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: 88

1992 AMC 12/AHSME, 29
Tags: probability , search , algebra , polynomial , binomial theorem
An "unfair" coin has a $2/3$ probability of turning up heads. If this coin is tossed $50$ times, what is the probability that the total number of heads is even? $ \textbf{(A)}\ 25\left(\frac{2}{3}\right)^{50}\qquad\textbf{(B)}\ \frac{1}{2}\left(1 - \frac{1}{3^{50}}\right)\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{1}{2}\left(1 + \frac{1}{3^{50}}\right)\qquad\textbf{(E)}\ \frac{2}{3} $

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 \[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\]

2003 AMC 8, 4
Tags: algebra , system of equations , binomial theorem
A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted $7$ children and $19$ wheels. How many tricycles were there? $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

2014 Bundeswettbewerb Mathematik, 2
Tags: binomial theorem , algebra
For all positive integers $m$ and $k$ with $m\ge k$, define $a_{m,k}=\binom{m}{k-1}-3^{m-k}$. Determine all sequences of real numbers $\{x_1, x_2, x_3, \ldots\}$, such that each positive integer $n$ satisfies the equation \[a_{n,1}x_1+ a_{n,2}x_2+ \cdots + a_{n,n}x_n = 0\]

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: $\textbf{(A)}\ 4\qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

2014 ELMO Shortlist, 3
Tags: algebra , polynomial , induction , binomial theorem , number theory
Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$. [i]Proposed by Michael Kural[/i]

2002 Balkan MO, 2
Tags: induction , binomial theorem , algebra
Let the sequence $ \{a_n\}_{n\geq 1}$ be defined by $ a_1 \equal{} 20$, $ a_2 \equal{} 30$ and $ a_{n \plus{} 2} \equal{} 3a_{n \plus{} 1} \minus{} a_n$ for all $ n\geq 1$. Find all positive integers $ n$ such that $ 1 \plus{} 5a_n a_{n \plus{} 1}$ is a perfect square.

1969 IMO Longlists, 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 $a_n$

2002 National Olympiad First Round, 11
Tags: algebra , binomial theorem
What is the coefficient of $x^5$ in the expansion of $(1 + x + x^2)^9$? $ \textbf{a)}\ 1680 \qquad\textbf{b)}\ 882 \qquad\textbf{c)}\ 729 \qquad\textbf{d)}\ 450 \qquad\textbf{e)}\ 246 $

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?

1972 Canada National Olympiad, 7
Tags: quadratic , limit , algebra , binomial theorem
a) Prove that the values of $x$ for which $x=(x^2+1)/198$ lie between $1/198$ and $197.99494949\cdots$. b) Use the result of problem a) to prove that $\sqrt{2}<1.41\overline{421356}$. c) Is it true that $\sqrt{2}<1.41421356$?

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 $P(n+1)$

1975 AMC 12/AHSME, 5
Tags: algebra , polynomial , binomial theorem
The polynomial $ (x\plus{}y)^9$ is expanded in decreasing powers of $ x$. The second and third terms have equal values when evaluated at $ x\equal{}p$ and $ y\equal{}q$, where $ p$ and $ q$ are positive numbers whose sum is one. What is the value of $ p$? $ \textbf{(A)}\ 1/5 \qquad \textbf{(B)}\ 4/5 \qquad \textbf{(C)}\ 1/4 \qquad \textbf{(D)}\ 3/4 \qquad \textbf{(E)}\ 8/9$

1962 AMC 12/AHSME, 12
Tags: algebra , binomial theorem
When $ \left ( 1 \minus{} \frac{1}{a} \right ) ^6$ is expanded the sum of the last three coefficients is: $ \textbf{(A)}\ 22 \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ \minus{}10 \qquad \textbf{(E)}\ \minus{}11$

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 Online Math Open Problems, 30
Tags: modular arithmetic , integration , algebra , binomial theorem
Let $p = 2^{16}+1$ be an odd prime. Define $H_n = 1+ \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$. Compute the remainder when \[ (p-1)! \sum_{n = 1}^{p-1} H_n \cdot 4^n \cdot \binom{2p-2n}{p-n} \] is divided by $p$. [i]Proposed by Yang Liu[/i]

2003 Iran MO (3rd Round), 17
Tags: binomial theorem , algebra
A simple calculator is given to you. (It contains 8 digits and only does the operations +,-,*,/,$ \sqrt{\mbox{}}$) How can you find $ 3^{\sqrt{2}}$ with accuracy of 6 digits.

1989 Canada National Olympiad, 3
Tags: modular arithmetic , algebra , binomial theorem , divisibility tests , logarithm , number theory
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$?

2008 Alexandru Myller, 2
Tags: complex number , trigonometry , binomial theorem , binomial coefficient , number theory
There are no integers $ a,b,c $ that satisfy $ \left( a+b\sqrt{-3}\right)^{17}=c+\sqrt{-3} . $ [i]Dorin Andrica, Mihai Piticari[/i]

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

2012 Finnish National High School Mathematics Competition, 3
Tags: algebra , binomial theorem , number theory
Prove that for all integers $k\geq 2,$ the number $k^{k-1}-1$ is divisible by $(k-1)^2.$

1988 IMO Shortlist, 25
Tags: algebra , binomial theorem , number theory , perfect square
A positive integer is called a [b]double number[/b] if its decimal representation consists of a block of digits, not commencing with 0, followed immediately by an identical block. So, for instance, 360360 is a double number, but 36036 is not. Show that there are infinitely many double numbers which are perfect squares.

1990 India Regional Mathematical Olympiad, 4
Tags: modular arithmetic , algebra , binomial theorem , number theory
Find the remainder when $2^{1990}$ is divided by $1990.$

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

2002 USAMTS Problems, 4
Tags: algebra , binomial theorem
Let $f(n)$ be the number of ones that occur in the decimal representations of all the numbers from 1 to $n$. For example, this gives $f(8)=1$, $f(9)=1$, $f(10)=2$, $f(11)=4$, and $f(12)=5$. Determine the value of $f(10^{100})$.