Found problems: 85335
1986 IMO Longlists, 27
In an urn there are n balls numbered $1, 2, \cdots, n$. They are drawn at random one by one without replacement and the numbers are recorded. What is the probability that the resulting random permutation has only one local maximum?
A term in a sequence is a local maximum if it is greater than all its neighbors.
2016 BMT Spring, 10
An $m \times n$ rectangle is tiled with $\frac{mn}{2}$ $1 \times 2$ dominoes. The tiling is such that whenever the rectangle is partitioned into two smaller rectangles, there exists a domino that is part of the interior of both rectangles. Given $mn > 2$, what is the minimum possible value of $mn$?
For instance, the following tiling of a $4 \times 3$ rectangle doesn’t work because we can partition along the line shown, but that doesn’t necessarily mean other $4 \times 3$ tilings don’t work.
[img]https://cdn.artofproblemsolving.com/attachments/d/3/cb1fed407e45463950542b3cc64185892afdc5.png[/img]
2008 Junior Balkan Team Selection Tests - Romania, 4
Determine the maximum possible real value of the number $ k$, such that
\[ (a \plus{} b \plus{} c)\left (\frac {1}{a \plus{} b} \plus{} \frac {1}{c \plus{} b} \plus{} \frac {1}{a \plus{} c} \minus{} k \right )\ge k\]
for all real numbers $ a,b,c\ge 0$ with $ a \plus{} b \plus{} c \equal{} ab \plus{} bc \plus{} ca$.
1982 IMO Longlists, 57
Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that
\[\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),\]
where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than $4.$
2005 Sharygin Geometry Olympiad, 20
Let $I$ be the center of the sphere inscribed in the tetrahedron $ABCD, A ', B', C ', D'$ be the centers of the spheres circumscribed around the tetrahedra $IBCD, ICDA, IDAB, IABC$, respectively. Prove that the sphere circumscribed around $ABCD$ lies entirely inside the circumscribed around $A'B'C'D '$.
2004 South africa National Olympiad, 4
Let $A_1$ and $B_1$ be two points on the base $AB$ of isosceles triangle $ABC$ (with $\widehat{C}>60^\circ$) such that $\widehat{A_1CB_1}=\widehat{BAC}$. A circle externally tangent to the circumcircle of triangle $\triangle A_1B_1C$ is tangent also to rays $CA$ and $CB$ at points $A_2$ and $B_2$ respectively. Prove that $A_2B_2=2AB$.
2010 Kazakhstan National Olympiad, 4
It is given that for some $n \in \mathbb{N}$ there exists a natural number $a$, such that $a^{n-1} \equiv 1 \pmod{n}$ and that for any prime divisor $p$ of $n-1$ we have $a^{\frac{n-1}{p}} \not \equiv 1 \pmod{n}$.
Prove that $n$ is a prime.
2014 Harvard-MIT Mathematics Tournament, 1
Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12}$, find all possible values of $x$.
1997 German National Olympiad, 1
Prove that there are no perfect squares $a,b,c$ such that $ab-bc = a$.
1941 Moscow Mathematical Olympiad, 075
Prove that $1$ plus the product of any four consecutive integers is a perfect square.
2013 Hanoi Open Mathematics Competitions, 3
What is the largest integer not exceeding $8x^3 +6x - 1$, where $x =\frac12 \left(\sqrt[3]{2+\sqrt5} + \sqrt[3]{2-\sqrt5}\right)$ ?
(A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.
1952 Polish MO Finals, 6
In a circular tower with an internal diameter of $ 2$ m, there is a spiral staircase with a height of $ 6$ m. The height of each stair step is $ 0.15$ m. In the horizontal projection, the steps form adjacent circular sections with an angle of $ 18^\circ $. The narrower ends of the steps are mounted in a round pillar with a diameter of $ 0.64$ m, the axis of which coincides with the axis of the tower. Calculate the greatest length of a straight rod that can be moved up these stairs from the bottom to the top (do not take into account the thickness of the rod or the thickness of the boards from which the stairs are made).
2012 Middle European Mathematical Olympiad, 4
Let $ p>2 $ be a prime number. For any permutation $ \pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) ) $ of the set $ S = \{ 1, 2, \cdots , p \} $, let $ f( \pi ) $ denote the number of multiples of $ p $ among the following $ p $ numbers:
\[ \pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p) \]
Determine the average value of $ f( \pi) $ taken over all permutations $ \pi $ of $ S $.
2000 AMC 8, 19
Three circular arcs of radius $5$ units bound the region shown. Arcs $AB$ and $AD$ are quarter-circles, and arc $BCD$ is a semicircle. What is the area, in square units, of the region?
[asy]
pair A,B,C,D;
A = (0,0);
B = (-5,5);
C = (0,10);
D = (5,5);
draw(arc((-5,0),A,B,CCW));
draw(arc((0,5),B,D,CW));
draw(arc((5,0),D,A,CCW));
label("$A$",A,S);
label("$B$",B,W);
label("$C$",C,N);
label("$D$",D,E);
[/asy]
$\text{(A)}\ 25 \qquad \text{(B)}\ 10 + 5\pi \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 50 + 5\pi \qquad \text{(E)}\ 25\pi$
2015 Purple Comet Problems, 30
Cindy and Neil wanted to paint the side of a staircase in the six-square pattern shown below so that each
of the six squares is painted a solid color, and no two squares that share an edge are the same color. Cindy
draws all n patterns that can be colored using the four colors red, white, blue, and green. Neil looked at
these patterns and claimed that k of the patterns Cindy drew were incorrect because two adjacent squares
were colored with the same color. This is because Neil is color-blind and cannot distinguish red from
green. Find $n + k$. For picture go to http://www.purplecomet.org/welcome/practice
2004 National High School Mathematics League, 1
If the equation $x^2+4x\cos\theta+\cot\theta=0$ has a repeated root, where $\theta$ is an acute angle, then the radian of $\theta$ is
$\text{(A)}\frac{\pi}{6}\qquad\text{(B)}\frac{\pi}{12}\text{ or }\frac{5\pi}{12}\qquad\text{(C)}\frac{\pi}{6}\text{ or }\frac{5\pi}{12}\qquad\text{(D)}\frac{\pi}{12}$
2014 IMS, 4
Let $(X,d)$ be a metric space and $f:X \to X$ be a function such that $\forall x,y\in X : d(f(x),f(y))=d(x,y)$.
$\text{a})$ Prove that for all $x \in X$, $\lim_{n \rightarrow +\infty} \frac{d(x,f^n(x))}{n}$ exists, where $f^n(x)$ is $\underbrace{f(f(\cdots f(x)}_{n \text{times}} \cdots ))$.
$\text{b})$ Prove that the amount of the limit does [b][u]not[/u][/b] depend on choosing $x$.
1996 IMO Shortlist, 7
Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1$ and
\[ f \left( x \plus{} \frac{13}{42} \right) \plus{} f(x) \equal{} f \left( x \plus{} \frac{1}{6} \right) \plus{} f \left( x \plus{} \frac{1}{7} \right).\]
Prove that $ f$ is a periodic function (that is, there exists a non-zero real number $ c$ such $ f(x\plus{}c) \equal{} f(x)$ for all $ x \in \mathbb{R}$).
2019 Korea Junior Math Olympiad., 1
Each integer coordinates are colored with one color and at least 5 colors are used to color every integer coordinates. Two integer coordinates $(x, y)$ and $(z, w)$ are colored in the same color if $x-z$ and $y-w$ are both multiples of 3. Prove that there exists a line that passes through exactly three points when five points with different colors are chosen randomly.
1980 VTRMC, 1
Let $*$ denote a binary operation on a set $S$ with the property that $$(w*x)*(y*z) = w * z$$ for all $w,x,y,z\in S.$ Show
(a) If $a*b=c,$ then $c*c = c.$
(b) If $a*b=c,$ then $a*x=c*x$ for all $x\in S.$
2020 SAFEST Olympiad, 5
Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\]
Define the set $A$ by
\[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\]
Prove that, if $A$ is not empty, then
\[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]
2002 All-Russian Olympiad Regional Round, 10.8
what maximal number of colors can we use in order to color squares of 10 *10 square board so that each
row or column contains squares of at most 5 different colors?
1997 South africa National Olympiad, 6
Six points are connected in pairs by lines, each of which is either red or blue. Every pair of points is joined. Determine whether there must be a closed path having four sides all of the same colour. (A path is closed if it begins and ends at the same point.)
2015 Azerbaijan JBMO TST, 1
Let $x,y$ and $z$ be non-negative real numbers satisfying the equation $x+y+z=xyz$. Prove that $2(x^2+y^2+z^2)\geq3(x+y+z)$.
2023 Thailand October Camp, 1
Let $C$ be a finite set of chords in a circle such that each chord passes through the midpoint of some other chord. Prove that any two of these chords intersect inside the circle.