Olympiad problems

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]

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} $

2002 AIME Problems, 7

Tags: modular arithmetic , algebra , binomial theorem

The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $ x, y,$ and $ r$ with $ |x| > |y|,$ \[ (x \plus{} y)^r \equal{} x^r \plus{} rx^{r \minus{} 1}y \plus{} \frac {r(r \minus{} 1)}2x^{r \minus{} 2}y^2 \plus{} \frac {r(r \minus{} 1)(r \minus{} 2)}{3!}x^{r \minus{} 3}y^3 \plus{} \cdots \] What are the first three digits to the right of the decimal point in the decimal representation of $ \left(10^{2002} \plus{} 1\right)^{10/7}?$

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]

2008 Alexandru Myller, 2

Tags: binomial coefficient , complex number , number theory , trigonometry , binomial theorem

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]

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.$

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 $(k-1)^2.$

2009 AMC 10, 6

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$

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 $\left(2^{2007}-1\right)\displaystyle\sum_{j=1}^{2006}\dfrac{1}{2-\omega^j}$.

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 , binomial theorem , combinatorics , algebra

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?

1971 AMC 12/AHSME, 13

Tags: algebra , binomial theorem

If $(1.0025)^{10}$ is evaluated correct to $5$ decimal places, then the digit in the fifth decimal place is $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }5\qquad \textbf{(E) }8$

1977 AMC 12/AHSME, 10

Tags: pascal triangle , algebra , binomial theorem

If $(3x-1)^7 = a_7x^7 + a_6x^6 + \cdots + a_0$, then $a_7 + a_6 + \cdots + a_0$ equals \[ \text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 64 \qquad \text{(D)}\ -64 \qquad \text{(E)}\ 128 \]