Found problems: 85335
2017-2018 SDPC, 3
Let $n > 2$ be a fixed positive integer. For a set $S$ of $n$ points in the plane, let $P(S)$ be the set of perpendicular bisectors of pairs of distinct points in $S$. Call set $S$ [i]complete[/i] if no two (distinct) pairs of points share the same perpendicular bisector, and every pair of lines in $P(S)$ intersects. Let $f(S)$ be the number of distinct intersection points of pairs of lines in $P(S)$.
(a) Find all complete sets $S$ such that $f(S) = 1$.
(b) Let $S$ be a complete set with $n$ points. Show that if $f(S)>1$, then $f(S) \geq n$.
2002 Iran Team Selection Test, 2
$n$ people (with names $1,2,\dots,n$) are around a table. Some of them are friends. At each step 2 friend can change their place. Find a necessary and sufficient condition for friendship relation between them that with these steps we can always reach to all of posiible permutations.
2015 AMC 12/AHSME, 21
A circle of radius $r$ passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16$. The set of all possible values of $r$ is an interval $[a,b)$. What is $a+b$?
$\textbf{(A) }5\sqrt2+4\qquad\textbf{(B) }\sqrt{17}+7\qquad\textbf{(C) }6\sqrt2+3\qquad\textbf{(D) }\sqrt{15}+8\qquad\textbf{(E) }12$
2022 ELMO Revenge, 1
In terms of $p$ and $k$, compute the number of solutions in positive integers to the equation
$ab+bc+ca=p^{2k}$ satisfying $a\leq b\leq c$ where $p$ is a fixed prime and $k$ is a fixed positive integer.
[i]Proposed by Alexander Wang[/i]
2001 Federal Competition For Advanced Students, Part 2, 3
Let be given a semicircle with the diameter $AB$, and points $C,D$ on it such that $AC = CD$. The tangent at $C$ intersects the line $BD$ at $E$. The line $AE$ intersects the arc of the semicircle at $F$. Prove that $CF < FD$.
2008 China Team Selection Test, 3
Determine the greatest positive integer $ n$ such that in three-dimensional space, there exist n points $ P_{1},P_{2},\cdots,P_{n},$ among $ n$ points no three points are collinear, and for arbitary $ 1\leq i < j < k\leq n$, $ P_{i}P_{j}P_{k}$ isn't obtuse triangle.
1988 IMO Shortlist, 25
A positive integer is called a [b]double number[/b] if its decimal representation consists of a block of digits, not commencing with 0, followed immediately by an identical block. So, for instance, 360360 is a double number, but 36036 is not. Show that there are infinitely many double numbers which are perfect squares.
2011 Math Prize For Girls Problems, 14
If $0 \le p \le 1$ and $0 \le q \le 1$, define $F(p, q)$ by
\[
F(p, q) = -2pq + 3p(1-q) + 3(1-p)q - 4(1-p)(1-q).
\]
Define $G(p)$ to be the maximum of $F(p, q)$ over all $q$ (in the interval $0 \le q \le 1$). What is the value of $p$ (in the interval $0 \le p \le 1$) that minimizes $G(p)$?
2001 Bundeswettbewerb Mathematik, 3
Let $ ABC$ an acute triangle with circumcircle center $ O.$ The line $ (BO)$ intersects the circumcircle again in $ D,$ and the extension of the altitude from $ A$ intersects the circle in $ E.$ Prove that the quadrilateral $ BECD$ and the triangle $ ABC$ have the same area.
2003 Bundeswettbewerb Mathematik, 3
Consider a cyclic quadrilateral $ABCD$, and let $S$ be the intersection of $AC$ and $BD$.
Let $E$ and $F$ the orthogonal projections of $S$ on $AB$ and $CD$ respectively.
Prove that the perpendicular bisector of segment $EF$ meets the segments $AD$ and $BC$ at their midpoints.
2020 AIME Problems, 1
Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$.
2018 Brazil Undergrad MO, 4
Consider the property that each a element of a group $G$ satisfies $a ^ 2 = e$, where e is the identity element of the group. Which of the following statements is not always valid for a
group $G$ with this property?
(a) $G$ is commutative
(b) $G$ has infinite or even order
(c) $G$ is Noetherian
(d) $G$ is vector space over $\mathbb{Z}_2$
Brazil L2 Finals (OBM) - geometry, 2012.3
Let be a triangle $ ABC $, the midpoint of the $ AC $ and $ N $ side, and the midpoint of the $ AB $ side. Let $ r $ and $ s $ reflect the straight lines $ BM $ and $ CN $ on the straight $ BC $, respectively. Also define $ D $ and $ E $ as the intersection of the lines $ r $ and $ s $ and the line $ MN $, respectively. Let $ X $ and $ Y $ be the intersection points between the circumcircles of the triangles $ BDM $ and $ CEN $, $ Z $ the intersection of the lines $ BE $ and $ CD $ and $ W $ the intersection between the lines $ r $ and $ s $. Prove that $ XY, WZ $, and $ BC $ are concurrents.
2011 Sharygin Geometry Olympiad, 1
Altitudes $AA_1$ and $BB_1$ of triangle ABC meet in point $H$. Line $CH$ meets the semicircle with diameter $AB$, passing through $A_1, B_1$, in point $D$. Segments $AD$ and $BB_1$ meet in point $M$, segments $BD$ and $AA_1$ meet in point $N$. Prove that the circumcircles of triangles $B_1DM$ and $A_1DN$ touch.
2017 Tournament Of Towns, 5
There is a set of control weights, each of them weighs a non-integer number of grams. Any
integer weight from $1$ g to $40$ g can be balanced by some of these weights (the control
weights are on one balance pan, and the measured weight on the other pan).What is the
least possible number of the control weights?
[i](Alexandr Shapovalov)[/i]
OMMC POTM, 2022 7
Find all ordered triples of positive integers $(a,b,c)$ where $$\left(a+\frac{1}{a}\right)\left(b+\frac{1}{b}\right)=c+\frac{1}{c}.$$
[i]Proposed by vsamc[/i]
2021 Sharygin Geometry Olympiad, 22
A convex polyhedron and a point $K$ outside it are given. For each point $M$ of a polyhedron construct a ball with diameter $MK$. Prove that there exists a unique point on a polyhedron which belongs to all such balls.
1982 All Soviet Union Mathematical Olympiad, 338
Cucumber river in the Flower city has parallel banks with the distance between them $1$ metre. It has some islands with the total perimeter $8$ metres. Mr. Know-All claims that it is possible to cross the river in a boat from the arbitrary point, and the trajectory will not exceed $3$ metres. Is he right?
2012 India IMO Training Camp, 1
Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\]
[i]Proposed by Warut Suksompong, Thailand[/i]
2024 Harvard-MIT Mathematics Tournament, 9
Compute the sum of all positive integers $n$ such that $n^2-3000$ is a perfect square.
2002 Junior Balkan Team Selection Tests - Romania, 1
Let $m,n > 1$ be integer numbers. Solve in positive integers $x^n+y^n = 2^m$.
VII Soros Olympiad 2000 - 01, 11.1
Prove that for any $a$ the function $y (x) = \cos (\cos x) + a \cdot \sin (\sin x)$ is periodic.
Find its smallest period in terms of $a$.
Kvant 2024, M2778
A parabola and a hyperbola are drawn on the coordinate plane. The graphs intersect at three points $A, B, C$ and the axis of the parabola is the asymptote of the hyperbola. Prove that the intersection point of the medians of the triangle $ABC$ lies on the axis of the parabola.
[i]From the folklore[/i]
1990 AMC 8, 17
A straight concrete sidewalk is to be $3$ feet wide, $60$ feet long, and $3$ inches thick. How many cubic yards of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards?
$ \text{(A)}\ 2\qquad\text{(B)}\ 5\qquad\text{(C)}\ 12\qquad\text{(D)}\ 20\qquad\text{(E)}\ \text{more than 20} $
2008 AIME Problems, 13
Let
\[ p(x,y) \equal{} a_0 \plus{} a_1x \plus{} a_2y \plus{} a_3x^2 \plus{} a_4xy \plus{} a_5y^2 \plus{} a_6x^3 \plus{} a_7x^2y \plus{} a_8xy^2 \plus{} a_9y^3.
\]Suppose that
\begin{align*}p(0,0) &\equal{} p(1,0) \equal{} p( \minus{} 1,0) \equal{} p(0,1) \equal{} p(0, \minus{} 1) \\&\equal{} p(1,1) \equal{} p(1, \minus{} 1) \equal{} p(2,2) \equal{} 0.\end{align*}
There is a point $ \left(\tfrac {a}{c},\tfrac {b}{c}\right)$ for which $ p\left(\tfrac {a}{c},\tfrac {b}{c}\right) \equal{} 0$ for all such polynomials, where $ a$, $ b$, and $ c$ are positive integers, $ a$ and $ c$ are relatively prime, and $ c > 1$. Find $ a \plus{} b \plus{} c$.