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

1983 AIME Problems, 13
Tags: probability , algebra , binomial theorem
For $\{1, 2, 3, \dots, n\}$ and each of its nonempty subsets a unique [b]alternating sum[/b] is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. (For example, the alternating sum for $\{1, 2, 4, 6, 9\}$ is $9 - 6 + 4 - 2 + 1 = 6$ and for $\{5\}$ it is simply 5.) Find the sum of all such alternating sums for $n = 7$.

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 $n$.

2010 AMC 10, 25
Tags: polynomial , number theory , least common multiple , floor function , binomial theorem , algebra
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!$

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 \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

2011 USA Team Selection Test, 5
Tags: induction , modular arithmetic , function , limit , algebra , polynomial , binomial theorem
Let $c_n$ be a sequence which is defined recursively as follows: $c_0 = 1$, $c_{2n+1} = c_n$ for $n \geq 0$, and $c_{2n} = c_n + c_{n-2^e}$ for $n > 0$ where $e$ is the maximal nonnegative integer such that $2^e$ divides $n$. Prove that \[\sum_{i=0}^{2^n-1} c_i = \frac{1}{n+2} {2n+2 \choose n+1}.\]

2010 Stanford Mathematics Tournament, 8
Tags: algebra , polynomial , function , binomial theorem
Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=3^k$ for $0\le k \le n$. Find $P(n+1)$

2001 Manhattan Mathematical Olympiad, 4
Tags: algebra , binomial theorem
How many digits has the number $2^{100}$?

1950 AMC 12/AHSME, 16
Tags: algebra , binomial theorem
The number of terms in the expansion of $ [(a\plus{}3b)^2(a\minus{}3b)^2]^2$ when simplified is: $\textbf{(A)}\ 4\qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

2002 National High School Mathematics League, 8
Tags: binomial theorem , arithmetic sequence , calculus , integration , algebra
Consider the expanded form of $\left(x+\frac{1}{2\sqrt[4]{x}}\right)^n$, put all items in number (from high power to low power). If the coefficients of the first three items are arithmetic sequence, then the number of items with an integral power is________.

1989 Canada National Olympiad, 3
Tags: modular arithmetic , algebra , binomial theorem , number theory , divisibility tests , logarithm
Define $ \{ a_n \}_{n\equal{}1}$ as follows: $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$ is the sum of the digits of $ a_{n\minus{}1}$. What is the value of $ a_5$?

2023 CCA Math Bonanza, T7
Tags: algebra , binomial theorem
The positive integer equal to the expression \[ \sum_{i=0}^{9} \left(i+(-9)^i\right)8^{9-i} \binom{9}{i}\] is divisible by exactly six distinct primes. Find the sum of these six distinct prime factors. [i]Team #7[/i]

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 \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

2015 Switzerland Team Selection Test, 2
Tags: binomial coefficient , inequalities , polynomial , binomial theorem , algebra
Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]

1969 AMC 12/AHSME, 16
Tags: algebra , binomial theorem
When $(a-b)^n$, $n\geq 2$, $ab\neq 0$, is expanded by the binomial theorem, it is found that , when $a=kb$, where $k$ is a positive integer, the sum of the second and third terms is zero. Then $n$ equals: $\textbf{(A) }\tfrac12k(k-1)\qquad \textbf{(B) }\tfrac12k(k+1)\qquad \textbf{(C) }2k-1\qquad \textbf{(D) }2k\qquad \textbf{(E) }2k+1$

2000 AIME Problems, 9
Tags: algebra , system of equations , binomial theorem , logarithm
The system of equations \begin{eqnarray*}\log_{10}(2000xy) - (\log_{10}x)(\log_{10}y) & = & 4 \\ \log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\ \log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\ \end{eqnarray*} has two solutions $ (x_{1},y_{1},z_{1})$ and $ (x_{2},y_{2},z_{2}).$ Find $ y_{1} + y_{2}.$

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 $

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

1986 AIME Problems, 11
Tags: algebra , polynomial , ratio , binomial theorem
The polynomial $1-x+x^2-x^3+\cdots+x^{16}-x^{17}$ may be written in the form $a_0+a_1y+a_2y^2+\cdots +a_{16}y^{16}+a_{17}y^{17}$, where $y=x+1$ and thet $a_i$'s are constants. Find the value of $a_2$.

PEN Q Problems, 6
Tags: binomial coefficient , polynomial , calculus , derivative , binomial theorem , algebra
Prove that for a prime $p$, $x^{p-1}+x^{p-2}+ \cdots +x+1$ is irreducible in $\mathbb{Q}[x]$.

2000 National High School Mathematics League, 8
Tags: algebra , binomial theorem
Define $a_n$: the coefficient of then item $x$ in $(3-\sqrt{x})^n$, where $n$ is a positive integer. Then $\lim_{n\to\infty}\left(\frac{3^2}{a_2}+\frac{3^3}{a_3}+\cdots+\frac{3^n}{a_n}\right)=$________.

1954 AMC 12/AHSME, 10
Tags: algebra , polynomial , binomial theorem
The sum of the numerical coefficients in the expansion of the binomial $ (a\plus{}b)^8$ is: $ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 7$

1986 Kurschak Competition, 3
Tags: function , modular arithmetic , algebra , binomial theorem , combinatorics
A and B plays the following game: they choose randomly $k$ integers from $\{1,2,\dots,100\}$; if their sum is even, A wins, else B wins. For what values of $k$ does A and B have the same chance of winning?

PEN A Problems, 22
Tags: polynomial , binomial theorem , divisibility theory , algebra
Prove that the number \[\sum_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}\] is not divisible by $5$ for any integer $n\geq 0$.

1985 IMO Longlists, 88
Tags: polynomial , calculus , binomial theorem , algebra
Determine the range of $w(w + x)(w + y)(w + z)$, where $x, y, z$, and $w$ are real numbers such that \[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\]

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 $g(x)$.