Found problems: 1679
2020 Candian MO, 3#
okay this one is from Prof. Mircea Lascu from Zalau, Romaniaand Prof. V. Cartoaje from Ploiesti, Romania. It goes like this: given being a triangle ABC for every point M inside we construct the points A[size=67]M[/size], B[size=67]M[/size], C[size=67]M[/size] on the circumcircle of the triangle ABC such that A, A[size=67]M[/size], M are collinear and so on. Find the locus of these points M for which the area of the triangle A[size=67]M[/size] B[size=67]M[/size] C[size=67]M[/size] is bigger than the area of the triangle ABC.
2009 AIME Problems, 15
Let $ \overline{MN}$ be a diameter of a circle with diameter $ 1$. Let $ A$ and $ B$ be points on one of the semicircular arcs determined by $ \overline{MN}$ such that $ A$ is the midpoint of the semicircle and $ MB\equal{}\frac35$. Point $ C$ lies on the other semicircular arc. Let $ d$ be the length of the line segment whose endpoints are the intersections of diameter $ \overline{MN}$ with the chords $ \overline{AC}$ and $ \overline{BC}$. The largest possible value of $ d$ can be written in the form $ r\minus{}s\sqrt{t}$, where $ r$, $ s$, and $ t$ are positive integers and $ t$ is not divisible by the square of any prime. Find $ r\plus{}s\plus{}t$.
2002 AIME Problems, 2
The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\frac{1}{2}\left(\sqrt{p}-q\right),$ where $p$ and $q$ are positive integers. Find $p+q.$
[asy]
size(250);real x=sqrt(3);
int i;
draw(origin--(14,0)--(14,2+2x)--(0,2+2x)--cycle);
for(i=0; i<7; i=i+1) {
draw(Circle((2*i+1,1), 1)^^Circle((2*i+1,1+2x), 1));
}
for(i=0; i<6; i=i+1) {
draw(Circle((2*i+2,1+x), 1));
}[/asy]
2004 Romania Team Selection Test, 14
Let $O$ be a point in the plane of the triangle $ABC$. A circle $\mathcal{C}$ which passes through $O$ intersects the second time the lines $OA,OB,OC$ in $P,Q,R$ respectively. The circle $\mathcal{C}$ also intersects for the second time the circumcircles of the triangles $BOC$, $COA$ and $AOB$ respectively in $K,L,M$.
Prove that the lines $PK,QL$ and $RM$ are concurrent.
1995 National High School Mathematics League, 4
Color all points on a plane in red or blue. Prove that there exists two similar triangles, their similarity ratio is $1995$, and apexes of both triangles are in the same color.
Indonesia Regional MO OSP SMA - geometry, 2014.4
Let $\Gamma$ be the circumcircle of triangle $ABC$. One circle $\omega$is tangent to $\Gamma$ at $A$ and tangent to $BC$ at $N$. Suppose that the extension of $AN$ crosses $\Gamma$ again at $E$. Let $AD$ and $AF$ be respectively the line of altitude $ABC$ and diameter of $\Gamma$, show that $AN \times AE = AD \times AF = AB \times AC$
1988 AMC 12/AHSME, 2
Triangles $ABC$ and $XYZ$ are similar, with $A$ corresponding to $X$ and $B$ to $Y$. If $AB=3$, $BC=4$, and $XY=5$, then $YZ$ is:
$ \textbf{(A)}\ 3\frac 3 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 6\frac 1 4 \qquad \textbf{(D)}\ 6\frac 2 3 \qquad \textbf{(E)}\ 8$
1978 Canada National Olympiad, 4
The sides $AD$ and $BC$ of a convex quadrilateral $ABCD$ are extended to meet at $E$. Let $H$ and $G$ be the midpoints of $BD$ and $AC$, respectively. Find the ratio of the area of the triangle $EHG$ to that of the quadrilateral $ABCD$.
2008 Sharygin Geometry Olympiad, 5
(I.Bogdanov) A section of a regular tetragonal pyramid is a regular pentagon. Find the ratio of its side to the side of the base of the pyramid.
2010 AIME Problems, 12
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $ 8: 7$. Find the minimum possible value of their common perimeter.
1968 AMC 12/AHSME, 11
If an arc of $60^\circ$ on circle I has the same length as an arc of $45^\circ$ on circle II, the ratio of the area of circle I to that of circle II is:
$\textbf{(A)}\ 16:9 \qquad
\textbf{(B)}\ 9:16 \qquad
\textbf{(C)}\ 4:3 \qquad
\textbf{(D)}\ 3:4 \qquad
\textbf{(E)}\ \text{None of these} $
2006 Switzerland Team Selection Test, 3
An airport contains 25 terminals which are two on two connected by tunnels. There is exactly 50 main tunnels which can be traversed in the two directions, the others are with single direction. A group of four terminals is called [i]good[/i] if of each terminal of the four we can arrive to the 3 others by using only the tunnels connecting them. Find the maximum number of good groups.
2011 Morocco National Olympiad, 4
Let $ABC$ be a triangle with area $1$ and $P$ the middle of the side $[BC]$. $M$ and $N$ are two points of $[AB]-\left \{ A,B \right \} $ and $[AC]-\left \{ A,C \right \}$ respectively such that $AM=2MB$ and$CN=2AN$. The two lines $(AP)$ and $(MN)$ intersect in a point $D$. Find the area of the triangle $ADN$.
2007 AMC 10, 23
A pyramid with a square base is cut by a plane that is parallel to its base and is $ 2$ units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid?
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ 2 \plus{} \sqrt{2}\qquad
\textbf{(C)}\ 1 \plus{} 2\sqrt{2}\qquad
\textbf{(D)}\ 4\qquad
\textbf{(E)}\ 4 \plus{} 2\sqrt{2}$
2014 Iran MO (3rd Round), 5
Can an infinite set of natural numbers be found, such that for all triplets $(a,b,c)$ of it we have $abc + 1 $ perfect square?
(20 points )
2010 Tournament Of Towns, 1
Alex has a piece of cheese. He chooses a positive number $a\neq 1$ and cuts the piece into several pieces one by one. Every time he chooses a piece and cuts it in the same ratio $1:a.$ His goal is to divide the cheese into two piles of equal masses. Can he do it?
2006 Federal Math Competition of S&M, Problem 3
Show that for an arbitrary tetrahedron there are two planes such that the ratio of the areas of the projections of the tetrahedron onto the two planes is not less than $\sqrt2$.
Kyiv City MO Juniors Round2 2010+ geometry, 2016.8.1
In a right triangle, the point $O$ is the center of the circumcircle. Another circle of smaller radius centered at the point $O$ touches the larger leg and the altitude drawn from the top of the right angle. Find the acute angles of a right triangle and the ratio of the radii of the circumscribed and smaller circles.
2007 Princeton University Math Competition, 8
What is the area of the region defined by $x^2+3y^2 \le 4$ and $y^2+3x^2 \le 4$?
2025 Euler Olympiad, Round 1, 4
Given any arc $AB$ on a circle and points $C$ and $D$ on segment $AB$, such that $$CD = DB = 2AC.$$ Find the ratio $\frac{CM}{MD}$, where $M$ is a point on arc $AB$, such that $\angle CMD$ is maximized.
[img]https://i.imgur.com/NfjRpgP.png[/img]
[i]
Proposed by Andria Gvaramia, Georgia [/i]
1966 IMO, 6
Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points in the interiors of the sides $ BC$, $ CA$, $ AB$ of this triangle. Prove that the area of at least one of the three triangles $ AQR$, $ BRP$, $ CPQ$ is less than or equal to one quarter of the area of triangle $ ABC$.
[i]Alternative formulation:[/i] Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Prove that
$ \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$,
where the abbreviation $ \left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $ P_1P_2P_3$.
1992 China Team Selection Test, 1
A triangle $ABC$ is given in the plane with $AB = \sqrt{7},$ $BC = \sqrt{13}$ and $CA = \sqrt{19},$ circles are drawn with centers at $A,B$ and $C$ and radii $\frac{1}{3},$ $\frac{2}{3}$ and $1,$ respectively. Prove that there are points $A',B',C'$ on these three circles respectively such that triangle $ABC$ is congruent to triangle $A'B'C'.$
2002 AMC 12/AHSME, 7
If an arc of $ 45^\circ$ on circle $ A$ has the same length as an arc of $ 30^\circ$ on circle $ B$, then the ratio of the area of circle $ A$ to the area of circle $ B$ is
$ \textbf{(A)}\ \frac {4}{9} \qquad \textbf{(B)}\ \frac {2}{3} \qquad \textbf{(C)}\ \frac {5}{6} \qquad \textbf{(D)}\ \frac {3}{2} \qquad \textbf{(E)}\ \frac {9}{4}$
2011 Bosnia Herzegovina Team Selection Test, 1
In triangle $ABC$ it holds $|BC|= \frac{1}{2}(|AB|+|AC|)$. Let $M$ and $N$ be midpoints of $AB$ and $AC$, and let $I$ be the incenter of $ABC$. Prove that $A, M, I, N$ are concyclic.
2010 Tournament Of Towns, 2
Pete has an instrument which can locate the midpoint of a line segment, and also the point which divides the line segment into two segments whose lengths are in a ratio of $n : (n + 1)$, where $n$ is any positive integer. Pete claims that with this instrument, he can locate the point which divides a line segment into two segments whose lengths are at any given rational ratio. Is Pete right?