Olympiad problems

2005 iTest, 37

How many zeroes appear at the end of $209$ factorial?

1987 Czech and Slovak Olympiad III A, 4

Tags: inequality , prime number , inequalities , factorial , algebra , number theory

Given an integer $n\ge3$ consider positive integers $x_1,\ldots,x_n$ such that $x_1<x_2<\cdots<x_n<2x_1$. If $p$ is a prime and $r$ is a positive integer such that $p^r$ divides the product $x_1\cdots x_n$, prove that $$\frac{x_1\cdots x_n}{p^r}>n!.$$

2007 Putnam, 3

Tags: distinguishability , binomial coefficient , factorial , function , counting , probability

Let $ k$ be a positive integer. Suppose that the integers $ 1,2,3,\dots,3k \plus{} 1$ are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by $ 3$ ? Your answer should be in closed form, but may include factorials.

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]

2000 China Team Selection Test, 2

Tags: algebra , finite differences , induction , polynomial , combinatorics , factorial

Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate \[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]

2024 Olimphíada, 1

Tags: least common multiple , number theory , factorial

Find all pairs of positive integers $(m,n)$ such that $$lcm(1,2,\dots,n)=m!$$ where $lcm(1,2,\dots,n)$ is the smallest positive integer multiple of all $1,2,\dots n-1$ and $n$.

2022 CIIM, 6

Tags: factorial , number theory

Prove that $\tau ((n+1)!) \leq 2 \tau (n!)$ for all positive integers $n$.

2004 Mediterranean Mathematics Olympiad, 1

Tags: prime factorization , factorial , number theory , inequalities

Find all natural numbers $m$ such that \[1! \cdot 3! \cdot 5! \cdots (2m-1)! = \biggl( \frac{m(m+1)}{2}\biggr) !.\]

2000 Tuymaada Olympiad, 4

Tags: factorial , number theory

Prove that no number of the form $10^{-n}$, $n\geq 1,$ can be represented as the sum of reciprocals of factorials of different positive integers.

2005 AMC 10, 22

Tags: factorial

For how many positive integers $ n$ less than or equal to $ 24$ is $ n!$ evenly divisible by $ 1 \plus{} 2 \plus{} \dots \plus{} n$? $ \textbf{(A)}\ 8\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 17\qquad \textbf{(E)}\ 21$

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 $0! = 1$.) $\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$

2000 Harvard-MIT Mathematics Tournament, 26

Tags: factorial

What are the last $3$ digits of $1!+2!+\cdots +100!$

2008 AMC 8, 13

Tags: factorial

Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than $100$ pounds or more than $150$ pounds. So the boxes are weighed in pairs in every possible way. The results are $122$, $125$ and $127$ pounds. What is the combined weight in pounds of the three boxes? $\textbf{(A)}\ 160\qquad \textbf{(B)}\ 170\qquad \textbf{(C)}\ 187\qquad \textbf{(D)}\ 195\qquad \textbf{(E)}\ 354$

2008 AIME Problems, 6

Tags: factorial , induction , ratio

The sequence $ \{a_n\}$ is defined by \[ a_0 \equal{} 1,a_1 \equal{} 1, \text{ and } a_n \equal{} a_{n \minus{} 1} \plus{} \frac {a_{n \minus{} 1}^2}{a_{n \minus{} 2}}\text{ for }n\ge2. \]The sequence $ \{b_n\}$ is defined by \[ b_0 \equal{} 1,b_1 \equal{} 3, \text{ and } b_n \equal{} b_{n \minus{} 1} \plus{} \frac {b_{n \minus{} 1}^2}{b_{n \minus{} 2}}\text{ for }n\ge2. \]Find $ \frac {b_{32}}{a_{32}}$.

1940 Moscow Mathematical Olympiad, 062

Tags: 3-digit , factorial , number theory

Find all $3$-digit numbers $\overline {abc}$ such that $\overline {abc} = a! + b! + c! $.

1983 Austrian-Polish Competition, 8

Tags: sequence , divide , product , factorial , number theory , recurrence relation , divisible

(a) Prove that $(2^{n+1}-1)!$ is divisible by $ \prod_{i=0}^n (2^{n+1-i}-1)^{2^i }$, for every natural number n (b) Define the sequence ($c_n$) by $c_1=1$ and $c_{n}=\frac{4n-6}{n}c_{n-1}$ for $n\ge 2$. Show that each $c_n$ is an integer.

2018 Balkan MO Shortlist, N2

Tags: number theory , factorial

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $$n!+f(m)!|f(n)!+f(m!)$$ for all $m,n\in\mathbb{N}$ [i]Proposed by Valmir Krasniqi and Dorlir Ahmeti, Albania[/i]

2023 Bangladesh Mathematical Olympiad, P1

Tags: diophantine equation , number theory , factorial

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, $5!=5\times4\times3\times2\times1=120$.

2014 AMC 10, 8

Tags: factorial

Which of the following numbers is a perfect square? $ \textbf{(A)}\ \dfrac{14!15!}2\qquad\textbf{(B)}\ \dfrac{15!16!}2\qquad\textbf{(C)}\ \dfrac{16!17!}2\qquad\textbf{(D)}\ \dfrac{17!18!}2\qquad\textbf{(E)}\ \dfrac{18!19!}2 $

2016 Taiwan TST Round 2, 1

Tags: divisibility , factorial , number theory

Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

2020 AMC 8 -, 12

Tags: factorial

For a positive integer $n,$ the factorial notation $n!$ represents the product of the integers from $n$ to $1.$ (For example, $6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1.$) What value of $N$ satisfies the following equation? $$5! \cdot 9! = 12 \cdot N!$$ $\textbf{(A) }10 \qquad \textbf{(B) }11 \qquad \textbf{(C) }12 \qquad \textbf{(D) }13 \qquad \textbf{(E) }14$

1953 Moscow Mathematical Olympiad, 256

Tags: factorial , equation , algebra

Find roots of the equation $$1 -\frac{x}{1}+ \frac{x(x - 1)}{2!} -... +\frac{ (-1)^nx(x-1)...(x - n + 1)}{n!}= 0$$

1982 Tournament Of Towns, (025) 3

Tags: diophantine equation , factorial , diophantine , number theory

Prove that the equation $m!n! = k!$ has infinitely many solutions in which $m, n$ and $k$ are natural numbers greater than unity .

2017 CCA Math Bonanza, I1

Tags: factorial

Find the integer $n$ such that $6!\times7!=n!$. [i]2017 CCA Math Bonanza Individual Round #1[/i]

2001 IMO Shortlist, 2

Tags: global , factorial , combinatorics , modular arithmetic

Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$ be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of $\{1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist permutations $a \neq b$ of $\{1,2,\ldots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.