Found problems: 85335
Does there exist an integer $ n > 1$ which satisfies the following condition? The set of positive integers can be partitioned into $ n$ nonempty subsets, such that an arbitrary sum of $ n \minus{} 1$ integers, one taken from each of any $ n \minus{} 1$ of the subsets, lies in the remaining subset.
Given any positive integer $n$, prove that:
[b](a)[/b] Every $n$ points in the closed unit square $[0,1]\times [0,1]$ can be joined by a path of length less than $2\sqrt{n}+4$; and
[b](b)[/b] There exist $n$ points in the closed unit square $[0,1]\times [0,1]$ that cannot be joined by a path of length less than $\sqrt{n}-1$.
Let $a_1,a_2,...,a_n,b_1,b_2,...,b_n,p$ be real numbers with $p >- 1$. Prove that
$$\sum_{i=1}^{n}(a_i-b_i)\left(a_i (a_1^2+a_2^2+...+a_n^2)^{p/2}-b_i (b_1^2+b_2^2+...+b_n^2)^{p/2}\right)\ge 0$$
Determine the smallest positive integer, $n$, which satisfies the equation $n^3+2n^2 = b$, where $b$ is the square of an odd integer.
Given a acute triangle $PA_1B_1$ is inscribed in the circle $\Gamma$ with radius $1$. for all integers $n \ge 1$ are defined:
$C_n$ the foot of the perpendicular from $P$ to $A_nB_n$
$O_n$ is the center of $\odot (PA_nB_n)$
$A_{n+1}$ is the foot of the perpendicular from $C_n$ to $PA_n$
$B_{n+1} \equiv PB_n \cap O_nA_{n+1}$
If $PC_1 =\sqrt{2}$, find the length of $PO_{2015}$
[hide=Source]Cono Sur Olympiad - 2015 - Day 1 - Problem 3[/hide]
Denote by $g_n$ the number of connected graphs of degree $n$ whose vertices are labeled with numbers $1,2,...,n$. Prove that $g_n \ge (\frac{1}{2}).2^{\frac{n(n-1)}{2}}$.
[b][u]Note[/u][/b]:If you prove that for $c < \frac{1}{2}$, $g_n \ge c.2^{\frac{n(n-1)}{2}}$, you will earn some point!
[i]proposed by Seyed Reza Hosseini and Mohammad Amin Ghiasi[/i]
Find all integers $a, b, c, d$ such that $$\begin{cases} ab - 2cd = 3 \\ ac + bd = 1\end{cases}$$
Let $f(x)=\int_0^ x \frac{1}{1+t^2}dt.$ For $-1\leq x<1$, find $\cos \left\{2f\left(\sqrt{\frac{1+x}{1-x}}\right)\right\}.$
[i]2011 Ritsumeikan University entrance exam/Science and Technology[/i]
Let $N_{15}$ be the answer to problem 15.
For a positive integer $x$ expressed in base ten, let $x'$ be the result of swapping its first and last digits (for example, if $x = 2024$, then $x' = 4022$). Let $C$ be the number of $N_{15}$-digit positive integers $x$ with a nonzero leading digit that satisfy the property that both $x$ and $x'$ are divisible by $11$ (note: $x'$ is allowed to have a leading digit of zero). Compute the sum of the digits of $C$ when $C$ is expressed in base ten.
There are $n$($\ge 2$) clubs $A_1,A_2,...A_n$ in Korean Mathematical Society. Prove that there exist $n-1$ sets $B_1,B_2,...B_{n-1}$ that satisfy the condition below.
(1) $A_1\cup A_2\cup \cdots A_n=B_1\cup B_2\cup \cdots B_{n-1}$
(2) for any $1\le i<j\le n-1$, $B_i\cap B_j=\emptyset, -1\le\left\vert B_i \right\vert -\left\vert B_j \right\vert\le 1$
(3) for any $1\le i \le n-1$, there exist $A_k,A_j $($1\le k\le j\le n$)such that $B_i\subseteq A_k\cup A_j$
Let $p(x)=x^5+x$ and $q(x)=x^5+x^2$, Find al pairs $(w,z)\in \mathbb{C}\times\mathbb{C}$, $w\not=z$ for which $p(w)=p(z),q(w)=q(z)$.
Find the free coefficient of the polynomial $P(x)$ with integer coefficients, knowing that it is less than $1000$ in absolute value and that $P(19) = P(94) = 1994$.
You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.
Prove that there are infinitely natural numbers $n$ such that $n$ can't be written as a sum of two positive integers with prime factors less than $1394$.
For positive integers $a,n$, define $F_n(a)=q+r$, where $a=qn+r$ ($q,r$ are nonnegative integers, $0\leq q<n$). Find the largest integer $A$, there are positive integers $n_1,n_2,n_3,n_4,n_5,n_6$, for all positive integer $a\leq A$, $F_{n_6}(F_{n_5}(F_{n_4}(F_{n_3}(F_{n_2}(F_{n_1}(a))))))=1$.
In a box there are $n$ balls, each colored in one of the following colors: green, red, blue or yellow. It is known that among any $28$ balls in the box at least one is green. Among any $26$ balls at least one is red. Among any $24$ balls at least one is blue. Among any $23$ balls at least one is yellow. Find the largest possible value of the number $n$.
Let $BD$ be the external bisector of a triangle $ABC$ with $AB > BC$; $K$ and $K_1$ be the touching points of side $AC$ with the incircle and the excircle centered at $I$ and $I_1$ respectively. The lines $BK$ and $DI_1$ meet at point $X$, and the lines $BK_1$ and $DI$ meet at point $Y$. Prove that $XY \perp AC$.
Given a triangle $ \triangle{ABC} $. Denote its incircle and circumcircle by $ \omega, \Omega $, respectively. Assume that $ \omega $ tangents the sides $ AB, AC $ at $ F, E $, respectively. Then, let the intersections of line $ EF $ and $ \Omega $ to be $ P,Q $. Let $ M $ to be the mid-point of $ BC $. Take a point $ R $ on the circumcircle of $ \triangle{MPQ} $, say $ \Gamma $, such that $ MR \perp EF $. Prove that the line $ AR $, $ \omega $ and $ \Gamma $ intersect at one point.
Let $x$ be a real number in the interval $\left(0,\dfrac{\pi}{2}\right)$ such that $\dfrac{1}{\sin x \cos x}+2\cot 2x=\dfrac{1}{2}$. Evaluate $\dfrac{1}{\sin x \cos x}-2\cot 2x$.
Prove that among the vertices of any convex nonagon, three can be found forming an obtuse triangle, none of whose sides coincide with the sides of the nonagon.
[i] Proposed by A. Yuran [/i]
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $P$ and $Q$ be points in the half plane defined by $BC$ containing $A$, such that $BP$ and $CQ$ are tangents to $\Gamma$ and $PB = BC = CQ$. Let $K$ and $L$ be points on the external bisector of the angle $\angle CAB$ , such that $BK = BA, CL = CA$. Let $M$ be the intersection point of the lines $PK$ and $QL$. Prove that $MK=ML$.
Prove that for each prime number $p > 5$ there exists a positive integer n such that $p^n$ ends in $001$ in decimal representation.
Given that $x_1>0$ and $x_2=4x_1$ are solutions to $ax^2+bx+c$ and that $3a=2(c-b)$, what is $x_1$?
Do there exist irrational numbers $a,b$ both greater than $1$, such that $\lfloor{a^m}\rfloor\neq \lfloor{b^n}\rfloor$ for all $m,n\in\mathbb{N}$ ?
We are given a polygon, a line $l$ and a point $P$ on $l$ in general position: all lines containing a side of the polygon meet $l$ at distinct points diering from $P$.
We mark each vertex of the polygon the sides meeting which, extended away from the vertex, meet the
line $l$ on opposite sides of $P$. Show that $P$ lies inside the polygon if and only if on each side of $l$ there are an odd number of marked vertices.
[i]O. Musin[/i]