MathDB

2009 Iran MO (3rd Round)

Part of Iran MO (3rd Round)

Subcontests

(8)
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Rolling a Ball

A ball is placed on a plane and a point on the ball is marked. Our goal is to roll the ball on a polygon in the plane in a way that it comes back to where it started and the marked point comes to the top of it. Note that We are not allowed to rotate without moving, but only rolling. Prove that it is possible.
Time allowed for this problem was 90 minutes.

Iran(3rd round)2009

5-Two circles S1 S_1 and S2 S_2 with equal radius and intersecting at two points are given in the plane.A line l l intersects S1 S_1 at B,D B,D and S2 S_2 at A,C A,C(the order of the points on the line are as follows:A,B,C,D A,B,C,D).Two circles W1 W_1 and W2 W_2 are drawn such that both of them are tangent externally at S1 S_1 and internally at S2 S_2 and also tangent to l l at both sides.Suppose W1 W_1 and W2 W_2 are tangent.Then PROVE AB \equal{} CD.
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Inequality over functions

Does there exists two functions f,g:R→Rf,g :\mathbb{R}\rightarrow \mathbb{R} such that: ∀x≠y:∣f(x)−f(y)∣+∣g(x)−g(y)∣>1\forall x\not =y : |f(x)-f(y)|+|g(x)-g(y)|>1
Time allowed for this problem was 75 minutes.

Iran(3rd round)2009

4-Point P P is taken on the segment BC BC of the scalene triangle ABC ABC such that AP≠AB,AP≠AC AP \neq AB,AP \neq AC.l1,l2 l_1,l_2 are the incenters of triangles ABP,ACP ABP,ACP respectively. circles W1,W2 W_1,W_2 are drawn centered at l1,l2 l_1,l_2 and with radius equal to l1P,l2P l_1P,l_2P,respectively. W1,W2 W_1,W_2 intersects at P P and Q Q. W1 W_1 intersects AB AB and BC BC at Y_1( \mbox{the intersection closer to B}) and X1 X_1,respectively. W2 W_2 intersects AC AC and BC BC at Y_2(\mbox{the intersection closer to C}) and X2 X_2,respectively.PROVE THE CONCURRENCY OF PQ,X1Y1,X2Y2 PQ,X_1Y_1,X_2Y_2.
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Stable Permutation

Permutation π\pi of {1,…,n}\{1,\dots,n\} is called stable if the set {π(k)−k∣k=1,…,n}\{\pi (k)-k|k=1,\dots,n\} is consisted of exactly two different elements. Prove that the number of stable permutation of {1,…,n}\{1,\dots,n\} equals to σ(n)−τ(n)\sigma (n)-\tau (n) in which σ(n)\sigma (n) is the sum of positive divisors of nn and τ(n)\tau (n) is the number of positive divisors of nn.
Time allowed for this problem was 75 minutes.

Iran(3rd round)2009

2-There is given a trapezoid ABCD ABCD.We have the following properties: AD\parallel{}BC,DA \equal{} DB \equal{} DC,\angle BCD \equal{} 72^\circ. A point K K is taken on BD BD such that AD \equal{} AK,K \neq D.Let M M be the midpoint of CD CD.AM AM intersects BD BD at N N.PROVE BK \equal{} ND.
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Geometric Inequalities

Suppose n>2n>2 and let A1,…,AnA_1,\dots,A_n be points on the plane such that no three are collinear. (a) Suppose M1,…,MnM_1,\dots,M_n be points on segments A1A2,A2A3,…,AnA1A_1A_2,A_2A_3,\dots ,A_nA_1 respectively. Prove that if B1,…,BnB_1,\dots,B_n are points in triangles M2A2M1,M3A3M2,…,M1A1MnM_2A_2M_1,M_3A_3M_2,\dots ,M_1A_1M_n respectively then ∣B1B2∣+∣B2B3∣+⋯+∣BnB1∣≤∣A1A2∣+∣A2A3∣+⋯+∣AnA1∣|B_1B_2|+|B_2B_3|+\dots+|B_nB_1| \leq |A_1A_2|+|A_2A_3|+\dots+|A_nA_1| Where ∣XY∣|XY| means the length of line segment between XX and YY.
(b) If XX, YY and ZZ are three points on the plane then by HXYZH_{XYZ} we mean the half-plane that it's boundary is the exterior angle bisector of angle XYZ^\hat{XYZ} and doesn't contain XX and ZZ ,having YY crossed out. Prove that if C1,…,CnC_1,\dots ,C_n are points in HAnA1A2,HA1A2A3,…,HAn−1AnA1{H_{A_nA_1A_2},H_{A_1A_2A_3},\dots,H_{A_{n-1}A_nA_1}} then ∣A1A2∣+∣A2A3∣+⋯+∣AnA1∣≤∣C1C2∣+∣C2C3∣+⋯+∣CnC1∣|A_1A_2|+|A_2A_3|+\dots +|A_nA_1| \leq |C_1C_2|+|C_2C_3|+\dots+|C_nC_1|
Time allowed for this problem was 2 hours.

Iran(3rd round)2009

1-Let △ABC \triangle ABC be a triangle and (O) (O) its circumcircle. D D is the midpoint of arc BC BC which doesn't contain A A. We draw a circle W W that is tangent internally to (O) (O) at D D and tangent to BC BC.We draw the tangent AT AT from A A to circle W W.P P is taken on AB AB such that AP \equal{} AT.P P and T T are at the same side wrt A A.PROVE \angle APD \equal{} 90^\circ.