Found problems: 85335
Given a regular convex $2m$- sided polygon $P$, show that there is a $2m$-sided polygon $\pi$ with the same vertices as $P$ (but in different order) such that $\pi$ has exactly one pair of parallel sides.
Let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be two finite sequences consisting of $2n$ real different numbers. Rearranging each of the sequences in increasing order we obtain $a_1',a_2',\ldots,a_n'$ and $b_1',b_2',\ldots,b_n'$. Prove that
\[\max_{1\le i\le n}|a_i-b_i|\ge\max_{1\le i\le n}|a_i'-b_i'|.\]
Prove that the sum $1^3 + 2^3 +...+ n^3$ is a perfect square for all $n$.
Let $ABC$ be an acute, non isosceles with $I$ is its incenter. Denote $D, E$ as tangent points of $(I)$ on $AB,AC$, respectively. The median segments respect to vertex $A$ of triangles $ABE$ and $ACD$ meet$ (I)$ at$ P,Q,$ respectively. Take points $M, N$ on the line $DE$ such that $AM \parallel BE$ and $AN \parallel C D$ respectively.
a) Prove that $A$ lies on the radical axis of $(MIP)$ and $(NIQ)$.
b) Suppose that the orthocenter $H$ of triangle $ABC$ lies on $(I)$. Prove that there exists a line which is tangent to three circles of center $A, B, C$ and all pass through $H$.
Triangle $ABC$ is right angled at $A$. The circle with center $A$ and radius $AB$ cuts $BC$ and $AC$ internally at $D$ and $E$ respectively. If $BD = 20$ and $DC = 16$, determine $AC^2$.
Each point with integral coordinates in the plane is coloured white or blue. Prove that one can choose a colour so that for every positive integer $ n $ there exists a triangle of area $ n $ having its vertices of the chosen colour.
Consider a group $ G $ which has the property that any element of it, with the exception of the identity, has order $ p\ge 2. $ Prove that
[b]a)[/b] $ p $ is prime.
[b]b)[/b] $ G $ is commutative if any subset of $ G $ having $ p^2-1 $ elements contains at least $ p $ elements that commute between themselves pairwise.
[b]T[/b]wo ordered pairs $(a,b)$ and $(c,d)$, where $a,b,c,d$ are real numbers, form a basis of the coordinate plane if $ad \neq bc$. Determine the number of ordered quadruples $(a,b,c)$ of integers between $1$ and $3$ inclusive for which $(a,b)$ and $(c,d)$ form a basis for the coordinate plane.
Points $ A_1$, $ A_2$, $ ...$ are placed on a circle with center $ O$ such that $ \angle OA_n A_{n\plus{}1}\equal{}35^\circ$ and $ A_n\neq A_{n\plus{}2}$ for all positive integers $ n$. What is the smallest $ n>1$ for which $ A_n\equal{}A_1$?
$\frac{2 \sqrt 6}{\sqrt 2 + \sqrt 3 + \sqrt 5}$ equals
A. $\sqrt 2 + \sqrt 3 - \sqrt 5$
B. $4 - \sqrt 2 - \sqrt 3$
C. $\sqrt 2 + \sqrt 3 + \sqrt 6 - 5$
D. $\frac{1}{2} (\sqrt 2 + \sqrt 5 - \sqrt 3)$
E. $\frac{1}{3} (\sqrt 3 + \sqrt 5 - \sqrt 2)$
Let the sides of two rectangles be $ \{a,b\}$ and $ \{c,d\},$ respectively, with $ a < c \leq d < b$ and $ ab < cd.$ Prove that the first rectangle can be placed within the second one if and only if
\[ \left(b^2 \minus{} a^2\right)^2 \leq \left(bc \minus{} ad \right)^2 \plus{} \left(bd \minus{} ac \right)^2.\]
If a positive sequence $\{a_n\}_{n\geq 1}$ satisfies $\int_0^{a_n} x^{n}\ dx=2$, then find $\lim_{n\to\infty} a_n.$
A square grid $2n \times 2n$ is constructed of matches (each match is a segment of length 1). By one move Peter can choose a vertex which (at this moment) is the endpoint of 3 or 4 matches and delete two matches whose union is a segment of length 2. Find the least possible number of matches that could remain after a number of Peter's moves.
Let $ABCD$ be a parallelogram. Suppose a line passing through $C$ and lying outside the parallelogram meets $AB$ and $AD$ produced at $E$ and $F$ respectively. Show that \[ AC^2 + CE \cdot CF = AB \cdot AE + AD \cdot AF . \]
Solve in real numbers the equation:
$ \frac{1}{x}\plus{}\frac{1}{x\plus{}2}\minus{}\frac{1}{x\plus{}4}\minus{}\frac{1}{x\plus{}6}\minus{}\frac{1}{x\plus{}8}\minus{}\frac{1}{x\plus{}10}\plus{}\frac{1}{x\plus{}12}\plus{}\frac{1}{x\plus{}14}\equal{}0.$
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect at $E$. Denotes by $S_1,S_2$ and $S$ the areas of the triangles $ABE$, $CDE$ and the quadrilateral $ABCD$ respectively. Prove that $\sqrt{S_1}+\sqrt{S_2}\le \sqrt{S}$ . When equality is reached?
Let $(a,\ b)$ be a point on the curve $y=\frac{x}{1+x}\ (x\geq 0).$ Denote $U$ the volume of the figure enclosed by the curve , the $x$ axis and the line $x=a$, revolved around the the $x$ axis and denote $V$ the volume of the figure enclosed by the curve , the $y$ axis and th line $y=b$, revolved around the $y$ axis. What's the relation of $U$ and $V?$
1978 Chuo university entrance exam/Science and Technology
Let $a_n$ and $b_n$ to be two sequences defined as below:
$i)$ $a_1 = 1$
$ii)$ $a_n + b_n = 6n - 1$
$iii)$ $a_{n+1}$ is the least positive integer different of $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$.
Determine $a_{2009}$.
Let $G=(V,E)$ be a tree graph with $n$ vertices, and let $P$ be a set of $n$ points in the plane with no three points collinear. Is it true that for any choice of graph $G$ and set $P$, we can embed $G$ in $P$, i.e., we can find a bijection $f:V\to P$ such that when we draw line segment $[f(x),f(y)]$ for all $(x,y)\in E$, no two such segments intersect each other?
Given are natural numbers $ a,b$ and $ n$ such that $ a^2\plus{}2nb^2$ is a complete square. Prove that the number $ a^2\plus{}nb^2$ can be written as a sum of squares of $ 2$ natural numbers.
Find all the possible values of the sum of the digits of all the perfect squares.
[Commented by djimenez]
[b]Comment: [/b]I would rewrite it as follows:
Let $f: \mathbb{N}\rightarrow \mathbb{N}$ such that $f(n)$ is the sum of all the digits of the number $n^2$. Find the image of $f$ (where, by image it is understood the set of all $x$ such that exists an $n$ with $f(n)=x$).
Let $S$ be the set of all pairs of positive integers $(x, y)$ for which $2x^2 + 5y^2 \le 5+6xy$. Compute $\displaystyle\sum_{(x,y) \in S} (x+y+100)$.
[i]Proposed by Daniel Whatley[/i]
On the base $BC$ of the isosceles triangle $ABC$ chose a point $D$ and in each of the triangles $ABD$ and $ACD$ inscribe a circle. Then everything was wiped, leaving only two circles. It is known from which side of their line of centers
the apex $A$ is located . Use a compass and ruler to restore the triangle $ABC$ , if we know that :
a) $AD$ is angle bisector,
b) $AD$ is median.
Find all triples $(a,b,c)$ of real numbers satisfying the system
$\begin{cases}
a^3+b^2c=ac \\
b^3+c^2a=ba \\
c^3+a^2b=cb
\end{cases}$
The sequence $ \{a_n\}$ satisfies $ a_0 \equal{} 0, a_{n \plus{} 1} \equal{} ka_n \plus{} \sqrt {(k^2 \minus{} 1)a_n^2 \plus{} 1}, n \equal{} 0, 1, 2, \ldots$, where $ k$ is a fixed positive integer. Prove that all the terms of the sequence are integral and that $ 2k$ divides $ a_{2n}, n \equal{} 0, 1, 2, \ldots$.