Find the coefficient of the fourth term of the expansion of $(x+y)^{15}$.

For integers $k$ and $n$ such that $1\le k<n$, let $C^n_k=\frac{n!}{k!(n-k)!}$. Then $\left(\frac{n-2k-1}{k+1}\right)C^n_k$ is an integer $\textbf{(A) }\text{for all }k\text{ and }n\qquad$ $\textbf{(B) }\text{for all even values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad$ $\textbf{(C) }\text{for all odd values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad$ $\textbf{(D) }\text{if }k=1\text{ or }n-1,\text{ but not for all odd values }k\text{ and }n\qquad$ $\textbf{(E) }\text{if }n\text{ is divisible by }k,\text{ but not for all even values }k\text{ and }n$

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$. [i]Victor Wang.[/i]

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 $k$.

Let $a_n = \frac{1}{\binom{2n}{n}}, \forall n \leq 1$. a) Show that $\sum\limits_{n=1}^{+\infty}a_nx^n$ converges for all $x \in (-4, 4)$ and that the function $f(x) = \sum\limits_{n=1}^{+\infty}a_nx^n$ satisfies the differential equation $x(x - 4)f'(x) + (x + 2)f(x) = -x$. b) Prove that $\sum\limits_{n=1}^{+\infty}\frac{1}{\binom{2n}{n}} = \frac{1}{3} + \frac{2\pi\sqrt{3}}{27}$.

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.

Let all positive integers $n$ satisfy the following condition: for each non-negative integers $k, m$ with $k + m \le n$, the numbers $\binom{n-k}{m}$ and $\binom{n-m}{k}$ leave the same remainder when divided by $2$. (Poland) PS. The translation was done using Google translate and in case it is not right, there is the original text in Slovak

Let be four positive integers $m, n, p, q$, with $p < m$ given and $q < n$. Take four points $A(0; 0), B(p; 0), C (m; q)$ and $D(m; n)$ in the coordinate plane. Consider the paths $f$ from $A$ to $D$ and the paths $g$ from $B$ to $C$ such that when going along $f$ or $g$, one goes only in the positive directions of coordinates and one can only change directions (from the positive direction of one axe coordinate into the the positive direction of the other axe coordinate) at the points with integral coordinates. Let $S$ be the number of couples $(f, g)$ such that $f$ and $g$ have no common points. Prove that \[S = \binom{n}{m+n} \cdot \binom{q}{m+q-p} - \binom{q}{m+q} \cdot \binom{n}{m+n-p}.\]

In town $ A,$ there are $ n$ girls and $ n$ boys, and each girl knows each boy. In town $ B,$ there are $ n$ girls $ g_1, g_2, \ldots, g_n$ and $ 2n \minus{} 1$ boys $ b_1, b_2, \ldots, b_{2n\minus{}1}.$ The girl $ g_i,$ $ i \equal{} 1, 2, \ldots, n,$ knows the boys $ b_1, b_2, \ldots, b_{2i\minus{}1},$ and no others. For all $ r \equal{} 1, 2, \ldots, n,$ denote by $ A(r),B(r)$ the number of different ways in which $ r$ girls from town $ A,$ respectively town $ B,$ can dance with $ r$ boys from their own town, forming $ r$ pairs, each girl with a boy she knows. Prove that $ A(r) \equal{} B(r)$ for each $ r \equal{} 1, 2, \ldots, n.$

Square $ABCD$ is divided into $n^2$ equal small squares by lines parallel to its sides.A spider starts from $A$ and moving only rightward or upwards,tries to reach $C$.Every "movement" of the spider consists of $k$ steps rightward and $m$ steps upwards or $m$ steps rightward and $k$ steps upwards(it can follow any possible order for the steps of each "movement").The spider completes $l$ "movements" and afterwards it moves without limitation (it still moves rightwards and upwards only).If $n=m\cdot l$,find the number of the possible paths the spider can follow to reach $C$.Note that $n,m,k,l\in \mathbb{N^{*}}$ with $k<m$.

Prove that for any natural numbers $n,r$ with $r + 3 \le n $the binomial coefficients $n \choose r$, $n \choose r+1$, $n \choose r+2 $, $n \choose r+3 $ cannot be successive terms of an arithmetic progression.

Let $\mathbb{Z}^+$ denote the set of all positive integers. Find all surjective functions $f:\mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ that satisfy all of the following conditions: for all $a,b,c \in \mathbb{Z}^+$, (i)$f(a,b) \leq a+b$; (ii)$f(a,f(b,c))=f(f(a,b),c)$ (iii)Both $\binom{f(a,b)}{a}$ and $\binom{f(a,b)}{b}$ are odd numbers.(where $\binom{n}{k}$ denotes the binomial coefficients)

Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \] cannot be divided by $5$.

For each positive integer $n$, set $x_n=\binom{2n}{n}$. a. Prove that if $\frac{2017^k}{2}<n<2017^k$ for some positive integer $k$ then $2017$ divides $x_n$. b. Find all positive integer $h>1$ such that there exists positive integers $N,T$ such that $(x_n)_{n>N}$ is periodic mod $h$ with period $T$.

Prove that for any prime $p\ge5$, the number $$\sum_{0<k<\frac{2p}3}\binom pk$$is divisible by $p^2$.

Let $n \geq 2$ be a given integer $a)$ Prove that one can arrange all the subsets of the set $\{1,2... ,n\}$ as a sequence of subsets $A_{1}, A_{2},\cdots , A_{2^{n}}$, such that $|A_{i+1}| = |A_{i}| + 1$ or $|A_{i}| - 1$ where $i = 1,2,3,\cdots , 2^{n}$ and $A_{2^{n} + 1} = A_{1}$ $b)$ Determine all possible values of the sum $\sum \limits_{i = 1}^{2^n} (-1)^{i}S(A_{i})$ where $S(A_{i})$ denotes the sum of all elements in $A_{i}$ and $S(\emptyset) = 0$, for any subset sequence $A_{1},A_{2},\cdots ,A_{2^n}$ satisfying the condition in $a)$

Ted flips seven fair coins. there are relatively prime positive integers $m$ and $n$ so that $\frac{m}{n}$ is the probability that Ted flips at least two heads given that he flips at least three tails. Find $m+n$.

Suppose that $m=nq$, where $n$ and $q$ are positive integers. Prove that the sum of binomial coefficients \[\sum_{k=0}^{n-1}{ \gcd(n, k)q \choose \gcd(n, k)}\] is divisible by $m$.

Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

Prove that the number of odd binomial coefficients in any finite binomial expansion is a power of $2.$

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

Evaluate $$ \sum_{k=0}^{n} \binom{n}{k}k^{2}.$$

For any polynomial $P(x)=a_0+a_1x+\ldots+a_kx^k$ with integer coefficients, the number of odd coefficients is denoted by $o(P)$. For $i-0,1,2,\ldots$ let $Q_i(x)=(1+x)^i$. Prove that if $i_1,i_2,\ldots,i_n$ are integers satisfying $0\le i_1<i_2<\ldots<i_n$, then: \[ o(Q_{i_{1}}+Q_{i_{2}}+\ldots+Q_{i_{n}})\ge o(Q_{i_{1}}). \]

There is a rectangular plot of size $1 \times n$. This has to be covered by three types of tiles - red, blue and black. The red tiles are of size $1 \times 1$, the blue tiles are of size $1 \times 1$ and the black tiles are of size $1 \times 2$. Let $t_n$ denote the number of ways this can be done. For example, clearly $t_1 = 2$ because we can have either a red or a blue tile. Also $t_2 = 5$ since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile. [list=a] [*]Prove that $t_{2n+1} = t_n(t_{n-1} + t_{n+1})$ for all $n > 1$. [*]Prove that $t_n = \sum_{d \ge 0} \binom{n-d}{d}2^{n-2d}$ for all $n >0$. [/list] Here, \[ \binom{m}{r} = \begin{cases} \dfrac{m!}{r!(m-r)!}, &\text{ if $0 \le r \le m$,} \\ 0, &\text{ otherwise} \end{cases}\] for integers $m,r$.