Found problems: 85335
Given are 10 quadric equations $x^2+a_1x+b_1=0$, $x^2+a_2x+b_2=0$,..., $x^2+a_{10}x+b_{10}=0$.
It is known that each of these equations has two distinct real roots and the set of all solutions is ${1,2,...10,-1,-2...,-10}$. Find the minimum value of $b_1+b_2+...+b_{10}$
Three circles in a plane have the sides of a triangle as their diameters. Prove that there is a point that is in the interior of all three circles.
A triangle has a fixed base $AB$ that is $2$ inches long. The median from $A$ to side $BC$ is $ 1\frac{1}{2}$ inches long and can have any position emanating from $A$. The locus of the vertex $C$ of the triangle is:
$\textbf{(A)}\ \text{A straight line }AB,1\dfrac{1}{2}\text{ inches from }A \qquad\\
\textbf{(B)}\ \text{A circle with }A\text{ as center and radius }2\text{ inches} \qquad\\
\textbf{(C)}\ \text{A circle with }A\text{ as center and radius }3\text{ inches} \qquad\\
\textbf{(D)}\ \text{A circle with radius }3\text{ inches and center }4\text{ inches from }B\text{ along } BA \qquad\\
\textbf{(E)}\ \text{An ellipse with }A\text{ as focus}$
Mr. Jones sold two pipes at $ \$ 1.20$ each. Based on the cost, his profit one was $ 20 \%$ and his loss on the other was $ 20 \%$. On the sale of the pipes, he:
$ \textbf{(A)}\ \text{broke even} \qquad\textbf{(B)}\ \text{lost } 4\text{ cents} \qquad\textbf{(C)}\ \text{gained } 4\text{ cents} \qquad\textbf{(D)}\ \text{lost } 10 \text{ cents} \qquad\textbf{(E)}\ \text{gained } 10 \text{ cents}$
Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties:
$(i)$ For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$
$(ii)$ There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$
[i]In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.[/i]
Let $ ABC$ be an acute triangle and $ CL$ be the angle bisector of $ \angle ACB$. The point $ P$ lies on the segment $CL$ such that $ \angle APB\equal{}\pi\minus{}\frac{_1}{^2}\angle ACB$. Let $ k_1$ and $ k_2$ be the circumcircles of the triangles $ APC$ and $ BPC$. $ BP\cap k_1\equal{}Q, AP\cap k_2\equal{}R$. The tangents to $ k_1$ at $ Q$ and $ k_2$ at $ B$ intersect at $ S$ and the tangents to $ k_1$ at $ A$ and $ k_2$ at $ R$ intersect at $ T$. Prove that $ AS\equal{}BT.$
In a soccer championship $2004$ teams are subscribed. Because of the extremely large number of teams the usual rules of the championship are modified as follows:
a) any two teams can play against one each other at most one game;
b) from any $4$ teams, $3$ of them play against one each other.
How many days are necessary to make such a championship, knowing that each team can play at most one game per day?
Find all functions $f : N\rightarrow N - \{1\}$ satisfying $f (n)+ f (n+1)= f (n+2) +f (n+3) -168$ for all $n \in N$
.
Pentagon $ABCDE$ is such that all five diagonals $AC, BD, CE, DA,$ and $EB$ lie entirely within the pentagon. If the area of each of the triangles $ABC, BCD, CDE,$ and $DEA$ is equal to $1$ and the area of triangle $EAB$ is equal to $2$, the area of the pentagon $ABCDE$ is closest to
$ \mathrm{(A) \ } 4.42 \qquad \mathrm{(B) \ } 4.44 \qquad \mathrm {(C) \ } 4.46 \qquad \mathrm{(D) \ } 4.48 \qquad \mathrm{(E) \ } 4.5$
A [i]slab[/i] is the set of points strictly between two parallel planes. Prove that a countable sequence of slabs, the sum of whose thicknesses converges, cannot fill space.
Let $a_0,a_1,a_2...$ be a sequence of natural numbers with the following property: $a_n^2$ divides $a_{n-1} a_{n+1}$ for $\forall$ $n\in \mathbb{N}$. Prove that, if for some natural $k\geq 2$ the numbers $a_1$ and $a_k$ are coprime, then $a_1$ divides $a_0$.
Let $S$ be the set of integers that represent the number of intersections of some four distinct lines in the plane. List the elements of $S$ in ascending order.
Let be two real numbers $ b>a>0, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ with $ x_2>x_1>0 $ and such that
$$ ax_{n+2}+bx_n\ge (a+b)x_{n+1} , $$
for any natural numbers $ n. $
Prove that $ \lim_{n\to\infty } x_n=\infty . $
[i]Dan Popescu[/i]
Ten volleyball teams played a tournament; every two teams met exactly once. The winner of the play gets 1 point, the loser gets 0 (there are no draws in volleyball). If the team that scored $n$-th has $x_{n}$ points ($n=1, \dots, 10$), prove that $x_{1}+2x_{2}+\dots+10x_{10}\geq 165$.
[i]Proposed by D. Teryoshin[/i]
Let $\phi (x, u)$ be the smallest positive integer $n$ so that $2^u$ divides $x^n + 95$ if it exists, or $0$ if no such positive integer exists. Determine$ \sum_{i=0}^{255} \phi(i, 8)$.
The sequence $ (x_n)^{\infty}_{n\equal{}1}$ is defined by $ x_1 \equal{} 1$ and $ x_{n\plus{}1} \equal{}\frac{(2 \plus{} \cos 2\alpha)x_n \minus{} \cos^2\alpha}{(2 \minus{} 2 \cos 2\alpha)x_n \plus{} 2 \minus{} \cos 2\alpha}$, for all $ n \in\mathbb{N}$, where $ \alpha$ is a given real parameter. Find all values of $ \alpha$ for which the sequence $ (y_n)$ given by $ y_n \equal{} \sum_{k\equal{}1}^{n}\frac{1}{2x_k\plus{}1}$ has a finite limit when $ n \to \plus{}\infty$ and find that limit.
Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a [i]ship[/i] a figure made up of unit squares connected by common edges. We call a [i]fleet[/i] a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the greatest natural number that, for each its representation as a sum of positive integers, there exists a fleet such that the summands are exactly the numbers of squares contained in individual ships.
Euclid has a tool called splitter which can only do the following two types of operations :
• Given three non-collinear marked points $X,Y,Z$ it can draw the line which forms the interior angle bisector of $\angle{XYZ}$.
• It can mark the intersection point of two previously drawn non-parallel lines .
Suppose Euclid is only given three non-collinear marked points $A,B,C$ in the plane . Prove that Euclid can use the splitter several times to draw the centre of circle passing through $A,B$ and $C$.
[i]Proposed by Shankhadeep Ghosh[/i]
Carl, James, Saif, and Ted play several games of two-player For The Win on the Art of Problem Solving website. If, among these games, Carl wins $5$ and loses $0,$ James wins $4$ and loses $2,$ Saif wins $1$ and loses $6,$ and Ted wins $4,$ how many games does Ted lose?
Let $A_{11}$ denote the answer to problem $11$. Determine the smallest prime $p$ such that the arithmetic sequence $p,p+A_{11},p+2A_{11},\cdots$ begins with the largest number of primes.
There is just one triple of possible $(A_{10},A_{11},A_{12})$ of answers to these three problems. Your team will receive credit only for answers matching these. (So, for example, submitting a wrong answer for problem $11$ will not alter the correctness of your answer to problem $12$.)
Given $1990$ piles of stones, containing $1, 2, 3, ... , 1990$ stones. A move is to take an equal number of stones from one or more piles. How many moves are needed to take all the stones?
On planet Polyped, every creature has either $6$ legs or $10$ legs. In a room with $20$ creatures and $156$ legs, how many of the creatures have $6$ legs?
$x_{n+1}= \left ( 1+\frac2n \right )x_n+\frac4n$, for every positive integer $n$. If $x_1=-1$, what is $x_{2000}$?
$ \textbf{(A)}\ 1999998
\qquad\textbf{(B)}\ 2000998
\qquad\textbf{(C)}\ 2009998
\qquad\textbf{(D)}\ 2000008
\qquad\textbf{(E)}\ 1999999
$
Let $ABC$ be a triangle with circumcenter $O$. Let $A_1$ be the midpoint of side $BC$. Ray $AA_1$ meet the circumcircle of triangle $ABC$ again at $A_2$ (other than A). Let $Q_a$ be the foot of the perpendicular from $A_1$ to line $AO$. Point $P_a$ lies on line $Q_aA_1$ such that $P_aA_2 \perp A_2O$. Define points $P_b$ and $P_c$ analogously. Prove that points $P_a$, P_b$, and $P_c$ lie on a line.
There are 8 coins, 7 of which are real, which weigh the same, and one is fake, which differs in weight from the rest. Cup scales without weights mean that if you put equal weights on their cups, then any of the cups can outweigh, but if the loads are different in mass, then the cup with a heavier load is definitely overpowered. How to definitely identify a counterfeit coin in four weighings and establish is it lighter or heavier than the others?