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

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Found problems: 88

2020 AIME Problems, 12
Tags: factorization , number theory , binomial theorem
Let $n$ be the least positive integer for which $149^n - 2^n$ is divisible by $3^3 \cdot 5^5 \cdot 7^7$. Find the number of positive divisors of $n$.

1962 Putnam, B1
Tags: binomial theorem , binomial
Let $x^{(n)}=x(x-1)\cdots (x-n+1)$ for $n$ a positive integer and let $x^{(0)}=1.$ Prove that $$(x+y)^{(n)}= \sum_{k=0}^{n} \binom{n}{k} x^{(k)} y^{(n-k)}.$$

2011 AMC 8, 22
Tags: algebra , binomial theorem
What is the tens digit of $7^{2011}$? $ \textbf{(A)}0\qquad\textbf{(B)}1\qquad\textbf{(C)}3\qquad\textbf{(D)}4\qquad\textbf{(E)}7 $

2001 USA Team Selection Test, 2
Tags: integration , function , calculus , algebra , binomial theorem , combinatorics
Express \[ \sum_{k=0}^n (-1)^k (n-k)!(n+k)! \] in closed form.

2010 Stanford Mathematics Tournament, 10
Tags: algebra , binomial theorem
Compute the base 10 value of $14641_{99}$

1969 IMO Shortlist, 61
Tags: function , algebra , binomial theorem , number theory , power of 2
$(SWE 4)$ Let $a_0, a_1, a_2, \cdots$ be determined with $a_0 = 0, a_{n+1} = 2a_n + 2^n$. Prove that if $n$ is power of $2$, then so is $a_n$

1994 AMC 12/AHSME, 21
Tags: algebra , binomial theorem
Find the number of counter examples to the statement: \[``\text{If N is an odd positive integer the sum of whose digits is 4 and none of whose digits is 0, then N is prime}."\] $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

1993 All-Russian Olympiad, 4
Tags: induction , function , algebra , polynomial , ring theory , vector , binomial theorem
If $ \{a_k\}$ is a sequence of real numbers, call the sequence $ \{a'_k\}$ defined by $ a_k' \equal{} \frac {a_k \plus{} a_{k \plus{} 1}}2$ the [i]average sequence[/i] of $ \{a_k\}$. Consider the sequences $ \{a_k\}$; $ \{a_k'\}$ - [i]average sequence[/i] of $ \{a_k\}$; $ \{a_k''\}$ - average sequence of $ \{a_k'\}$ and so on. If all these sequences consist only of integers, then $ \{a_k\}$ is called [i]Good[/i]. Prove that if $ \{x_k\}$ is a [i]good[/i] sequence, then $ \{x_k^2\}$ is also [i]good[/i].

2010 AMC 10, 25
Tags: algebra , polynomial , number theory , least common multiple , floor function , binomial theorem
Let $ a>0$, and let $ P(x)$ be a polynomial with integer coefficients such that \[ P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and}\] \[ P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.}\] What is the smallest possible value of $ a$? $ \textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$

2017 India National Olympiad, 6
Tags: binomial theorem , geometry , algebra , heron's formula , binomial coefficient , integer
Let $n\ge 1$ be an integer and consider the sum $$x=\sum_{k\ge 0} \dbinom{n}{2k} 2^{n-2k}3^k=\dbinom{n}{0}2^n+\dbinom{n}{2}2^{n-2}\cdot{}3+\dbinom{n}{4}2^{n-k}\cdot{}3^2 + \cdots{}.$$ Show that $2x-1,2x,2x+1$ form the sides of a triangle whose area and inradius are also integers.

2009 AMC 10, 6
Tags: binomial theorem , algebra
Kiana has two older twin brothers. The product of their ages is $ 128$. What is the sum of their three ages? $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 24$

1990 Dutch Mathematical Olympiad, 3
Tags: polynomial , induction , binomial theorem , algebra
A polynomial $ f(x)\equal{}ax^4\plus{}bx^3\plus{}cx^2\plus{}dx$ with $ a,b,c,d>0$ is such that $ f(x)$ is an integer for $ x \in \{ \minus{}2,\minus{}1,0,1,2 \}$ and $ f(1)\equal{}1$ and $ f(5)\equal{}70$. $ (a)$ Show that $ a\equal{}\frac{1}{24}, b\equal{}\frac{1}{4},c\equal{}\frac{11}{24},d\equal{}\frac{1}{4}$. $ (b)$ Prove that $ f(x)$ is an integer for all $ x \in \mathbb{Z}$.

2014 India IMO Training Camp, 1
Tags: binomial theorem , pigeonhole principle , algebra
Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

2014 India IMO Training Camp, 1
Tags: algebra , binomial theorem , pigeonhole principle
Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

2005 MOP Homework, 5
Tags: function , integration , calculus , binomial theorem , algebra
Show that for nonnegative integers $m$ and $n$, $\frac{\dbinom{m}{0}}{n+1}-\frac{\dbinom{m}{1}}{n+2}+...+(-1)^m\frac{\dbinom{m}{m}}{n+m+1}$ $=\frac{\dbinom{n}{0}}{m+1}-\frac{\dbinom{n}{1}}{m+2}+...+(-1)^n\frac{\dbinom{n}{n}}{m+n+1}$.

2006 AMC 12/AHSME, 24
Tags: inequalities , geometry , analytic geometry , algebra , binomial theorem
The expression \[ (x \plus{} y \plus{} z)^{2006} \plus{} (x \minus{} y \minus{} z)^{2006} \]is simplified by expanding it and combining like terms. How many terms are in the simplified expression? $ \textbf{(A) } 6018 \qquad \textbf{(B) } 671,676 \qquad \textbf{(C) } 1,007,514 \qquad \textbf{(D) } 1,008,016 \qquad \textbf{(E) } 2,015,028$

1991 IMO Shortlist, 18
Tags: binomial theorem , number theory , divisibility , algebra
Find the highest degree $ k$ of $ 1991$ for which $ 1991^k$ divides the number \[ 1990^{1991^{1992}} \plus{} 1992^{1991^{1990}}.\]

1977 AMC 12/AHSME, 10
Tags: binomial theorem , pascal triangle , algebra
If $(3x-1)^7 = a_7x^7 + a_6x^6 + \cdots + a_0$, then $a_7 + a_6 + \cdots + a_0$ equals \[ \text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 64 \qquad \text{(D)}\ -64 \qquad \text{(E)}\ 128 \]

2012 India Regional Mathematical Olympiad, 2
Tags: algebra , polynomial , vieta , inequalities , binomial theorem , sum of roots
Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ be a polynomial of degree $n\geq 3.$ Knowing that $a_{n-1}=-\binom{n}{1}$ and $a_{n-2}=\binom{n}{2},$ and that all the roots of $P$ are real, find the remaining coefficients. Note that $\binom{n}{r}=\frac{n!}{(n-r)!r!}.$

2014 Brazil Team Selection Test, 2
Tags: algebra , binomial theorem , pigeonhole principle
Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

2000 Hungary-Israel Binational, 2
Tags: algebra , binomial theorem , number theory
Prove or disprove: For any positive integer $k$ there exists an integer $n > 1$ such that the binomial coeffcient $\binom{n}{i}$ is divisible by $k$ for any $1 \leq i \leq n-1.$

2009 Math Prize For Girls Problems, 10
Tags: modular arithmetic , algebra , binomial theorem , recurrence
When the integer $ {\left(\sqrt{3} \plus{} 5\right)}^{103} \minus{} {\left(\sqrt{3} \minus{} 5\right)}^{103}$ is divided by 9, what is the remainder?

2014 Harvard-MIT Mathematics Tournament, 28
Tags: algebra , polynomial , binomial theorem
Let $f(n)$ and $g(n)$ be polynomials of degree $2014$ such that $f(n)+(-1)^ng(n)=2^n$ for $n=1,2,\ldots,4030$. Find the coefficient of $x^{2014}$ in $g(x)$.

2010 AMC 12/AHSME, 21
Tags: algebra , polynomial , number theory , least common multiple , floor function , binomial theorem
Let $ a>0$, and let $ P(x)$ be a polynomial with integer coefficients such that \[ P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and}\] \[ P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.}\] What is the smallest possible value of $ a$? $ \textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$

2000 Federal Competition For Advanced Students, Part 2, 1
Tags: binomial theorem , algebra
The sequence an is defined by $a_0 = 4, a_1 = 1$ and the recurrence formula $a_{n+1} = a_n + 6a_{n-1}$. The sequence $b_n$ is given by \[b_n=\sum_{k=0}^n \binom nk a_k.\] Find the coefficients $\alpha,\beta$ so that $b_n$ satisfies the recurrence formula $b_{n+1} = \alpha b_n + \beta b_{n-1}$. Find the explicit form of $b_n$.