Found problems: 85335
2016 CHMMC (Fall), 15
In a $5 \times 5$ grid of squares, how many nonintersecting pairs rectangles of rectangles are there? (Note sharing a vertex or edge still means the rectangles intersect.)
2013 AIME Problems, 10
Given a circle of radius $\sqrt{13}$, let $A$ be a point at a distance $4 + \sqrt{13}$ from the center $O$ of the circle. Let $B$ be the point on the circle nearest to point $A$. A line passing through the point $A$ intersects the circle at points $K$ and $L$. The maximum possible area for $\triangle BKL$ can be written in the form $\tfrac{a-b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers, $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.
2007 Sharygin Geometry Olympiad, 15
In a triangle $ABC$, let $AA', BB'$ and $CC'$ be the bisectors. Suppose $A'B' \cap CC' =P$ and $A'C' \cap BB'= Q$. Prove that $\angle PAC = \angle QAB$.
2020 Vietnam Team Selection Test, 6
In the scalene acute triangle $ABC$, $O$ is the circumcenter. $AD, BE, CF$ are three altitudes. And $H$ is the orthocenter. Let $G$ be the reflection point of $O$ through $BC$. Draw the diameter $EK$ in $\odot (GHE)$, and the diameter $FL$ in $\odot (GHF)$.
a) If $AK, AL$ and $DE, DF$ intersect at $U, V$ respectively, prove that $UV\parallel EF$.
b) Suppose $S$ is the intersection of the two tangents of the circumscribed circle of $\triangle ABC$ at $B$ and $C$. $T$ is the intersection of $DS$ and $HG$. And $M,N$ are the projection of $H$ on $TE,TF$ respectively. Prove that $M,N,E,F$ are concyclic.
2015 NIMO Summer Contest, 15
Suppose $x$ and $y$ are real numbers such that \[x^2+xy+y^2=2\qquad\text{and}\qquad x^2-y^2=\sqrt5.\] The sum of all possible distinct values of $|x|$ can be written in the form $\textstyle\sum_{i=1}^n\sqrt{a_i}$, where each of the $a_i$ is a rational number. If $\textstyle\sum_{i=1}^na_i=\frac mn$ where $m$ and $n$ are positive realtively prime integers, what is $100m+n$?
[i] Proposed by David Altizio [/i]
2010 ITAMO, 4
In a trapezium $ABCD$, the sides $AB$ and $CD$ are parallel and the angles $\angle ABC$ and $\angle BAD$ are acute. Show that it is possible to divide the triangle $ABC$ into 4 disjoint triangle $X_1. . . , X_4$ and the triangle $ABD$ into 4 disjoint triangles $Y_1,. . . , Y_4$ such that the triangles $X_i$ and $Y_i$ are congruent for all $i$.
1991 IMTS, 3
On a 8 x 8 board we place $n$ dominoes, each covering two adjacent squares, so that no more dominoes can be placed on the remaining squares. What is the smallest value of $n$ for which the above statement is true?
1992 IMO Shortlist, 6
Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]
2009 USAMTS Problems, 5
The cubic equation $x^3+2x-1=0$ has exactly one real root $r$. Note that $0.4<r<0.5$.
(a) Find, with proof, an increasing sequence of positive integers $a_1 < a_2 < a_3 < \cdots$ such that
\[\frac{1}{2}=r^{a_1}+r^{a_2}+r^{a_3}+\cdots.\]
(b) Prove that the sequence that you found in part (a) is the unique increasing sequence with the above property.
2007 Princeton University Math Competition, 7
A set of points $P_i$ [i]covers[/i] a polygon if for every point in the polygon, a line can be drawn inside the polygon to at least one $P_i$. Points $A_1, A_2, \cdots, A_n$ in the plane form a $2007$-gon, not necessarily convex. Find the minimum value of $n$ such that for any such polygon, we can pick $n$ points inside it that cover the polygon.
1986 All Soviet Union Mathematical Olympiad, 431
Given two points inside a convex dodecagon (twelve sides) situated $10$ cm far from each other. Prove that the difference between the sum of distances, from the point to all the vertices, is less than $1$ m for those points.
2017 Saudi Arabia BMO TST, 2
Solve the following equation in positive integers $x, y$: $x^{2017} - 1 = (x - 1)(y^{2015}- 1)$
Russian TST 2018, P3
Let $a < b$ be positive integers. Prove that there is a positive integer $n{}$ and a polynomial of the form \[\pm1\pm x\pm x^2\pm\cdots\pm x^n,\]divisible by the polynomial $1+x^a+x^b$.
JOM 2015 Shortlist, A2
Let $ a, b, c $ be positive real numbers greater or equal to $ 3 $. Prove that $$ 3(abc+b+2c)\ge 2(ab+2ac+3bc) $$ and determine all equality cases.
1997 Austrian-Polish Competition, 7
(a) Prove that $p^2 + q^2 + 1 > p(q + 1)$ for any real numbers $p, q$, .
(b) Determine the largest real constant $b$ such that the inequality $p^2 + q^2 + 1 \ge bp(q + 1)$ holds for all real numbers $p, q$
(c) Determine the largest real constant c such that the inequality $p^2 + q^2 + 1 \ge cp(q + 1)$ holds for all integers $p, q$.
2010 China Team Selection Test, 2
Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$,
and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.
1899 Eotvos Mathematical Competition, 1
The points $A_0, A_1, A_2, A_3, A_4$ divide a unit circle (circle of radius $1$) into five equal parts. Prove that the chords $A_0, A_1, A_0, A_2$ satisfy $$(A_0A_1 \cdot A_0A_2)^2= 5$$
2012 Princeton University Math Competition, A8
If $n$ is an integer such that $n \ge 2^k$ and $n < 2^{k+1}$, where $k = 1000$, compute the following:
$$n - \left( \lfloor \frac{n -2^0}{2^1} \rfloor + \lfloor \frac{n -2^1}{2^2} \rfloor + ...+ \lfloor \frac{n -2^{k-1}}{2^k} \rfloor \right)$$
2021 Iranian Geometry Olympiad, 1
With putting the four shapes drawn in the following figure together make a shape with at least two reflection symmetries.
[img]https://cdn.artofproblemsolving.com/attachments/6/0/8ace983d3d9b5c7f93b03c505430e1d2d189fd.png[/img]
[i]Proposed by Mahdi Etesamifard - Iran[/i]
1975 Kurschak Competition, 2
Prove or disprove: given any quadrilateral inscribed in a convex polygon, we can find a rhombus inscribed in the polygon with side not less than the shortest side of the quadrilateral.
2001 District Olympiad, 2
Consider the number $n=123456789101112\ldots 99100101$.
a)Find the first three digits of the number $\sqrt{n}$.
b)Compute the sum of the digits of $n$.
c)Prove that $\sqrt{n}$ isn't rational.
[i]Valer Pop[/i]
2006 Silk Road, 4
A family $L$ of 2006 lines on the plane is given in such a way that it doesn't contain
parallel lines and it doesn't contain three lines with a common point.We say that
the line $l_1\in L$ is [i]bounding[/i] the line $l_2\in L$,if all intersection points
of the line $l_2$ with other lines from $L$ lie on the one side of the line $l_1$.
Prove that in the family $L$ there are two lines $l$ and $l'$ such that the following
2 conditions are satisfied simultaneously:
[b]1)[/b] The line $l$ is bounding the line $l'$;
[b]2)[/b] the line $l'$ is not bounding the line $l$.
2020 German National Olympiad, 1
Let $k$ be a circle with center $M$ and let $B$ be another point in the interior of $k$. Determine those points $V$ on $k$ for which $\measuredangle BVM$ becomes maximal.
2008 Miklós Schweitzer, 5
Let $A$ be an infinite subset of the set of natural numbers, and denote by $\tau_A(n)$ the number of divisors of $n$ in $A$. Construct a set $A$ for which
$$\sum_{n\le x}\tau_A(n)=x+O(\log\log x)$$
and show that there is no set for which the error term is $o(\log\log x)$ in the above formula.
(translated by Miklós Maróti)
2008 Cuba MO, 4
Determine all functions $f : R \to R$ such that $f(xy + f(x)) =xf(y) + f(x)$ for all real numbers $x, y$.