MathDB

3

Part of 2012 Iran Team Selection Test

Problems(5)

Tangent circles and parallelogram

Source: Iran TST 2012-Second exam-1st day-P3

5/12/2012
Suppose ABCDABCD is a parallelogram. Consider circles w1w_1 and w2w_2 such that w1w_1 is tangent to segments ABAB and ADAD and w2w_2 is tangent to segments BCBC and CDCD. Suppose that there exists a circle which is tangent to lines ADAD and DCDC and externally tangent to w1w_1 and w2w_2. Prove that there exists a circle which is tangent to lines ABAB and BCBC and also externally tangent to circles w1w_1 and w2w_2.
Proposed by Ali Khezeli
geometryparallelogramgeometric transformationhomothetytrigonometrygeometry proposed
a subset of points in the plane

Source: Iran TST 2012 -first day- problem 3

4/23/2012
Let nn be a positive integer. Let SS be a subset of points on the plane with these conditions:
i)i) There does not exist nn lines in the plane such that every element of SS be on at least one of them.
ii)ii) for all X∈SX \in S there exists nn lines in the plane such that every element of S−XS - {X} be on at least one of them.
Find maximum of ∣S∣\mid S\mid.
Proposed by Erfan Salavati
algebrapolynomialfunctionvectorinductioncombinatorics proposedcombinatorics
Geometric inequality

Source: Iran TST 2012-First exam-2nd day-P6

4/24/2012
The pentagon ABCDEABCDE is inscirbed in a circle ww. Suppose that wa,wb,wc,wd,wew_a,w_b,w_c,w_d,w_e are reflections of ww with respect to sides AB,BC,CD,DE,EAAB,BC,CD,DE,EA respectively. Let A′A' be the second intersection point of wa,wew_a,w_e and define B′,C′,D′,E′B',C',D',E' similarly. Prove that 2≤SA′B′C′D′E′SABCDE≤3,2\le \frac{S_{A'B'C'D'E'}}{S_{ABCDE}}\le 3, where SXS_X denotes the surface of figure XX.
Proposed by Morteza Saghafian, Ali khezeli
inequalitiesgeometrygeometric transformationreflectionlimitanalytic geometrycomplex numbers
Ugly equation in integers!

Source: Iran TST 2012-Third exam-1st day-P3

5/15/2012
Find all integer numbers xx and yy such that: (y3+xy−1)(x2+x−y)=(x3−xy+1)(y2+x−y).(y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y).
Proposed by Mahyar Sefidgaran
number theory proposednumber theory
Two triangles having the same orthocenter

Source: Iran TST 2012-Third exam-2nd day-P6

5/16/2012
Let OO be the circumcenter of the acute triangle ABCABC. Suppose points A′,B′A',B' and C′C' are on sides BC,CABC,CA and ABAB such that circumcircles of triangles AB′C′,BC′A′AB'C',BC'A' and CA′B′CA'B' pass through OO. Let ℓa\ell_a be the radical axis of the circle with center B′B' and radius B′CB'C and circle with center C′C' and radius C′BC'B. Define ℓb\ell_b and ℓc\ell_c similarly. Prove that lines ℓa,ℓb\ell_a,\ell_b and ℓc\ell_c form a triangle such that it's orthocenter coincides with orthocenter of triangle ABCABC.
Proposed by Mehdi E'tesami Fard
geometrycircumcirclegeometric transformationreflectionincenterpower of a pointradical axis