MathDB

2

Part of 2000 Iran MO (3rd Round)

Problems(6)

A1o1\o2o3

Source: 17-th Iranian Mathematical Olympiad 1999/2000

12/14/2005
Isosceles triangles A3A1O2A_3A_1O_2 and A1A2O3A_1A_2O_3 are constructed on the sides of a triangle A1A2A3A_1A_2A_3 as the bases, outside the triangle. Let O1O_1 be a point outside ΔA1A2A3\Delta A_1A_2A_3 such that ∠O1A3A2=12∠A1O3A2\angle O_1A_3A_2 =\frac 12\angle A_1O_3A_2 and ∠O1A2A3=12∠A1O2A3\angle O_1A_2A_3 =\frac 12\angle A_1O_2A_3. Prove that A1O1⊥O2O3A_1O_1\perp O_2O_3, and if TT is the projection of O1O_1 onto A2A3A_2A_3, then A1O1O2O3=2O1TA2A3\frac{A_1O_1}{O_2O_3} = 2\frac{O_1T}{A_2A_3}.
geometrypower of a pointradical axisgeometry proposed
A,b,c

Source: 17-th Iranian Mathematical Olympiad 1999/2000

12/14/2005
Suppose that a,b,ca, b, c are real numbers such that for all positive numbers x1,x2,…,xnx_1,x_2,\dots,x_n we have (1n∑i=1nxi)a(1n∑i=1nxi2)b(1n∑i=1nxi3)c≥1(\frac{1}{n}\sum_{i=1}^nx_i)^a(\frac{1}{n}\sum_{i=1}^nx_i^2)^b(\frac{1}{n}\sum_{i=1}^nx_i^3)^c\ge 1 Prove that vector (a,b,c)(a, b, c) is a nonnegative linear combination of vectors (−2,1,0)(-2,1,0) and (−1,2,−1)(-1,2,-1).
vectorinequalities proposedinequalities
space geometry 2

Source: 17-th Iranian Mathematical Olympiad 1999/2000

2/29/2004
Call two circles in three-dimensional space pairwise tangent at a point P P if they both pass through P P and lines tangent to each circle at P P coincide. Three circles not all lying in a plane are pairwise tangent at three distinct points. Prove that there exists a sphere which passes through the three circles.
geometry3D geometryspherecircumcircleIran
Set

Source: 17-th Iranian Mathematical Olympiad 1999/2000

12/14/2005
Let AA and BB be arbitrary finite sets and let f:A⟶Bf: A\longrightarrow B and g:B⟶Ag: B\longrightarrow A be functions such that gg is not onto. Prove that there is a subset SS of AA such that AS=g(Bf(S))\frac{A}{S}=g(\frac{B}{f(S)}).
functionalgebra proposedalgebra
Not hard

Source: Iran 2000

5/31/2004
Find all f:N ⟶\longrightarrow N that: a) f(m)=1⟺m=1f(m)=1 \Longleftrightarrow m=1 b) d=gcd(m,n)f(m⋅n)=f(m)⋅f(n)f(d)d=gcd(m,n) f(m\cdot n)= \frac{f(m)\cdot f(n)}{f(d)} c) f2000(m)=f(m) f^{2000}(m)=f(m)
functionnumber theory proposednumber theory
Mn=ae+af

Source: 17-th Iranian Mathematical Olympiad 1999/2000

12/14/2005
Circles C1 C_1 and C2 C_2 with centers at O1 O_1 and O2 O_2 respectively meet at points A A and B B. The radii O1B O_1B and O2B O_2B meet C1 C_1 and C2 C_2 at F F andE E. The line through B B parallel to EF EF intersects C1 C_1 again at M M and C2 C_2 again at N N. Prove that MN \equal{} AE \plus{} AF.
geometryrectangletrapezoidgeometry proposed