IMO Problems and Solutions 2020

Problem 1.

Consider the convex quadrilateral ABCD. The point P is in the interior of ABCD. The following ratio equalities hold:
∠PAD : ∠PBA : ∠DPA = 1 : 2 : 3 = ∠CBP : ∠BAP : ∠BPC.
Prove that the following three lines meet in a point: the internal bisectors of angles ∠ADP and ∠PCB and the perpendicular bisector of segment AB.

Solution:

Math Olympiad

Let, ABCD be a convex Quadrilateral. The point is an interior point of ABCD.

∠PAD : ∠PBA : ∠DPA = x : 2x : 3x and ∠CBP : ∠BAP : ∠BPC = y : 2y : 3y [Figure: 1]

Now, draw a line segment PE such that PE = AE and draw PF such that PF = BF. [Figure: 2]

In ∆AEP, ∠DEP = ∠PAE + ∠APE = x + x = 2x. [Exterior angle property]

Therefore, points A, E, P, B is cyclic. [Cyclic Quadrilateral]

In ∆BFP, ∠CPF = ∠PBF + ∠BPF = y + y = 2y. [Exterior angle property]

Therefore, points A, B, F, P is cyclic. [Cyclic Quadrilateral]

So, we have A, E, P, F, B is lies on the same circle.

Now, ∆PED and ∆PCE are isosceles triangle. Therefore, the bisector of CG & DH are passes through the circle center. Here, DH is the internal bisectors of angles ∠ADP, CG is the internal bisectors of angles ∠PCB.

The bisectors of ∠ADP, ∠PCB & MN (perpendicular bisector of AB) meet each other at a point O. [Circle property]

Hence, Internal bisectors of angles ∠ADP and ∠PCB and the perpendicular bisector of segment AB meet in a point.

Problem 2.

The real numbers a, b, c, d is such that a ≥ b ≥ c ≥ d > 0 and a + b + c + d = 1.
Prove that
(a + 2b + 3c + 4d) aa bb cc dd < 1.
Problem 3.

There are 4n pebbles of weights 1, 2, 3, …, 4n. Each pebble is coloured in one of n colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied:
• The total weights of both piles are the same.
• Each pile contains two pebbles of each colour.

Problem 4.

There is an integer n > 1. There are n2 stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, A and B, operates k cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The k cable cars of A have k different starting points and k different finishing points, and a cable car
which starts higher also finishes higher. The same conditions hold for B. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer k for which one can guarantee that there are two stations
that are linked by both companies.
Problem 5.

A deck of n > 1 cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards. For which n does it follow that the numbers on the cards are all equal?
Problem 6.

Prove that there exists a positive constant c such that the following statement is true: Consider an integer n > 1, and a set S of n points in the plane such that the distance between
any two different points in S is at least 1. It follows that there is a line separating S such that the distance from any point of S to is at least cn1/3. (A line ℓ separates a set of points S if some segment joining two points in S crosses .)
Note. Weaker results with cn1/3 replaced by cn may be awarded points depending on the value of the constant α > 1/3.