Found problems: 85335
Real numbers $ a_{1}$, $ a_{2}$, $ \ldots$, $ a_{n}$ are given. For each $ i$, $ (1 \leq i \leq n )$, define
\[ d_{i} \equal{} \max \{ a_{j}\mid 1 \leq j \leq i \} \minus{} \min \{ a_{j}\mid i \leq j \leq n \}
\]
and let $ d \equal{} \max \{d_{i}\mid 1 \leq i \leq n \}$.
(a) Prove that, for any real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$,
\[ \max \{ |x_{i} \minus{} a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*)
\]
(b) Show that there are real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*).
[i]Author: Michael Albert, New Zealand[/i]
The five real numbers $v,w,x,y,s$ satisfy the system of equations
\begin{align*}
v&=wx+ys,\\
v^2&=w^2x+y^2s,\\
v^3&=w^3x+y^3s.
\end{align*}
Show that at least two of them are equal.
Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$?
$\textbf{(A) }-10\qquad\textbf{(B) }-6\qquad\textbf{(C) }0\qquad\textbf{(D) }6\qquad \textbf{(E) }10$
Let $P(x)$ be a quadratic polynomial with complex coefficients whose $x^2$ coefficient is $1$. Suppose the equation $P(P(x))=0$ has four distinct solutions, $x=3,4,a,b$. Find the sum of all possible values of $(a+b)^2$.
The altitudes of the acute-angled non-isosceles triangle $ ABC $ intersect at the point $ H $. On the segment $ C_1H $, where $ CC_1 $ is the altitude of the triangle, the point $ K $ is marked. Points $ L $ and $ M $ are the feet of perpendiculars from point $ K $ on straight lines $ AC $ and $ BC $, respectively. The lines $ AM $ and $ BL $ intersect at $ N $. Prove that $ \angle ANK = \angle HNL $.
Determine all 2nd degree polynomials with integer coefficients of the form $P(X)=aX^{2}+bX+c$, that satisfy: $P(a)=b$, $P(b)=a$, with $a\neq b$.
Let $\vartriangle ABC$ be a triangle with $AB = 5$, $BC = 8$, and, $CA = 7$. Let the center of the $A$-excircle be $O$, and let the $A$-excircle touch lines $BC$, $CA$, and,$ AB$ at points $X, Y$ , and, $Z$, respectively. Let $h_1$, $h_2$, and, $h_3$ denote the distances from $O$ to lines $XY$ , $Y Z$, and, ZX, respectively. If $h^2_1+ h^2_2+ h^2_3$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m + n$.
$n$ cars are racing. At first they have a particular order. At each moment a car may overtake another car. No two overtaking actions occur at the same time, and except moments a car is passing another, the cars always have an order.
A set of overtaking actions is called "small" if any car overtakes at most once.
A set of overtaking actions is called "complete" if any car overtakes exactly once.
If $F$ is the set of all possible orders of the cars after a small set of overtaking actions and $G$ is the set of all possible orders of the cars after a complete set of overtaking actions, prove that
\[\mid F\mid=2\mid G\mid\]
(20 points)
[i]Proposed by Morteza Saghafian[/i]
Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$. A straight line $L$ is parallel to $BC$ and tangent to the incircle. Suppose $L$ intersects $IO$ at $X$, and select $Y$ on $L$ such that $YI$ is perpendicular to $IO$. Prove that $A$, $X$, $O$, $Y$ are cyclic.
[i]Proposed by Telv Cohl[/i]
Any two of the real numbers $a_1,a_2,a_3,a_4,a_5$ differ by no less than $1$. There exists some real number $k$ satisfying
\[a_1+a_2+a_3+a_4+a_5=2k\]\[a_1^2+a_2^2+a_3^2+a_4^2+a_5^2=2k^2\]
Prove that $k^2\ge 25/3$.
Let $r,g,b$ be non negative integers and $\Gamma$ be a connected graph with $r+g+b+1$ vertices. Its edges are colored in red green and blue. It turned out that $\Gamma $ contains
A spanning tree with exactly $r$ red edges.
A spanning tree with exactly $g$ green edges.
A spanning tree with exactly $b$ blue edges.
Prove that $\Gamma$ contains a spanning tree with exactly $r$ red edges, $g$ green edges and $b$ blue edges.
[b]Q6.[/b] At $3:00$ AM, the temperature was $13^o$ below zero. By none it has risen to $32^o$. What is the average hourly increase in temperature ?
Triangle $ABC$ is given with its centroid $G$ and cicumcentre $O$ is such that $GO$ is perpendicular to $AG$. Let $A'$ be the second intersection of $AG$ with circumcircle of triangle $ABC$. Let $D$ be the intersection of lines $CA'$ and $AB$ and $E$ the intersection of lines $BA'$ and $AC$. Prove that the circumcentre of triangle $ADE$ is on the circumcircle of triangle $ABC$.
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x, y, z,$
\[f(x+y^2+z)=f(f(x))+yf(y)+f(z). \]
One rainy afternoon you write the number $1$ once, the number $2$ twice, the number $3$ three times, and so forth until you have written the number $99$ ninety-nine times. What is the $2005$ th digit that you write?
Let $E$ be the set of all bijective mappings from $\mathbb R$ to $\mathbb R$ satisfying
\[f(t) + f^{-1}(t) = 2t, \qquad \forall t \in \mathbb R,\]
where $f^{-1}$ is the mapping inverse to $f$. Find all elements of $E$ that are monotonic mappings.
On the screen of a computer there is an $2^n\times 2^n$ board. On each cell of the main diagonal there is a file. At each step, we may select some files and move them to the left, on their respective rows, by the same distance. What is the minimum number of necessary moves in order to put all files on the first column?
[i]Proposed by Vlad Spătaru[/i]
Let $a_j (j = 1, 2, \ldots, n)$ be entirely arbitrary numbers except that no one is equal to unity. Prove \[ a_1 + \sum^n_{i=2} a_i \prod^{i-1}_{j=1} (1 - a_j) = 1 - \prod^n_{j=1} (1 - a_j).\]
Prove that if $m$ and $s$ are integers with $ms=2000^{2001}$, then the equation $mx^2-sy^2=3$ has no integer solutions.
Source: 2018 Canadian Open Math Challenge Part B Problem 2
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Let ABCD be a square with side length 1. Points $X$ and $Y$ are on sides $BC$ and $CD$ respectively such that the areas of triangels $ABX$, $XCY$, and $YDA$ are equal. Find the ratio of the area of $\triangle AXY$ to the area of $\triangle XCY$.
[center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZi9lLzAzZjhhYzU0N2U0MGY2NGZlODM4MWI4Njg2MmEyMjhlY2M3ZjgzLnBuZw==&rn=YjIuUE5H[/img][/center]
Given triangle $ ABC$ with $ C$ a right angle, show that the diameter of the incenter is $ a\plus{}b\minus{}c$, where $ a\equal{}BC$, $ b\equal{}CA$, and $ c\equal{}AB$.
Let $S$ be the set of points $A$ in the Cartesian plane such that the four points $A$, $(2, 3)$, $(-1, 0)$, and $(0, 6)$ form the vertices of a parallelogram. Let $P$ be the convex polygon whose vertices are the points in $S$. What is the area of $P$?
$P\in A_n(\mathbb R)=\{M_{n\times n}|M^2=M\}$. Which of the following are true?
$\textbf{(A)}~P^T=P,\forall P\in A_n(\mathbb R)$
$\textbf{(B)}~\exists P\ne0,P\in A_n(\mathbb R)\text{ with }\operatorname{tr}(P)=0$
$\textbf{(C)}~\exists X_{n\times r}\text{ such that }Px=X\text{ for }r=\operatorname{rank}(P)$
Find the area of inscribed convex octagon, if the length of four sides is $2$, and length of other four sides is $ 6\sqrt {2}$.
$\textbf{(A)}\ 120 \qquad\textbf{(B)}\ 24 \plus{} 68\sqrt {2} \qquad\textbf{(C)}\ 88\sqrt {2} \qquad\textbf{(D)}\ 124 \qquad\textbf{(E)}\ 72\sqrt {3}$
Compute
$$\lim_{x\rightarrow1}\frac{x^3-1}{x-1}.$$