Olympiad problems

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 $1000$?

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.

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 \[a_{n,1}x_1+ a_{n,2}x_2+ \cdots + a_{n,n}x_n = 0\]

PEN L Problems, 1

Tags: linear recurrence , induction , algebra , binomial theorem

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 $n$.

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 Putnam, 6

Tags: linear algebra , matrix , algebra , polynomial , modular arithmetic , binomial theorem

Let $p$ be a prime number. Prove that the determinant of the matrix \[ \begin{bmatrix}x & y & z\\ x^p & y^p & z^p \\ x^{p^2} & y^{p^2} & z^{p^2} \end{bmatrix} \] is congruent modulo $p$ to a product of polynomials of the form $ax+by+cz$, where $a$, $b$, and $c$ are integers. (We say two integer polynomials are congruent modulo $p$ if corresponding coefficients are congruent modulo $p$.)

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$

1983 AIME Problems, 6

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

Let $a_n = 6^n + 8^n$. Determine the remainder on dividing $a_{83}$ by 49.

1985 IMO Longlists, 54

Tags: number theory , greatest common divisor , induction , algorithm , modular arithmetic , algebra , binomial theorem

Set $S_n = \sum_{p=1}^n (p^5+p^7)$. Determine the greatest common divisor of $S_n$ and $S_{3n}.$

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 $\sqrt{2}<1.41421356$?

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? $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

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].

2005 MOP Homework, 5

Tags: function , integration , calculus , binomial theorem , algebra

Show that for nonnegative integers $m$ and $n$, $\frac{\dbinom{m}{0}}{n+1}-\frac{\dbinom{m}{1}}{n+2}+...+(-1)^m\frac{\dbinom{m}{m}}{n+m+1}$ $=\frac{\dbinom{n}{0}}{m+1}-\frac{\dbinom{n}{1}}{m+2}+...+(-1)^n\frac{\dbinom{n}{n}}{m+n+1}$.

2014 Taiwan TST Round 2, 4

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

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 $

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 $p^3$.

1987 AIME Problems, 12

Tags: inequalities , algebra , binomial theorem

Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/1000$. Find $n$.

2001 National High School Mathematics League, 5

Tags: algebra , binomial theorem

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 $\text{(A)}3^{333}\qquad\text{(B)}3^{666}\qquad\text{(C)}3^{999}\qquad\text{(D)}3^{2001}$

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 $f(10^{100})$.

1990 India Regional Mathematical Olympiad, 4

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

Find the remainder when $2^{1990}$ is divided by $1990.$

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.

2014 USAJMO, 1

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

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

2009 AMC 12/AHSME, 5

Tags: algebra , binomial theorem

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$

1999 Romania Team Selection Test, 7

Tags: calculus , integration , arithmetic sequence , geometric sequence , geometric series , algebra , binomial theorem

Prove that for any integer $n$, $n\geq 3$, there exist $n$ positive integers $a_1,a_2,\ldots,a_n$ in arithmetic progression, and $n$ positive integers in geometric progression $b_1,b_2,\ldots,b_n$ such that \[ b_1 < a_1 < b_2 < a_2 <\cdots < b_n < a_n . \] Give an example of two such progressions having at least five terms. [i]Mihai Baluna[/i]

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 $a_n$