Found problems: 85335
2009 German National Olympiad, 2
Find all positive interger $ n$ so that $ n^3\minus{}5n^2\plus{}9n\minus{}6$ is perfect square number.
2021 Taiwan TST Round 3, C
A city is a point on the plane. Suppose there are $n\geq 2$ cities. Suppose that for each city $X$, there is another city $N(X)$ that is strictly closer to $X$ than all the other cities. The government builds a road connecting each city $X$ and its $N(X)$; no other roads have been built. Suppose we know that, starting from any city, we can reach any other city through a series of road.
We call a city $Y$ [i]suburban[/i] if it is $N(X)$ for some city $X$. Show that there are at least $(n-2)/4$ suburban cities.
[i]Proposed by usjl.[/i]
2012 Argentina National Olympiad Level 2, 6
Let $k$ be a positive integer. There are $2k$ pieces arranged in a row. A [i]move[/i] consists of swapping two adjacent pieces. Several moves must be made so that each piece passes through both the first and the last position. What is the minimum number of moves required to achieve this?
2014 Tuymaada Olympiad, 4
A $k\times \ell$ 'parallelogram' is drawn on a paper with hexagonal cells (it consists of $k$ horizontal rows of $\ell$ cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs.
Juniors) How many vertical sides can there be in this set?
Seniors) How many ways are there to do that?
[asy]
size(120);
defaultpen(linewidth(0.8));
path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle;
for(int i=0;i<=3;i=i+1)
{
for(int j=0;j<=2;j=j+1)
{
real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2;
draw(shift(shiftx,shifty)*hex);
}
}
[/asy]
[i](T. Doslic)[/i]
2017 Princeton University Math Competition, A1/B3
Triangle $ABC$ has $AB=BC=10$ and $CA=16$. The circle $\Omega$ is drawn with diameter $BC$. $\Omega$ meets $AC$ at points $C$ and $D$. Find the area of triangle $ABD$.
2000 Miklós Schweitzer, 6
Suppose the real line is decomposed into two uncountable Borel sets. Prove that a suitable translated copy of the first set intersects the second in an uncountable set.
2019 Sharygin Geometry Olympiad, 7
Let $AH_A$, $BH_B$, $CH_C$ be the altitudes of the acute-angled $\Delta ABC$. Let $X$ be an arbitrary point of segment $CH_C$, and $P$ be the common point of circles with diameters $H_CX$ and BC, distinct from $H_C$. The lines $CP$ and $AH_A$ meet at point $Q$, and the lines $XP$ and $AB$ meet at point $R$. Prove that $A, P, Q, R, H_B$ are concyclic.
2018 PUMaC Live Round, 1.3
Let a sequence be defined as follows: $a_0=1$, and for $n>0$, $a_n$ is $\tfrac{1}{3}a_{n-1}$ and is $\tfrac{1}{9}a_{n-1}$ with probability $\tfrac{1}{2}$. If the expected value of $\textstyle\sum_{n=0}^{\infty}a_n$ can be expressed in simplest form as $\tfrac{p}{q}$, what is $p+q$?
2011 South East Mathematical Olympiad, 1
In triangle $ABC$ , $AA_0,BB_0,CC_0$ are the angle bisectors , $A_0,B_0,C_0$are on sides $BC,CA,AB,$ draw $A_0A_1//BB_0,A_0A_2//CC_0$ ,$A_1$ lies on $AC$ ,$A_2$ lies on $AB$ , $A_1A_2$ intersects $BC$ at $A_3$.$B_3$ ,$C_3$ are constructed similarly.Prove that : $A_3,B_3,C_3$ are collinear.
2009 AMC 10, 3
Which of the following is equal to $ 1\plus{}\frac{1}{1\plus{}\frac{1}{1\plus{}1}}$?
$ \textbf{(A)}\ \frac{5}{4} \qquad
\textbf{(B)}\ \frac{3}{2} \qquad
\textbf{(C)}\ \frac{5}{3} \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 3$
1987 National High School Mathematics League, 5
Two sets $M=\{x,xy,\lg(xy)\},N=\{0,|x|,y\}$, if $M=N$, then $(x+\frac{1}{y})+(x^2+\frac{1}{y^2})+\cdots+(x^{2001}+\frac{1}{y^{2001}})=$________.
2004 Putnam, B2
Let $m$ and $n$ be positive integers. Show that
$\frac{(m+n)!}{(m+n)^{m+n}} < \frac{m!}{m^m}\cdot\frac{n!}{n^n}$
2019 PUMaC Combinatorics A, 2
Keith has $10$ coins labeled $1$ through $10$, where the $i$th coin has weight $2^i$. The coins are all fair, so the probability of flipping heads on any of the coins is $\tfrac{1}{2}$. After flipping all of the coins, Keith takes all of the coins which land heads and measures their total weight, $W$. If the probability that $137\le W\le 1061$ is $\tfrac{m}{n}$ for coprime positive integers $m,n$, determine $m+n$.
Estonia Open Senior - geometry, 2010.1.4
Circle $c$ passes through vertices $A$ and $B$ of an isosceles triangle $ABC$, whereby line $AC$ is tangent to it. Prove that circle $c$ passes through the circumcenter or the incenter or the orthocenter of triangle $ABC$.
1971 AMC 12/AHSME, 28
Nine lines parallel to the base of a triangle divide the other sides each into $10$ equal segments and the area into $10$ distinct parts. If the area of the largest of these parts is $38$, then the area of the original triangle is
$\textbf{(A) }180\qquad\textbf{(B) }190\qquad\textbf{(C) }200\qquad\textbf{(D) }210\qquad \textbf{(E) }240$
2004 AIME Problems, 11
A right circular cone has a base with radius 600 and height $200\sqrt{7}$. A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is 125, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is $375\sqrt{2}$. Find the least distance that the fly could have crawled.
Kyiv City MO 1984-93 - geometry, 1989.7.3
The student drew a triangle $ABC$ on the board, in which $AB>BC$. On the side $AB$ is taken point $D$ such that $BD = AC$. Let points $E$ and $F$ be the midpoints of the segments $AD$ and $BC$ respectively. Then the whole picture was erased, leaving only dots $E$ and $F$. Restore triangle $ABC$.
2023 HMNT, 21
An integer $n$ is chosen uniformly at random from the set $\{1, 2, \ldots, 2023!\}.$ Compute the probability that $$\gcd(n^n+50, n+1)=1.$$
2015 HMNT, 4
Chords $AB$ and $CD$ of a circle are perpendicular and intersect at a point $P$. If $AP = 6, BP = 12$, and $CD = 22$, find the area of the circle.
2017 IMO Shortlist, G3
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
2010 Contests, 523
Prove the following inequality.
\[ \ln \frac {\sqrt {2009} \plus{} \sqrt {2010}}{\sqrt {2008} \plus{} \sqrt {2009}} < \int_{\sqrt {2008}}^{\sqrt {2009}} \frac {\sqrt {1 \minus{} e^{ \minus{} x^2}}}{x}\ dx < \sqrt {2009} \minus{} \sqrt {2008}\]
2014 Paraguay Mathematical Olympiad, 4
Nair and Yuli play the following game:
$1.$ There is a coin to be moved along a horizontal array with $203$ cells.
$2.$ At the beginning, the coin is at the first cell, counting from left to right.
$3.$ Nair plays first.
$4.$ Each of the players, in their turns, can move the coin $1$, $2$, or $3$ cells to the right.
$5.$ The winner is the one who reaches the last cell first.
What strategy does Nair need to use in order to always win the game?
1993 Nordic, 2
A hexagon is inscribed in a circle of radius $r$. Two of the sides of the hexagon have length $1$, two have length $2$ and two have length $3$. Show that $r$ satisfies the equation $2r^3 - 7r - 3 = 0$.
2010 Tournament Of Towns, 7
A square is divided into congruent rectangles with sides of integer lengths. A rectangle is important if it has at least one point in common with a given diagonal of the square. Prove that this diagonal bisects the total area of the important rectangles
1998 Greece JBMO TST, 6
Prove that if the number $A = 111 \cdots 1$ ($n$ digits) is prime, then $n$ is prime. Is the converse true?