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

2007 Harvard-MIT Mathematics Tournament, 8
Tags: calculus , algebra , polynomial , vieta , binomial theorem
Suppose that $\omega$ is a primitive $2007^{\text{th}}$ root of unity. Find $\left(2^{2007}-1\right)\displaystyle\sum_{j=1}^{2006}\dfrac{1}{2-\omega^j}$.

2014 Contests, 1
Tags: inequalities , algebra , polynomial , binomial coefficient , binomial theorem
Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]

2001 National High School Mathematics League, 5
Tags: binomial theorem , algebra
If $(1+x+x^2)^{1000}=a_0+a_1x+a_2x^2+\cdots+a_{2000}x^{2000}$ ($a_0,a_1,\cdots,a_{2000}$ are coefficients), then the value of $a_0+a_3+a_6+\cdots+a_{1998}$ is $\text{(A)}3^{333}\qquad\text{(B)}3^{666}\qquad\text{(C)}3^{999}\qquad\text{(D)}3^{2001}$

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 \[\sum_{i=0}^{2^n-1} c_i = \frac{1}{n+2} {2n+2 \choose n+1}.\]

2012 Iran MO (3rd Round), 3
Tags: induction , pigeonhole principle , geometry , perpendicular bisector , algebra , binomial theorem , combinatorics
Prove that if $n$ is large enough, among any $n$ points of plane we can find $1000$ points such that these $1000$ points have pairwise distinct distances. Can you prove the assertion for $n^{\alpha}$ where $\alpha$ is a positive real number instead of $1000$?

2007 AMC 8, 9
Tags: algebra , binomial theorem
To complete the grid below, each of the digits 1 through 4 must occur once in each row and once in each column. What number will occupy the lower right-hand square? \[ \begin{tabular}{|c|c|c|c|}\hline 1 & & 2 & \\ \hline 2 & 3 & & \\ \hline & &&4\\ \hline & &&\\ \hline\end{tabular} \] $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ \text{cannot be determined}$

2008 Putnam, B4
Tags: algebra , polynomial , modular arithmetic , function , group theory , binomial theorem
Let $ p$ be a prime number. Let $ h(x)$ be a polynomial with integer coefficients such that $ h(0),h(1),\dots, h(p^2\minus{}1)$ are distinct modulo $ p^2.$ Show that $ h(0),h(1),\dots, h(p^3\minus{}1)$ are distinct modulo $ p^3.$

1988 IMO Longlists, 76
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.

2014 USAJMO, 1
Tags: algebra , polynomial , binomial theorem , binomial coefficient , inequalities
Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]

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$

1983 AIME Problems, 13
Tags: probability , binomial theorem , algebra
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 $n = 7$.

2012 National Olympiad First Round, 8
Tags: binomial theorem , algebra
In how many different ways can one select two distinct subsets of the set $\{1,2,3,4,5,6,7\}$, so that one includes the other? $ \textbf{(A)}\ 2059 \qquad \textbf{(B)}\ 2124 \qquad \textbf{(C)}\ 2187 \qquad \textbf{(D)}\ 2315 \qquad \textbf{(E)}\ 2316$

PEN L Problems, 1
Tags: induction , algebra , binomial theorem , linear recurrence
An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{0}=0, \; a_{1}=1, \; a_{n+2}=2a_{n+1}+a_{n}\] Show that $2^{k}$ divides $a_{n}$ if and only if $2^{k}$ divides $n$.

2002 USA Team Selection Test, 2
Tags: algebra , modular arithmetic , binomial theorem , number theory , summation
Let $p>5$ be a prime number. For any integer $x$, define \[{f_p}(x) = \sum_{k=1}^{p-1} \frac{1}{(px+k)^2}\] Prove that for any pair of positive integers $x$, $y$, the numerator of $f_p(x) - f_p(y)$, when written as a fraction in lowest terms, is divisible by $p^3$.

1969 AMC 12/AHSME, 16
Tags: binomial theorem , algebra
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: $\textbf{(A) }\tfrac12k(k-1)\qquad \textbf{(B) }\tfrac12k(k+1)\qquad \textbf{(C) }2k-1\qquad \textbf{(D) }2k\qquad \textbf{(E) }2k+1$

2002 National High School Mathematics League, 8
Tags: algebra , binomial theorem , arithmetic sequence , calculus , integration
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________.

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

2003 China Team Selection Test, 3
Tags: induction , modular arithmetic , binomial theorem , algebra
Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence.

2003 China Team Selection Test, 3
Tags: induction , modular arithmetic , binomial theorem , algebra
Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence.

1990 Spain Mathematical Olympiad, 3
Tags: floor function , algebra , binomial theorem
Prove that $ \lfloor{(4+\sqrt11)^{n}}\rfloor $ is odd for every natural number n.

2014 All-Russian Olympiad, 3
Tags: algorithm , algebra , binomial theorem , combinatorics
In a country, mathematicians chose an $\alpha> 2$ and issued coins in denominations of 1 ruble, as well as $\alpha ^k$ rubles for each positive integer k. $\alpha$ was chosen so that the value of each coins, except the smallest, was irrational. Is it possible that any natural number of rubles can be formed with at most 6 of each denomination of coins?

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 $\lim_{n\to\infty}\left(\frac{3^2}{a_2}+\frac{3^3}{a_3}+\cdots+\frac{3^n}{a_n}\right)=$________.

1983 AIME Problems, 6
Tags: modular arithmetic , function , algebra , binomial theorem , number theory
Let $a_n = 6^n + 8^n$. Determine the remainder on dividing $a_{83}$ by 49.

2005 MOP Homework, 4
Tags: modular arithmetic , algebra , binomial theorem , number theory
Let $p$ be an odd prime. Prove that \[\sum^{p-1}_{k=1} k^{2p-1} \equiv \frac{p(p+1)}{2}\pmod{p^2}.\]

2013 IMO Shortlist, A2
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 \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]