Found problems: 119
Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \]
cannot be divided by $5$.
Let $\{a_n\}^\infty_{n=0}$ be the sequence of integers such that $a_0=1$, $a_1=1$, $a_{n+2}=2a_{n+1}-2a_n$. Decide whether
$$a_n=\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}\binom n{2k}(-1)^k.$$
Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)
Prove that
$$\sum_{k=0}^\infty x^k\frac{1+x^{2k+2}}{(1-x^{2k+2})^2}=\sum_{k=0}^\infty(-1)^k\frac{x^k}{(1-x^{k+1})^2}$$for all $x\in(-1,1)$.
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?
Prove that the following inequality holds with the exception of finitely many positive integers $n$:
\[\sum_{i=1}^n\sum_{j=1}^n gcd(i,j)>4n^2.\]
Evaluate $\sum_{k=0}^{37}(-1)^k\binom{75}{2k}$.
Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.
Suppose that $a_1 = 2$ and the sequence $(a_n)$ satisfies the recurrence relation \[\frac{a_n -1}{n-1}=\frac{a_{n-1}+1}{(n-1)+1}\] for all $n \ge 2.$ What is the greatest integer less than or equal to \[\sum^{100}_{n=1} a_n^2?\]
$\textbf{(A) } 338{,}550 \qquad \textbf{(B) } 338{,}551 \qquad \textbf{(C) } 338{,}552 \qquad \textbf{(D) } 338{,}553 \qquad \textbf{(E) } 338{,}554$
Evaluate the sum
$$\sum_{n=0}^\infty\operatorname{arctan}\left(\frac1{1+n+n^2}\right).$$
Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \]
[hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]
Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$
Prove that
$$\sum_{n=1}^\infty\frac{n^2}{(7n)!}=\frac1{7^3}\sum_{k=1}^2\sum_{j=0}^6e^{\cos(2\pi j/7)}\cdot\cos\left(\frac{2k\pi j}7+\sin\frac{2\pi j}7\right).$$
We have a deck of $90$ cards that are numbered from $10$ to $99$ (all two-digit numbers). How many sets of three or more different cards in this deck are there such that the number on one of them is the sum of the other numbers, and those other numbers are consecutive?
Let $\sum_{n=1}^\infty a_n$ be a convergent series of positive terms (so $a_i>0$ for all $i$) and set $b_n=\frac1{na_n^2}$ for $n\ge1$. Prove that $\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n}$ is convergent.
There are $\tfrac{n(n+1)}{2}$ distinct sums of two distinct numbers, if there are $n$ numbers.
For which $n \ (n \geq 3)$ do there exist $n$ distinct integers, such that those sums are $\tfrac{n(n-1)}{2}$ consecutive numbers?
Prove that for any prime $p\ge5$, the number
$$\sum_{0<k<\frac{2p}3}\binom pk$$is divisible by $p^2$.
$n(\geq 2)$ is a given integer and $T$ is set of all $n\times n$ matrices whose entries are elements of the set $S=\{1,\cdots,2017\}$. Evaluate the following value.
\[\sum_{A\in T} \text{det}(A)\]
Let $x_1, x_2,\geq , x_n$ be a positive numbers, $k \geq 1$. Then the following inequality is true: $$\left(x_1^k+x_2^k+\cdots +x_n^k\right)^{k+1}\geq \left(x_1^{k+1}+x_2^{k+1}\cdots +x_n^{k+1}\right)^k+2\left(\sum \limits_{1\leq i<j\leq n}x_i^kx_j\right)^k$$
Prove that $$\int \limits_0^{\infty} e^{3t}\frac{4e^{4t}(3t - 1) + 2e^{2t}(15t - 17) + 18(1 - t)}{\left(1 + e^{4t} - e^{2t}\right)^2}=12\sum \limits_{k=0}^{\infty}\frac{(-1)^k}{(2k + 1)^2}-10$$
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?
The value of
$$\left(1+\frac26+\frac{2\cdot5}{6\cdot12}+\frac{2\cdot5\cdot8}{6\cdot12\cdot18}+\ldots\right)^3$$
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?
Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)
Let $\{a_n\}$ be a monotonically decreasing sequence of positive real numbers with limit $0$. Let $\{b_n\}$ be a rearrangement of the sequence such that for every non-negative integer $m$, the terms $b_{3m+1}$, $b_{3m+2}$, $b_{3m+3}$ are a rearrangement of the terms $a_{3m+1}$, $a_{3m+2}$, $a_{3m+3}$. Prove or give a counterexample to the following statement: the series $\sum_{n=1}^\infty(-1)^nb_n$ is convergent.