4
votes
1answer
694 views

IMO 2014 problem 3, first day

Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such ...
2
votes
2answers
70 views

A question about 4 concyclic points

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles ...
1
vote
1answer
50 views

Parallelogram constructed through medians

Bdmo In $\Delta ABC$, Medians AD and CF intersect at P.Let Q be any point on AC.Construct QM and QN parallel to AD and CF respectively.Now the line joining M and N intersects CF and T and AD at ...
2
votes
3answers
87 views

Bisector of angle formed at the orthocentre passes through the circumcentre

BdMO 2012 In an acute angled triangle $ABC$, $\angle A= 60$. We have to prove that the bisector of one of the angles formed by the altitudes drawn from $B$ and $C$ passes through the center of the ...
1
vote
2answers
44 views

Maximize the inradius given the base and the area of the triangle

BdMO 2013 Secondary: A triangle has base of length 8 and area 12. What is the radius of the largest circle that can be inscribed in this triangle? Let $A,r,s$ denote the area,inradius and ...
1
vote
2answers
51 views

Connecting square vertexes with minimal road

I have four cities in $A=(0,0),B=(1,0),C=(1,1),D=(0,1)$. I am asked to build the shortest motorway to connect the cities. How can I do that? I was thinking that first I need some compactness argument ...
3
votes
1answer
124 views

I am looking for a proof of the “ begonia theorem”.

Let $D$, $E$, $F$ be points on respective (extended) sides $\overleftrightarrow{BC}$, $\overleftrightarrow{CA}$, $\overleftrightarrow{AB}$ of $\triangle ABC$, such that $\overleftrightarrow{AD}$, ...
2
votes
1answer
93 views

Area of triangle inside triangle

In triangle $ABC$ we choose 3 points $D,E,F$, such that $\overline{AD} = \frac 13 \overline{AB}, \overline{BE} = \frac 13 \overline{BC}, \overline{CF} = \frac 13 \overline{CA}$. Draw segments ...
1
vote
1answer
175 views

Maximizing the perimeter of a triangle inside a square

BdMO 2014: We have a square $ABCD$ of side length 5.We take a point $E$ on $AD$ and $F$ on $AB$ so that $\angle FCE=45^\circ$. What can be the maximum perimeter of $\triangle AEF$? I can ...
2
votes
3answers
148 views

Moscow Math Olympiad 1973

In every polyhedron there is at least one pair of faces with the same number of sides. Solution: Let $N$ be the greatest number of sides in a face of a given polyhedron. Then the number of ...
0
votes
0answers
60 views

Ratios in a rhombus

NOTE: I am NOT looking for a full answer,just a hint. Last problem on this question. BdMO 2013 Chittagong: Let $ABCD$ be a rhombus.Let $G$ be a point outside the rhombus such that GE is ...
2
votes
3answers
467 views

Book with lots of geometry theorems

I want to study geometry and was looking for some book that has lots of theorems and covers almost all Euclidean geometry that is needed for High School and Maths Olympiads. Thanks.
0
votes
2answers
83 views

Inequality regarding areas of triangles

BdMO Nationals 2013: There is a point O inside ∆ABC. After joining A,O; B,O and C,O extend those line and they will intersect BC, AC and AB at points D, E and F respectively. ...
3
votes
4answers
119 views

A Question regarding radius of circumcircle and sides of a triangle

NOTE: I am looking for a hint,not the whole solution. A question from BdMO Nationals 2012 Given triangle $ABC$, the square $PQRS$ is drawn such that $P$,$Q$ are on BC, $R$ is on $CA$ and $S$ on ...
0
votes
1answer
90 views

Figuring out an angle in an isosceles triangle

A problem from BdMO 2013: Let $ABC$ be an isoscles triangle with $AB=AC$.The bisector of $\angle B$ meets $AC$ at $D$.Given that $BC=BD+AD$,we need to figure out $\angle A$. If we consider ...
4
votes
1answer
202 views

Online Math Open Contest 2 Problem 50

In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be ...
2
votes
1answer
129 views

Lines $ MF, DE, QR$ in a triangle intersect at one point

In a triangle ABC, a circle is inscribed with center in $I$. The inscribed circle touches sides $BC,CA,AB$ in $D,E,F$ respectively. Join the point $C$ and $F$, $B$ and $E$. Let $Q$ and $R$ be the ...
4
votes
1answer
147 views

Straightedge-only construction of a perpendicular

There is a circle in the plane with a drawn diameter. Given a point inside the circle (not on the diameter or the circle), draw the perpendicular from the point to the diameter using only a ...
33
votes
3answers
1k views

A generalization of IMO 1983 problem 6

Note: This question has a bounty that will expire in just a few days. Let $a,b,c$ and $d$ be the lengths of the sides of a quadrilateral. Show that $$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$$ ...
0
votes
3answers
239 views

geometry problem on circles from a competition

Triangle $\triangle ABC$ is an equilateral triangle whose side is $16$. A circle meets the sides of the triangle at $6$ points: it intersects $AC$ at $G$ and $F$ and $|AG|=2$, $|GF|=13$, $|FC|=1$. ...
1
vote
1answer
258 views

Existence of Gergonne point, without Ceva theorem

The intersection at one point (called Gergonne point) of the lines from vertices of a triangle to contact points of the inscribed circle can be proved immediately using Ceva's theorem. Is there a ...
1
vote
2answers
379 views

A geometry problem from maths competition

I have a problem understanding the proof. Given an acute angle $A$. Choose an arbitrary point $P$ from the bisector of $A$ and another point $B$ from the side of angle $A$. Draw a line $l$ going ...
7
votes
2answers
517 views

Geometry IMO 1988

(IMO 1988/1) Consider two circles of radii $R$ and $r$ $(R > r)$ with the same center. Let $P$ be a fixed point on the smaller circle and $B$ a variable point on the larger circle. The line $BP$ ...
2
votes
2answers
631 views

Euclidean Geometry Intersection of Circles

Two circles intersect in the Cartesian Coordinate system at points $A$ and $B$. Point $A$ lies on the line $y=3$. Point $B$ lies on the line $y=12$. These two circles are also tangent to the x-axis at ...