Found problems: 85335
Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that \[ \sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}. \]
A convex quadrilateral $ ABCD$ with area $ 2002$ contains a point $ P$ in its interior such that $ PA \equal{} 24$, $ PB \equal{} 32$, $ PC \equal{} 28$, and $ PD \equal{} 45$. FInd the perimeter of $ ABCD$.
$ \textbf{(A)}\ 4\sqrt {2002}\qquad \textbf{(B)}\ 2\sqrt {8465}\qquad \textbf{(C)}\ 2\left(48 \plus{} \sqrt {2002}\right)$
$ \textbf{(D)}\ 2\sqrt {8633}\qquad \textbf{(E)}\ 4\left(36 \plus{} \sqrt {113}\right)$
Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors.
Suppose $M$ can be written as $\Pi_{k=1}^{n}p_k^{a_k}$ where the $p_k$’s are distinct primes and the $a_k$’s are positive integers.
Find $\Sigma_{k=1}^{n}(p_k + a_k)$
Let $ABC$ be an acute triangle with $BC = 4$ and $AC = 5$. Let $D$ be the midpoint of $BC$, $E$ be the foot of the altitude from $B$ to $AC$, and $F$ be the intersection of the angle bisector of $\angle BCA$ with segment $AB$. Given that $AD$, $BE$, and $CF$ meet at a single point $P$, compute the area of triangle $ABC$. Express your answer as a common fraction in simplest radical form.
Find all non-negative integers $ x,y,z$ such that $ 5^x \plus{} 7^y \equal{} 2^z$.
:lol:
([i]Daniel Kohen, University of Buenos Aires - Buenos Aires,Argentina[/i])
Find the least positive integer $N$ which is both a multiple of 19 and whose digits add to 23.
In $\triangle ABC$, $\angle B=90^\circ$, segment $AB>BC$. Now we have a $\triangle A_iBC(i=1,2,...,n)$ which is similiar to $\triangle ABC$ (the vertexs of them might not correspond). Find the maximum value of $n+2018$.
On an arbitrary triangle $ ABC$ let $ E$ be a point on the height from $ A$. Prove that $ (AC)^2 - (CE)^2 = (AB)^2 - (EB)^2$.
Let there be $ n \ge 2$ real numbers such that none of them is greater than the arithmetic mean of the other numbers. Prove that all the numbers are equal.
Let $(a_n)_{n \ge 1}$ be a sequence of real numbers satisfying the properties: [list=1]
[*] the sequence $x_n=\sum\limits_{k=1}^n a_k^2$ is convergent;
[*] the sequence $y_n=\sum\limits_{k=1}^n a_k$ is unbounded.
[/list]
Prove that the sequence $(b_n)_{n \ge 1}$ given by $b_n=\{y_n\}$ is divergent.
Note: $\{ x \}$ denotes the fractional part of $x.$
Let us call a triangle “almost right angle triangle” if one of its angles differs from $90^\circ$ by no more than $15^\circ$. Let us call a triangle “almost isosceles triangle” if two of its angles differs from each other by no more than $15^\circ$. Is it true that that any acute triangle is either “almost right angle triangle” or “almost isosceles triangle”?
[i](2 points)[/i]
Determine whether the inequality $$ \left|\sqrt{x^2+2x+5}-\sqrt{x^2-4x+8}\right|<3$$ is valid for all real numbers $x$.
A triplet of positive integers $(x, y, z)$ satisfying $x, y, z > 1$ and $x^3 - yz^3 = 2021$ is called [i]primary [/i] if at least two of the integers $x, y, z$ are prime numbers.
a) Find at least one primary triplet.
b) Show that there are infinitely many primary triplets.
For a nonnegative integer $n$, define $a_n$ to be the positive integer with decimal representation
\[1\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}1\mbox{.}\]
Prove that $\frac{a_n}{3}$ is always the sum of two positive perfect cubes but never the sum of two perfect squares.
[i](4th Middle European Mathematical Olympiad, Team Competition, Problem 7)[/i]
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
2008 boys and 2008 girls sit on 4016 chairs around a round table. Each boy brings a garland and each girl brings a chocolate. In an "activity", each person gives his/her goods to the nearest person on the left. After some activities, it turns out that all boys get chocolates and all girls get garlands. Find the number of possible arrangements.
Some $k{}$ vertices of a regular $n{}$-gon are colored red. We will call a coloring [i]uniform[/i] if for any $m$ the number of red vertices in any two sets of $m$ consecutive vertices of the $n{}$-gon coincide or differ by 1. Prove that a uniform coloring exists for any $k<n$ and is unique, up to rotations of the $n{}$-gon.
[i]Proposed by M. Kontsevich[/i]
Given a circle \( \omega \) and two points \( A \) and \( B \) outside \( \omega \), a quadrilateral \( PQRS \) is defined as[i] "good"[/i] if \( P, Q, R, S \) are four distinct points on \( \omega \) in order, and lines \( PQ \) and \( RS \) intersect at \( A \) and lines \( PS \) and \( QR \) intersect at \( B \).
For a quadrilateral \( T \), let \( S_T \) denote its area. If there exists a [i]good[/i] quadrilateral, prove that there exists [i]good[/i] quadrilateral \( T \) such that for any good quadrilateral $T_1 (T_1 \neq T)$, \( S_{T_1} < S_T \).
Consider the numbers arranged in the following way:
\[\begin{array}{ccccccc} 1 & 3 & 6 & 10 & 15 & 21 & \cdots \\ 2 & 5 & 9 & 14 & 20 & \cdots & \cdots \\ 4 & 8 & 13 & 19 & \cdots & \cdots & \cdots \\ 7 & 12 & 18 & \cdots & \cdots & \cdots & \cdots \\ 11 & 17 & \cdots & \cdots & \cdots & \cdots & \cdots \\ 16 & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array}\]
Find the row number and the column number in which the the number $20096$ occurs.
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
A $ 3 \times n$ grid is filled as follows. The first row consists of the numbers from $ 1$ to $ n$ arranged in ascending order. The second row is a cyclic shift of the top row: $ i,i\plus{}1,...,n,1,2,...,i\minus{}1$ for some $ i$. The third row has the numbers $ 1$ to $ n$ in some order so that in each of the $ n$ columns, the sum of the three numbers is the same. For which values of $ n$ is it possible to fill the grid in this way? For all such $ n$, determine the number of different ways of filling the grid.
Let $P$ and $Q$ be points on $AC$ and $AB$, respectively, of triangle $\triangle ABC$ such that $PB=PC$ and $PQ\perp AB$. Suppose $\frac{AQ}{QB}=\frac{AP}{PB}.$ Find $\angle CBA$, in degrees.
[i]Proposed by Nathan Ramesh
How many non-congruent right triangles are there such that the perimeter in $\text{cm}$ and the area in $\text{cm}^2$ are numerically equal?
$\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 4 \qquad \text{(E)} \ \text{infinitely many}$
Evaluate
\[\int_0^{\pi} \{(1-x\sin 2x)e^{\cos ^2 x}+(1+x\sin 2x)e^{\sin ^ 2 x}\}\ dx.\]
Let $n$ be a positive integer. In a village, $n$ boys and $n$ girls are living.
For the yearly ball, $n$ dancing couples need to be formed, each of which consists of one boy and one girl. Every girl submits a list, which consists of the name of the boy with whom she wants to dance the most, together with zero or more names of other boys with whom she wants to dance. It turns out that $n$ dancing couples can be formed in such a way that every girl is paired with a boy who is on her list.
Show that it is possible to form $n$ dancing couples in such a way that every girl is paired with a boy who is on her list, and at least one girl is paired with the boy with whom she wants to dance the most.