Found problems: 85335
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.
2019 Yasinsky Geometry Olympiad, p1
The sports ground has the shape of a rectangle $ABCD$, with the angle between the diagonals $AC$ and $BD$ is equal to $60^o$ and $AB >BC$. The trainer instructed Andriyka to go first $10$ times on the route $A-C-B-D-A$, and then $15$ more times along the route $A-D-A$. Andriyka performed the task, moving a total of $4.5$ km. What is the distance $AC$?
2020 Princeton University Math Competition, B1
Find all pairs of natural numbers $(n, k)$ with the following property:
Given a $k\times k$ array of cells, such that every cell contains one integer, there always exists a path from the left to the right edges such that the sum of the numbers on the path is a multiple of $n$.
Note: A path from the left to the right edge is a sequence of cells of the array $a_1, a_2, ... , a_m$ so that $a_1$ is a cell of the leftmost column, $a_m$ is the cell of the rightmost column, and $a_{i}$, $a_{i+1}$ share an edge for all $i = 1, 2, ... , m -1$.
2006 Purple Comet Problems, 9
Moving horizontally and vertically from point to point along the lines in the diagram below, how many routes are there from point $A$ to point $B$ which consist of six horizontal moves and six vertical moves?
[asy]
for(int i=0; i<=6;++i)
{
draw((i,i)--(6,i),black);
draw((i,i)--(i,0),black);
for(int a=i; a<=6;++a)
{
dot((a,i));
}
}
label("A",(0,0),W);
label("B",(6,6),N);
[/asy]
2007 China National Olympiad, 1
Let $O, I$ be the circumcenter and incenter of triangle $ABC$. The incircle of $\triangle ABC$ touches $BC, CA, AB$ at points $D, E, F$ repsectively. $FD$ meets $CA$ at $P$, $ED$ meets $AB$ at $Q$. $M$ and $N$ are midpoints of $PE$ and $QF$ respectively. Show that $OI \perp MN$.
2015 JHMT, 3
Consider a triangular pyramid $ABCD$ with equilateral base $ABC$ of side length $1$. $AD = BD =CD$ and $\angle ADB = \angle BDC = \angle ADC = 90^o$ . Find the volume of $ABCD$.
2024 USAMTS Problems, 4
During a lecture, each of $26$ mathematicians falls asleep exactly once, and stays asleep for a nonzero amount of time. Each mathematician is awake at the moment the lecture starts, and the moment the lecture finishes. Prove that there are either $6$ mathematicians such that no two are asleep at the same time, or $6$ mathematicians such that there is some point in time during which all $6$ are asleep.
1970 IMO Longlists, 34
In connection with a convex pentagon $ABCDE$ we consider the set of ten circles, each of which contains three of the vertices of the pentagon on its circumference. Is it possible that none of these circles contains the pentagon? Prove your answer.
1988 AMC 12/AHSME, 12
Each integer $1$ through $9$ is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. Which digit is most likely to be the units digit of the [b]sum[/b] of Jack's integer and Jill's integer?
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ \text{each digit is equally likely} $
1995 Turkey MO (2nd round), 5
Let $t(A)$ denote the sum of elements of a nonempty set $A$ of integers, and define $t(\emptyset)=0$. Find a set $X$ of positive integers such that for every integers $k$ there is a unique ordered pair of disjoint subsets $(A_{k},B_{k})$ of $X$ such that $t(A_{k})-t(B_{k}) = k$.
2004 Bundeswettbewerb Mathematik, 2
Consider a triangle whose sidelengths $a$, $b$, $c$ are integers, and which has the property that one of its altitudes equals the sum of the two others.
Then, prove that $a^2+b^2+c^2$ is a perfect square.