textbf{(F) }9$

2012 Math Prize for Girls Olympiad, 4

Tags: function , pigeonhole principle , factorial

Let $f$ be a function from the set of rational numbers to the set of real numbers. Suppose that for all rational numbers $r$ and $s$, the expression $f(r + s) - f(r) - f(s)$ is an integer. Prove that there is a positive integer $q$ and an integer $p$ such that \[ \Bigl\lvert f\Bigl(\frac{1}{q}\Bigr) - p \Bigr\rvert \le \frac{1}{2012} \, . \]

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: $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 0$

2019 LIMIT Category A, Problem 7

Tags: factorial , number theory

The digit in unit place of $1!+2!+\ldots+99!$ is $\textbf{(A)}~3$ $\textbf{(B)}~0$ $\textbf{(C)}~1$ $\textbf{(D)}~7$