Found problems: 166
1967 IMO Longlists, 17
Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that
\[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\]
is divisible by the product $c_1c_2\ldots c_n$.
1988 IMO Longlists, 3
Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial
\[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n.
\]
2012 Silk Road, 3
Let $n > 1$ be an integer.
Determine the greatest common divisor of the set of numbers $\left\{ \left( \begin{matrix}
2n \\
2i+1 \\
\end{matrix} \right):0 \le i \le n-1 \right\}$
i.e. the largest positive integer, dividing $\left( \begin{matrix}
2n \\
2i+1 \\
\end{matrix} \right)$ without remainder for every $i = 0, 1, ..., n–1$ .
(Here $\left( \begin{matrix}
m \\
l \\
\end{matrix} \right)=\text{C}_{m}^{l}=\frac{m\text{!}}{l\text{!}\left( m-l \right)\text{!}}$ is binomial coefficient.)
PEN S Problems, 4
If $x$ is a real number such that $x^2 -x$ is an integer, and for some $n \ge 3$, $x^n -x$ is also an integer, prove that $x$ is an integer.
2013 Bundeswettbewerb Mathematik, 4
Consider the Pascal's triangle in the figure where the binomial coefficients are arranged in the usual manner. Select any binomial coefficient from anywhere except the right edge of the triangle and labet it $C$. To the right of $C$, in the horizontal line, there are $t$ numbers, we denote them as $a_1,a_2,\cdots,a_t$, where $a_t = 1$ is the last number of the series. Consider the line parallel to the left edge of the triangle containing $C$, there will only be $t$ numbers diagonally above $C$ in that line. We successively name them as $b_1,b_2,\cdots,b_t$, where $b_t = 1$. Show that
\[b_ta_1-b_{t-1}a_2+b_{t-2}a_3-\cdots+(-1)^{t-1}b_1a_t = 1\].
For example, Suppose you choose $\binom41 = 4$ (see figure), then $t = 3$, $a_1 = 6, a_2 = 4, a_3 = 1$ and $b_1 = 3, b_2 = 2, b_3 = 1$.
\[\begin{array}{ccccccccccc} & & & & & 1 & & & & & \\
& & & & 1 & & \underset{b_3}{1} & & & & \\
& & & 1 & & \underset{b_2}{2} & & 1 & & & \\
& & 1 & & \underset{b_1}{3} & & 3 & & 1 & & \\
& 1 & & \boxed{4} & & \underset{a_1}{6} & & \underset{a_2}{4} & & \underset{a_3}{1} & \\
\ldots & & \ldots & & \ldots & & \ldots & & \ldots & & \ldots \\
\end{array}\]
2015 Romania Team Selection Tests, 2
Given an integer $k \geq 2$, determine the largest number of divisors the binomial coefficient $\binom{n}{k}$ may have in the range $n-k+1, \ldots, n$ , as $n$ runs through the integers greater than or equal to $k$.
2014 Balkan MO Shortlist, C3
Let $n$ be a positive integer. A regular hexagon with side length $n$ is divided into equilateral triangles with side length $1$ by lines parallel to its sides.
Find the number of regular hexagons all of whose vertices are among the vertices of those equilateral triangles.
[i]UK - Sahl Khan[/i]
1972 Putnam, A1
Show that $\binom{n}{m},\binom{n}{m+1},\binom{n}{m+2}$ and $\binom{n}{m+3}$ cannot be in arithmetic progression, where $n,m>0$ and $n\geq m+3$.
2010 ISI B.Stat Entrance Exam, 8
Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]
PEN H Problems, 57
Show that the equation ${n \choose k}=m^{l}$ has no integral solution with $l \ge 2$ and $4 \le k \le n-4$.
1990 IMO Longlists, 44
Prove that for any positive integer $n$, the number of odd integers among the binomial coefficients $\binom nh \ ( 0 \leq h \leq n)$ is a power of 2.
2004 Romania Team Selection Test, 7
Let $a,b,c$ be 3 integers, $b$ odd, and define the sequence $\{x_n\}_{n\geq 0}$ by $x_0=4$, $x_1=0$, $x_2=2c$, $x_3=3b$ and for all positive integers $n$ we have
\[ x_{n+3} = ax_{n-1}+bx_n + cx_{n+1} . \]
Prove that for all positive integers $m$, and for all primes $p$ the number $x_{p^m}$ is divisible by $p$.
2017 Greece Team Selection Test, 2
Prove that the number $A=\frac{(4n)!}{(2n)!n!}$ is an integer and divisible by $2^{n+1}$,
where $n$ is a positive integer.
PEN E Problems, 16
Prove that for any prime $p$ in the interval $\left]n, \frac{4n}{3}\right]$, $p$ divides \[\sum^{n}_{j=0}{{n}\choose{j}}^{4}.\]
2017 India National Olympiad, 6
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.
1923 Eotvos Mathematical Competition, 2
If $$s_n = 1 + q + q^2 +... + q^n$$ and $$ S_n = 1 +\frac{1 + q}{2}+ \left( \frac{1 + q}{2}\right)^2 +... + \left( \frac{1 + q}{2}\right)^n,$$ prove that $${n + 1 \choose 1}+{n + 1 \choose 2} s_1 + {n + 1 \choose 3} s_2 + ... + {n + 1 \choose n + 1} s_n = 2^nS_n$$