Found problems: 85335
2023 Durer Math Competition (First Round), 1
Find all positive integers $n$ such that $$\lfloor \sqrt{n} \rfloor +
\left\lfloor \frac{n}{\lfloor \sqrt{n} \rfloor} \right \rfloor> 2\sqrt{n}.$$
If $k$ is a real number, then $\lfloor k \rfloor$ means the floor of $k$, this is the greatest integer less than or equal to $k$.
2011 Sharygin Geometry Olympiad, 22
Let $CX, CY$ be the tangents from vertex $C$ of triangle $ABC$ to the circle passing through the midpoints of its sides. Prove that lines $XY , AB$ and the tangent to the circumcircle of $ABC$ at point $C$ concur.
2002 Spain Mathematical Olympiad, Problem 2
In the triangle $ABC$, $A'$ is the foot of the altitude to $A$, and $H$ is the orthocenter.
$a)$ Given a positive real number $k = \frac{AA'}{HA'}$ , find the relationship between the angles $B$ and $C$, as a function of $k$.
$b)$ If $B$ and $C$ are fixed, find the locus of the vertice $A$ for any value of $k$.
2024 AMC 12/AHSME, 22
The figure below shows a dotted grid $8$ cells wide and $3$ cells tall consisting of $1''\times1''$ squares. Carl places $1$-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks? [asy]
size(6cm);
for (int i=0; i<9; ++i) {
draw((i,0)--(i,3),dotted);
}
for (int i=0; i<4; ++i){
draw((0,i)--(8,i),dotted);
}
for (int i=0; i<8; ++i) {
for (int j=0; j<3; ++j) {
if (j==1) {
label("1",(i+0.5,1.5));
}}}
[/asy] $\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) }162\qquad\textbf{(E) }196$
1996 Tournament Of Towns, (514) 1
Consider three edges $a, b, c$ of a cube such that no two of these edges lie in one plane. Find the locus of points inside the cube which are equidistant from $a$, $b$ and $c$.
(V Proizvolov,)
2015 Korea Junior Math Olympiad, 4
Reals $a,b,c,x,y$ satisfy $a^2+b^2+c^2=x^2+y^2=1$. Find the maximum value of $$(ax+by)^2+(bx+cy)^2$$
2023 Math Prize for Girls Problems, 8
For a positive integer $n$, let $p(n)$ denote the number of distinct prime numbers that divide evenly into $n$. Determine the number of solutions, in positive integers $n$, to the inequality $\log_4 n \le p(n)$.
2021 Czech-Polish-Slovak Junior Match, 1
Consider a trapezoid $ABCD$ with bases $AB$ and $CD$ satisfying $| AB | > | CD |$. Let $M$ be the midpoint of $AB$. Let the point $P$ lie inside $ABCD$ such that $| AD | = | PC |$ and $| BC | = | PD |$. Prove that if $| \angle CMD | = 90^o$, then the quadrilaterals $AMPD$ and $BMPC$ have the same area.
2018 IFYM, Sozopol, 7
The rows $x_n$ and $y_n$ of positive real numbers are such that:
$x_{n+1}=x_n+\frac{1}{2y_n}$ and $y_{n+1}=y_n+\frac{1}{2x_n}$
for each positive integer $n$.
Prove that at least one of the numbers $x_{2018}$ and $y_{2018}$ is bigger than 44,9
1948 Putnam, B3
Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$ for all $n \in N$.
2011 Morocco National Olympiad, 2
Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$
($p$ and $q$ are two real parameters).
2023 Azerbaijan National Mathematical Olympiad, 2
Let $I$ be the incenter in the acute triangle $ABC.$ Rays $BI$ and $CI$ intersect the circumcircle of triangle $ABC$ at points $S$ and $T,$ respectively. The segment $ST$ intersects the sides $AB$ and $AC$ at points $K$ and $L,$ respectively. Prove that $AKIL$ is a rhombus.
2011 AIME Problems, 13
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\tfrac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive integers, and $r+s+t<1000$. Find $r+s+t$.
2019 Pan-African Shortlist, C1
A pawn is a chess piece which attacks the two squares diagonally in front if it. What is the maximum number of pawns which can be placed on an $n \times n$ chessboard such that no two pawns attack each other?
2013 F = Ma, 16
A very large number of small particles forms a spherical cloud. Initially they are at rest, have uniform mass density per unit volume $\rho_0$, and occupy a region of radius $r_0$. The cloud collapses due to gravitation; the particles do not interact with each other in any other way.
How much time passes until the cloud collapses fully? (The constant $0.5427$ is actually $\sqrt{\frac{3 \pi}{32}}$.)
$\textbf{(A) } \frac{0.5427}{r_0^2 \sqrt{G \rho_0}}\\ \\
\textbf{(B) } \frac{0.5427}{r_0 \sqrt{G \rho_0}}\\ \\
\textbf{(C) } \frac{0.5427}{\sqrt{r_0} \sqrt{G \rho_0}}\\ \\
\textbf{(D) } \frac{0.5427}{\sqrt{G \rho_0}}\\ \\
\textbf{(E) } \frac{0.5427}{\sqrt{G \rho_0}}r_0$
2011 Federal Competition For Advanced Students, Part 2, 3
Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ touch each outside at point $Q$. The other endpoints of the diameters through $Q$ are $P$ on $k_1$ and $R$ on $k_2$.
We choose two points $A$ and $B$, one on each of the arcs $PQ$ of $k_1$. ($PBQA$ is a convex quadrangle.)
Further, let $C$ be the second point of intersection of the line $AQ$ with $k_2$ and let $D$ be the second point of intersection of the line $BQ$ with $k_2$.
The lines $PB$ and $RC$ intersect in $U$ and the lines $PA$ and $RD$ intersect in $V$ .
Show that there is a point $Z$ that lies on all of these lines $UV$.
2007 Cuba MO, 6
Let the triangle $ABC$ be acute. Let us take in the segment $BC$ two points $F$ and $G$ such that $BG > BF = GC$ and an interior point$ P$ to the triangle on the bisector of $\angle BAC$. Then are drawn through $P$, $PD\parallel AB$ and $PE \parallel AC$, $D \in AC$ and $E \in AB$, $\angle FEP = \angle PDG$. prove that $\vartriangle ABC$ is isosceles.
2000 German National Olympiad, 5
(a) Let be given $2n$ distinct points on a circumference, $n$ of which are red and $n$ are blue. Prove that one can join these points pairwise by $n$ segments so that no two segments intersect and the endpoints of each segments have different colors.
(b) Show that the statement from (a) remains valid if the points are in an arbitrary position in the plane so that no three of them are collinear.
Cono Sur Shortlist - geometry, 1993.10
Let $\omega$ be the unit circle centered at the origin of $R^2$. Determine the largest possible value for the radius of the circle inscribed to the triangle $OAP$ where $ P$ lies the circle and $A$ is the projection of $P$ on the axis $OX$.
2002 Tournament Of Towns, 4
$x,y,z\in\left(0,\frac{\pi}{2}\right)$ are given. Prove that:
\[ \frac{x\cos x+y\cos y+z\cos z}{x+y+z}\le \frac{\cos x+\cos y+\cos z}{3} \]
2003 District Olympiad, 1
Let $(G,\cdot)$ be a finite group with the identity element, $e$. The smallest positive integer $n$ with the property that $x^{n}= e$, for all $x \in G$, is called the [i]exponent[/i] of $G$.
(a) For all primes $p \geq 3$, prove that the multiplicative group $\mathcal G_{p}$ of the matrices of the form $\begin{pmatrix}\hat 1 & \hat a & \hat b \\ \hat 0 & \hat 1 & \hat c \\ \hat 0 & \hat 0 & \hat 1 \end{pmatrix}$, with $\hat a, \hat b, \hat c \in \mathbb Z \slash p \mathbb Z$, is not commutative and has [i]exponent[/i] $p$.
(b) Prove that if $\left( G, \circ \right)$ and $\left( H, \bullet \right)$ are finite groups of [i]exponents[/i] $m$ and $n$, respectively, then the group $\left( G \times H, \odot \right)$ with the operation given by $(g,h) \odot \left( g^\prime, h^\prime \right) = \left( g \circ g^\prime, h \bullet h^\prime \right)$, for all $\left( g,h \right), \, \left( g^\prime, h^\prime \right) \in G \times H$, has the [i]exponent[/i] equal to $\textrm{lcm}(m,n)$.
(c) Prove that any $n \geq 3$ is the [i]exponent[/i] of a finite, non-commutative group.
[i]Ion Savu[/i]
2006 China Second Round Olympiad, 2
Suppose $log_x (2x^2+x-1)>log_x 2-1$. Then the range of $x$ is
${ \textbf{(A)}\ \frac{1}{2}<x<1\qquad\textbf{(B)}\ x>\frac{1}{2} \text{and} x \not= 1\qquad\textbf{(C)}\ x>1\qquad\textbf{(D)}}\ 0<x<1\qquad $
2004 IMO Shortlist, 6
Let $P$ be a convex polygon. Prove that there exists a convex hexagon that is contained in $P$ and whose area is at least $\frac34$ of the area of the polygon $P$.
[i]Alternative version.[/i] Let $P$ be a convex polygon with $n\geq 6$ vertices. Prove that there exists a convex hexagon with
[b]a)[/b] vertices on the sides of the polygon (or)
[b]b)[/b] vertices among the vertices of the polygon
such that the area of the hexagon is at least $\frac{3}{4}$ of the area of the polygon.
[i]Proposed by Ben Green and Edward Crane, United Kingdom[/i]
1985 Traian Lălescu, 1.2
For the triangles of fixed perimeter, find the maximum value of the product of the radius of the incircle with the radius of the excircle.
1998 IMO Shortlist, 1
Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.