#### Practice (14)

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Find all positive integers $n$ and $k_i$ $(1\le i \le n)$ such that $$k_1 + k_2 + \cdots + k_n = 5n-4$$ and $$\frac{1}{k_1} + \frac{1}{k_2} + \cdots + \frac{1}{k_n}=1$$

Solve in positive integers the equation $$3(xy+yz+zx)=4xyz$$

Solve in nonnegative integers the equation $$2^x -1 = xy$$

Let $a_1, a_2, \cdots, a_{2n+1}$ be a set of integers such that, if any one of them is removed, the remaining ones can be divided into two sets of $n$ integers with equal sums. Prove $a_1 = a_2 = \cdots = a_{2n+1}$.

Compute the value of $$\displaystyle\sum_{k=1}^n k^2\binom{n}{k}$$

Let $n$ be a positive integer. Show that the smallest integer that is larger than $(1+\sqrt{3})^{2n}$ is divisible by $2^{n+1}$.

There are $6$ points in the $3$-D space. No three points are on the same line and no four points are one the same plane. Hence totally $15$ segments can be created among these points. Show that if each of these $15$ segments is colored either black or white, there must exist a triangle whose sides are of same color.

Alan and Barbara play a game in which they take turns filling entries of an initially empty $1024$ by $1024$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all the entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?

$\textbf{Heaps of Beans}$

A game starts with four heaps of beans, containing $3$, $4$, $5$ and $6$ beans, respectively. The two players move alternately. A move consists of taking either one bean from a heap, provided at least two beans are left behind in that heap, or a complete heap of two or three beans. The player who takes the last bean wins. Does the first or second player have a winning strategy?

Show that if $n$ is an integer greater than $1$, then $(2^n-1)$ is not divisible by $n$.

$$|\sin x + \cos x + \tan x + \cot x + \sec x + \csc x|$$ where $x$ is a real number.

Let $p$ be an odd prime. Show that $$\sum_{j=0}^p\binom{p}{j}\binom{p+j}{j}\equiv 2^p +1 \pmod{p^2}$$

Let $\{ a_1, a_2, \cdots, a_{2n+1}\}$ be a set of integers such that after removing any element, the remaining ones can always be equally divided into two groups with equal sum. Show that all these $a_i$, $(1 \le i \le 2n+1)$ are equal.

Let $\lfloor{x}\rfloor$ be the largest integer not exceeding real number $x$. Show that $$\sum_{k=0}^{\lfloor{\frac{n-1}{2}}\rfloor}\left(\left(1-\frac{2k}{n}\right)\binom{n}{k}\right)^2=\frac{1}{n}\binom{2n-2}{n-1}$$

Let $m$ and $n$ be positive integers. Show that $$\frac{(m+n)!}{(m+n)^{m+n}}<\frac{m!}{m^m}\frac{n!}{n^n}$$

Let $\alpha(n)$ be the number of ways to write a positive integer $n$ as the sum of $1$s and $2$s. Let $\beta(n)$ be the number of ways to write $n$ as a sum of several integers greater than $1$. Different orders are treated as different. Prove $\alpha(n)=\beta(n+2)$.

For what pairs $(a, b)$ of positive real numbers does the the following improper integral converge?

$$\int_b^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\right)dx$$