Found problems: 25757
1993 Flanders Math Olympiad, 2
A jeweler covers the diagonal of a unit square with small golden squares in the following way:
- the sides of all squares are parallel to the sides of the unit square
- for each neighbour is their sidelength either half or double of that square (squares are neighbour if they share a vertex)
- each midpoint of a square has distance to the vertex of the unit square equal to $\dfrac12, \dfrac14, \dfrac18, ...$ of the diagonal. (so real length: $\times \sqrt2$)
- all midpoints are on the diagonal
(a) What is the side length of the middle square?
(b) What is the total gold-plated area?
[img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=281[/img]
2014 Purple Comet Problems, 20
Triangle $ABC$ has a right angle at $C$. Let $D$ be the midpoint of side $\overline{AC}$, and let $E$ be the intersection of $\overline{AC}$ and the bisector of $\angle ABC$. The area of $\triangle ABC$ is $144$, and the area of $\triangle DBE$ is $8$. Find $AB^2$.
1986 Federal Competition For Advanced Students, P2, 5
Show that for every convex $ n$-gon $ ( n \ge 4)$, the arithmetic mean of the lengths of its sides is less than the arithmetic mean of the lengths of all its diagonals.
2017 Harvard-MIT Mathematics Tournament, 2
Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $\ell$ be a line passing through two sides of triangle $ABC$. Line $\ell$ cuts triangle $ABC$ into two figures, a triangle and a quadrilateral, that have equal perimeter. What is the maximum possible area of the triangle?
2014 Benelux, 4
Let $ABCD$ be a square. Consider a variable point $P$ inside the square for which $\angle BAP \ge 60^\circ.$ Let $Q$ be the intersection of the line $AD$ and the perpendicular to $BP$ in $P$. Let $R$ be the intersection of the line $BQ$ and the perpendicular to $BP$ from $C$.
[list]
[*] [b](a)[/b] Prove that $|BP|\ge |BR|$
[*] [b](b)[/b] For which point(s) $P$ does the inequality in [b](a)[/b] become an equality?[/list]
1992 Tournament Of Towns, (347) 5
An angle with vertex $O$ and a point $A$ inside it are placed on a plane. Points $M$ and $N$ are chosen on different sides of the angle so that the angles $CAM$ and $CAN$ are equal. Prove that the straight line $MN$ always passes through a fixed point (or is always parallel to a fixed line).
(S Tokarev)
1998 Belarus Team Selection Test, 1
Two circles $S_1$ and $S_2$ intersect at different points $P,Q$. The arc of $S_1$ lying inside $S_2$ measures $2a$ and the arc of $S_2$ lying inside $S_1$ measures $2b$. Let $T$ be any point on $S_1$. Let $R,S$ be another points of intersection of $S_2$ with $TP$ and $TQ$ respectively. Let $a+2b<\pi$ . Find the locus of the intersection points of $PS$ and $RQ$.
S.Shikh
1999 Turkey MO (2nd round), 5
In an acute triangle $\vartriangle ABC$ with circumradius $R$, altitudes $\overline{AD},\overline{BE},\overline{CF}$ have lengths ${{h}_{1}},{{h}_{2}},{{h}_{3}}$, respectively. If ${{t}_{1}},{{t}_{2}},{{t}_{3}}$ are lengths of the tangents from $A,B,C$, respectively, to the circumcircle of triangle $\vartriangle DEF$, prove that
$\sum\limits_{i=1}^{3}{{{\left( \frac{t{}_{i}}{\sqrt{h{}_{i}}} \right)}^{2}}\le }\frac{3}{2}R$.
2019 Romania Team Selection Test, 2
Let $ABC$ be an acute triangle with $AB<BC$. Let $I$ be the incenter of $ABC$, and let $\omega$ be the circumcircle of $ABC$. The incircle of $ABC$ is tangent to the side $BC$ at $K$. The line $AK$ meets $\omega$ again at $T$. Let $M$ be the midpoint of the side $BC$, and let $N$ be the midpoint of the arc $BAC$ of $\omega$. The segment $NT$ intersects the circumcircle of $BIC$ at $P$. Prove that $PM\parallel AK$.
2013 Sharygin Geometry Olympiad, 4
A point $F$ inside a triangle $ABC$ is chosen so that $\angle AFB = \angle BFC = \angle CFA$. The line passing through $F$ and perpendicular to $BC$ meets the median from $A$ at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the points $A_1, B_1$, and $C_1$ are three vertices of some regular hexagon, and that the three remaining vertices of that hexagon lie on the sidelines of $ABC$.
2013 Iran Team Selection Test, 12
Let $ABCD$ be a cyclic quadrilateral that inscribed in the circle $\omega$.Let $I_{1},I_{2}$ and $r_{1},r_{2}$ be incenters and radii of incircles of triangles $ACD$ and $ABC$,respectively.assume that $r_{1}=r_{2}$. let $\omega'$ be a circle that touches $AB,AD$ and touches $\omega$ at $T$. tangents from $A,T$ to $\omega$ meet at the point $K$.prove that $I_{1},I_{2},K$ lie on a line.
2005 Sharygin Geometry Olympiad, 21
The planet Tetraincognito covered by ocean has the shape of a regular tetrahedron with an edge of $900$ km. What area of the ocean will the tsunami' cover $2$ hours after the earthquake with the epicenter in
a) the center of the face,
b) the middle of the edge,
if the tsunami propagation speed is $300$ km / h?
1999 USAMTS Problems, 5
In $\triangle ABC$, $AC>BC$, $CM$ is the median, and $CH$ is the altitude emanating from $C$, as shown in the figure on the right. Determine the measure of $\angle MCH$ if $\angle ACM$ and $\angle BCH$ each have measure $17^\circ$.
[asy]
size(200);
defaultpen(linewidth(0.8));
pair A=origin,B=(10,0),C=(7,5),M=(5,0),H=(7,0);
draw(A--C--B--cycle^^H--C--M);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,NE);
label("$M$",M,NW);
label("$H$",H,NE);
[/asy]
May Olympiad L1 - geometry, 2006.2
A rectangle of paper of $3$ cm by $9$ cm is folded along a straight line, making two opposite vertices coincide. In this way a pentagon is formed. Calculate it's area.
2023 CMIMC Geometry, 5
In trapezoid $ABCD, AB=3, BC=2, CD=5,$ and $\angle B = \angle C = 90^{\circ}.$ The angle bisectors of $\angle A$ and $\angle D$ intersect at a point $P$ in the interior of $ABCD.$ Compute $BP^2+CP^2.$
[i]Proposed by Kyle Lee[/i]
2014 IMS, 9
Let $G$ be a $2n-$vertices simple graph such that in any partition of the set of vertices of $G$ into two $n-$vertices sets $V_1$ and $V_2$, the number of edges from a vertex in $V_1$ to another vertex in $V_1$ is equal to the number of edges from a vertex in $V_2$ to another vertex in $V_2$. Prove that all the vertices have equal degrees.
1965 Vietnam National Olympiad, 2
$AB$ and $CD$ are two fixed parallel chords of the circle $S$. $M$ is a variable point on the circle. $Q$ is the intersection of the lines $MD$ and $AB$. $X$ is the circumcenter of the triangle $MCQ$.
Find the locus of $X$.
What happens to $X$ as $M$ tends to
(1) $D$,
(2) $C$?
Find a point $E$ outside the plane of $S$ such that the circumcenter of the tetrahedron $MCQE$ has the same locus as $X$.
Indonesia MO Shortlist - geometry, g7
Given a convex quadrilateral $ABCD$, such that $OA = \frac{OB \cdot OD}{OC + CD}$ where $O$ is the intersection of the two diagonals. The circumcircle of triangle $ABC$ intersects $BD$ at point $Q$. Prove that $CQ$ bisects $\angle ACD$
1997 Junior Balkan MO, 4
Determine the triangle with sides $a,b,c$ and circumradius $R$ for which $R(b+c) = a\sqrt{bc}$.
[i]Romania[/i]
2000 IberoAmerican, 3
A convex hexagon is called [i]pretty[/i] if it has four diagonals of length 1, such that their endpoints are all the vertex of the hexagon.
($a$) Given any real number $k$ with $0<k<1$ find a [i]pretty[/i] hexagon with area equal to $k$
($b$) Show that the area of any [i]pretty[/i] hexagon is less than 1.
2009 AMC 12/AHSME, 20
A convex polyhedron $ Q$ has vertices $ V_1,V_2,\ldots,V_n$, and $ 100$ edges. The polyhedron is cut by planes $ P_1,P_2,\ldots,P_n$ in such a way that plane $ P_k$ cuts only those edges that meet at vertex $ V_k$. In addition, no two planes intersect inside or on $ Q$. The cuts produce $ n$ pyramids and a new polyhedron $ R$. How many edges does $ R$ have?
$ \textbf{(A)}\ 200\qquad
\textbf{(B)}\ 2n\qquad
\textbf{(C)}\ 300\qquad
\textbf{(D)}\ 400\qquad
\textbf{(E)}\ 4n$
1979 IMO Longlists, 43
Let $a, b, c$ denote the lengths of the sides $BC,CA,AB$, respectively, of a triangle $ABC$. If $P$ is any point on the circumference of the circle inscribed in the triangle, show that $aPA^2+bPB^2+cPC^2$ is constant.
1998 All-Russian Olympiad Regional Round, 8.4
A set of $n\ge 9$ points is given on the plane. For any 9 it points can be selected from all circles so that all these points end up on selected circles. Prove that all n points lie on two circles
2022 Iran-Taiwan Friendly Math Competition, 3
Let $ABC$ be a scalene triangle with $I$ be its incenter. The incircle touches $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. $Y$, $Z$ are the midpoints of $DF$, $DE$ respectively, and $S$, $V$ are the intersections of lines $YZ$ and $BC$, $AD$, respectively. $T$ is the second intersection of $\odot(ABC)$ and $AS$. $K$ is the foot from $I$ to $AT$. Prove that $KV$ is parallel to $DT$.
[i]Proposed by ltf0501[/i]
2018 Hanoi Open Mathematics Competitions, 9
There are three polygons and the area of each one is $3$. They are drawn inside a square of area $6$. Find the greatest value of $m$ such that among those three polygons, we can always find two polygons so that the area of their overlap is not less than $m$.