IMO - SL-2008 - G4

In an acute triangle ABC segments BE and CF are altitudes. Two circles passing through the point A anf F and tangent to the line BC at the points P and Q so that B lies between C and Q. Prove that lines PE and QF intersect on the circumcircle of triangle AEF.

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IGO2016-P3

Find all positive integers N such that there exists a triangle which can be dissected into N similar quadrilaterals.

the duplication of the cube

There are three  famous  problems  of  antiquity.

1)the duplication  of  the  cube

2)the  trisection  of  the  general  angle

3)the  squaring  of the  circle

Which there wasn't any solution for them.

But now there are some solutions for these problems.

In this post I want you to find the answer of problem 1.

Help:

1)for this you want to make cube root of 2.

So try to draw a×f(2).(note that f(x)=cube root of x)

Contest#01

Hi everybody,

I'm holding a contest and if you want you can attend in it.

If you like you can send me your solutions by clicking here.

After 1 week the solutions will be on website and you can download it only if you are a user.so you can register here.

Download questions.

For sending solutions and other please click on following link:

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IGO 2016-P5

Let the circles ω and ωintersect in points A and B. Tangent to circle ω at A intersects ωin C and tangent to circle ωat A intersects ω in D. Suppose that the internal bisector of ∠CAD intersects ω and ωat E and F, respectively, and the external bisector of ∠CAD intersects ω and ωin X and Y, respectively. Prove that the perpendicular bisector of XY is tangent to the circumcircle of triangle BEF

3rd Iranian Geometry Olympiad-P4

Let ω be the circumcircle of right-angled triangle ABC (∠A = 90◦). Tangent to ω at point A intersects the line BC in point P. Suppose that M is the midpoint of (the smaller) arc AB, and PM intersects ω for the second time in Q. Tangent to ω at point Q intersects AC in K. Prove that ∠PKC = 90◦.

Proposed by Davood Vakili.

Tiling with two kinds of figures

We call rectagle of the size 1 x 2 domino. Rectangle of the 2 x 3 removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated.

It requires to tile rectangle of size 2008x 2010 by using dominoes and tetraminoes. What is the minimal number of dominoes should be used

IMO2005-P5

Let ABCD be a xed convex quadrilateral with BC = DA and BC not parallel with DA. Let two variable points E and F lie of the sides BC and DA, respectively and satisfy BE = DF. The lines AC and BDmeet at P, the lines BD and EF meet at Q, the lines EF and AC meet at R. Prove that the circumcircles of the triangles PQR, as E and F vary, have a common point other than P.

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To see solution please click here.

IMO 2005-P1

Six points are chosen on the sides of an equilateral triangle ABC: A1, A2 on BC, B1, B2 on CA and C1, C2 on AB, such that they are the vertices of a convex hexagon A1A2B1B2C1C2 with equal side lengths. Prove that the lines A1B2;B1C2 and C1A2 are concurrent.

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To see solutions click here.

34th United States of America Mathematical Olympiad-P3

Let ABC be an acute-angled triangle, and let P and Q be two points on side BC. Construct point C1 in such a way that convex quadrilateral APBC1 is cyclic, QC1 CA, and C1 and Q lie on opposite sides of line AB. Construct point B1 in such a way that convex quadrilateral APCB1 is cyclic, QB1 BA, and B1 and Q lie on opposite sides of line AC. Prove that points B1,C1,P, and Q lie on a circle.

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To see solution click here