Olympiad problems

1988 IMO Longlists, 76

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.

2017 India National Olympiad, 6

Tags: binomial coefficient , integer , binomial theorem , geometry , algebra , heron's formula

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.

2006 AMC 12/AHSME, 24

Tags: inequalities , geometry , analytic geometry , binomial theorem , algebra

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$

2002 Balkan MO, 2

Tags: algebra , induction , binomial theorem

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.

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!$

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

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?

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 $$(x+y)^{(n)}= \sum_{k=0}^{n} \binom{n}{k} x^{(k)} y^{(n-k)}.$$

2012 National Olympiad First Round, 8

Tags: algebra , binomial theorem

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$

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.

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.

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.

1962 AMC 12/AHSME, 12

Tags: binomial theorem , algebra

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$

2005 MOP Homework, 4

Tags: modular arithmetic , algebra , binomial theorem , number theory

Let $p$ be an odd prime. Prove that \[\sum^{p-1}_{k=1} k^{2p-1} \equiv \frac{p(p+1)}{2}\pmod{p^2}.\]

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 $b_n$.

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

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 \[ 1990^{1991^{1992}} \plus{} 1992^{1991^{1990}}.\]

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 $1 \leq i \leq n-1.$

2014 Contests, 1

Tags: binomial coefficient , inequalities , algebra , polynomial , binomial theorem

Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]

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 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?

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 $

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]

2007 AMC 8, 9

Tags: binomial theorem , algebra

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? \[ \begin{tabular}{|c|c|c|c|}\hline 1 & & 2 & \\ \hline 2 & 3 & & \\ \hline & &&4\\ \hline & &&\\ \hline\end{tabular} \] $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ \text{cannot be determined}$

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 $\binom{n}{r}=\frac{n!}{(n-r)!r!}.$