Olympiad problems

2008 Putnam, B4

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.$

1986 Kurschak Competition, 3

Tags: combinatorics , function , modular arithmetic , algebra , binomial theorem

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?

1969 IMO Shortlist, 61

Tags: algebra , number theory , function , binomial theorem , 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 $a_n$

1983 AIME Problems, 6

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

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

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

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$

PEN A Problems, 22

Tags: polynomial , algebra , 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 $n\geq 0$.

2010 AMC 10, 25

Tags: number theory , floor function , algebra , polynomial , least common multiple , 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!$

2001 USA Team Selection Test, 2

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

Express \[ \sum_{k=0}^n (-1)^k (n-k)!(n+k)! \] in closed form.

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 $n$.

2014 Contests, 1

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

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

2002 AIME Problems, 7

Tags: algebra , binomial theorem , modular arithmetic

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}?$

PEN Q Problems, 6

Tags: polynomial , calculus , derivative , binomial theorem , binomial coefficient , algebra

Prove that for a prime $p$, $x^{p-1}+x^{p-2}+ \cdots +x+1$ is irreducible in $\mathbb{Q}[x]$.