Hooray!
The team of Russian schoolchildren took second place!
The gold medals were won by Danila Demin from Sochi (36 points) and Alexey Lvov from Novosibirsk (36 points). Silver was taken by Ivan Gaidai-Turlov (25), Anton Sadovnichy (29) from Moscow, Danil Sibgatullin (29) from Moscow and Kazan, and Maxim Turevsky (30) from St. Petersburg.
The absolute winner of the Olympiad in the individual competition was a schoolboy from China Jinmin Li, who scored the maximum possible 42 points.
I recently published the texts of problems and some of them were solved by the readers of Habr in the comments.
Under the cut are some interesting statistics on the results of the Olympiad .
Our fellows!
Team Results
China is leading the way. The gap between Russia and the United States is 2 points.
It is interesting that the United States has a leader with a pronounced Asian surname, and the deputy. leader - with a pronounced Ukrainian name and surname.
Individual results
Chinese participants (1, 2, 3) by a wide margin. Representatives of many countries scored 36 points (4th place).
The absolute champion Jinmin Li from Chongqing. Respect.
Tasks
Problem 1
Inside the convex quadrilateral ABCD, there is a point P such that the equalities
∠PAD: ∠PBA: ∠DPA = 1: 2: 3 = ∠CBP: ∠BAP: ∠BPC hold.
Prove that the following three straight lines intersect at one point: the inner bisectors of the angles ∠ADP and ∠PCB and the midpoint perpendicular to the segment AB.
Problem 2
Given real numbers a, b, c, d such that a> b> c> d> 0 and a + b + c + d = 1.
Prove that
(a + 2b + 3c + 4d) a a b b c c d d <1.
Solution fromnovoselov here
Problem 3
There are 4n pebbles with masses 1, 2, 3, ..., 4n . Each of the pebbles is colored in one of n colors, and there are 4 pebbles of each color.
Prove that the stones can be divided into two heaps of equal total weight so that each heap contains two stones of each color.
Decision fromcelen here
Decision fromnovoselov here
Task 4
An integer n> 1 is given . There are n 2 funicular stations on the mountain slope at different heights. Each of the two funicular companies A and B owns k lifts. Each lift carries out a regular direct transfer from one of the stations to another, higher station. The k transfers of company A start at k different stations; they also end at k different stations; with a transfer that starts above and ends above. The same conditions are met for company B. We will say that two stations are connectedfunicular company, if you can get from the lower station to the upper one using one or more transfers of this company (other transfers between stations are prohibited). Find the smallest k for which there are known to be two stations connected by both companies.
Problem 5
There are n> 1 cards, each of which contains a positive integer.
It turned out that for any two cards the arithmetic mean of the numbers written on them is equal to the geometric mean of the numbers written on the cards of a certain set consisting of one or more cards. For which n does it follow that all the numbers written on the cards are equal?
Decision fromnovoselov here
Problem 6
Prove that there exists a positive constant c for which the following statement holds:
Let S be a set of n> 1 points of the plane in which the distance between any two points is at least 1. Then there is a line ℓ separating the set S such that the distance from any points S to ℓ are at least cn −1/3 .
(A straight line ℓ separates the set of points S if it intersects some segment whose ends belong to S.)
Remark. Weaker results with cn −1/3 replaced by cn −α can be estimated depending on the value of the constant α> 1/3 .
Statistics for solving the 6th problem. The Chinese showed themselves excellently. The Frenchman Vladimir Ivanov also had a good result.