Found problems: 1782
Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. If $n$ is a positive integer then prove that
\[ \frac{(3a)^n}{(b + 1)(c + 1)} + \frac{(3b)^n}{(c + 1)(a + 1)} + \frac{(3c)^n}{(a + 1)(b + 1)} \ge \frac{27}{16} \,. \]
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$,
\[f(a+d)+f(b-c)=f(a-d)+f(b+c).\]
[i]Calvin Deng.[/i]
A [i]base[/i] 10 [i]over-expansion[/i] of a positive integer $N$ is an expression of the form $N=d_k10^k+d_{k-1}10^{k-1}+\cdots+d_0 10^0$ with $d_k\ne 0$ and $d_i\in\{0,1,2,\dots,10\}$ for all $i.$ For instance, the integer $N=10$ has two base 10 over-expansions: $10=10\cdot 10^0$ and the usual base 10 expansion $10=1\cdot 10^1+0\cdot 10^0.$ Which positive integers have a unique base 10 over-expansion?
Find all positive integers $a$ such that for any positive integer $n\ge 5$ we have $2^n-n^2\mid a^n-n^a$.
Let $n$ be a positive odd integer and let $\theta$ be a real number such that $\theta/\pi$ is irrational. Set $a_{k}=\tan(\theta+k\pi/n),\ k=1,2\dots,n.$ Prove that
\[\frac{a_{1}+a_{2}+\cdots+a_{n}}{a_{1}a_{2}\cdots a_{n}}\]
is an integer, and determine its value.
It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has $k$ weights of integral values. What is the minimum $k$ for which there exist weights that satisfy this condition?
The sequence $ \{a_n\}$ of integers is defined by
\[ a_1 \equal{} 2, a_2 \equal{} 7
\]
and
\[ \minus{} \frac {1}{2} < a_{n \plus{} 1} \minus{} \frac {a^2_n}{a_{n \minus{} 1}} \leq \frac {}{}, n \geq 2.
\]
Prove that $ a_n$ is odd for all $ n > 1.$
Let $T$ a set with 2007 points on the plane, without any 3 collinear points.
Let $P$ any point which belongs to $T$.
Prove that the number of triangles that contains the point $P$ inside and
its vertices are from $T$, is even.
Let $f\colon\mathbb{N}\rightarrow\mathbb{N}^{\ast}$ be a strictly increasing function. Prove that:
[list=a]
[*]There exists a decreasing sequence of positive real numbers, $(y_{n})_{n\in\mathbb{N}}$, converging to $0$, such that $y_{n}\leq2y_{f(n)}$, for all $n\in\mathbb{N}$.
[*]If $(x_{n})_{n\in\mathbb{N}}$ is a decreasing sequence of real numbers, converging to $0$, then there exists a decreasing sequence of real numbers $(y_{n})_{n\in\mathbb{N}}$, converging to $0$, such that $x_{n}\leq y_{n} \leq2y_{f(n)}$, for all $n\in\mathbb{N}$.[/list]
Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number?
[i]Nanang Susyanto, Jogjakarta[/i]
Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that:
(i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and
(ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.
Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A\cup B=S,\ A\cap B=\emptyset,$ and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_{S}$ be the number of linear partitions of $S.$ For each positive integer $n,$ find the maximum of $L_{S}$ over all sets $S$ of $n$ points.
The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$.
Note: In all the triangles the three vertices do not lie on a straight line.
Let $ \mathcal{P}$ be a square and let $ n$ be a nonzero positive integer for which we denote by $ f(n)$ the maximum number of elements of a partition of $ \mathcal{P}$ into rectangles such that each line which is parallel to some side of $ \mathcal{P}$ intersects at most $ n$ interiors (of rectangles). Prove that
\[ 3 \cdot 2^{n\minus{}1} \minus{} 2 \le f(n) \le 3^n \minus{} 2.\]
Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate
\[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]
Let $ n$ and $ k$ be positive integers. Say that a permutation $ \sigma$ of $ \{1,2,\dots n\}$ is $ k$-[i]limited[/i] if $ |\sigma(i)\minus{}i|\le k$ for all $ i.$ Prove that the number of $ k$-limited permutations of $ \{1,2,\dots n\}$ is odd if and only if $ n\equiv 0$ or $ 1\pmod{2k\plus{}1}.$
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.)
A figure $\Phi$ composed of unit squares has the following property: if the squares of an $m \times n$ rectangle ($m,n$ are fixed) are filled with numbers whose sum is positive, the figure $\Phi$ can be placed within the rectangle (possibly after being rotated) so that the sum of the covered numbers is also positive. Prove that a number of such figures can be put on the $m\times n$ rectangle so that each square is covered by the same number of figures.
Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)
A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$.
Prove that $S$ can be covered by a strip of width $2$.
A positive integer $N$ is said to be $n-$ good if
(i) $N$ has at least $n$ distinct prime divisors, and
(ii) there exist distinct positive divisors $1, x_{2}, . . . , x_{n}$ whose sum is $N$ .
Show that there exists an $n-$ good number for each $n\geq 6$.
Let $n>1$ be an integer and $a,b,c$ be three complex numbers such that $a+b+c=0$ and $a^n+b^n+c^n=0$. Prove that two of $a,b,c$ have the same magnitude.
[i]Evan O'Dorney.[/i]
Let $(a_n)_{n\in \mathbb N}$ be the sequence of integers defined recursively by $a_1 = a_2 = 1, a_{n+2} = 7a_{n+1} - a_n - 2$ for $n \geq 1$. Prove that $a_n$ is a perfect square for every $n.$
Prove: \[|\sin a_1|+|\sin a_2|+|\sin a_3|+\ldots+|\sin a_n|+|\cos(a_1+a_2+a_3+\ldots+a_n)|\geq 1.\]
10.4, 11.3 Given $N\geq 3$ points enumerated with 1, 2, ..., $N$. Each two numbers are connected by mean of arrow from a lesser number to a greater one. A coloring of all arrows into red and blue is called [i]monochromatic[/i] iff for any numbers $A$ and $B$ there are [color=red]no[/color] two monochromatic paths from $A$ to $B$ of different colors. Find the number of monochromatic colorings.
([i]I. Bogdanov, G. Chelnokov[/i])