International Mathematical Olympiad for Schoolchildren 2015. Am I? If I want to? Will I decide?
I am writing an article for myself when I was a schoolboy.
On July 15, the 56th International Mathematical Olympiad ended in Chiang Mai (Thailand). The first place was taken by the United States, the second - China, the third - South Korea, the fourth - North Korea, the fifth - Vietnam, the sixth - Australia, the seventh - Iran.
')
At the International Mathematical Olympiad no more than six students represent each country. They must solve six problems, for each of which you can get a maximum of seven points. About half of the participants receive medals; between them gold, silver and bronze awards are distributed in a ratio of 1: 2: 3. The last time Russian schoolchildren took first place at this Olympiad in 2007.
There is an opinion that task 6 "was the most difficult task in the entire history of the International Mathematical Olympiad, which was first held in 1959".
Can anyone decide anything?
Presentation video:
Arrival day:
The opening ceremony:
Lectures:
At the 2015 Olympics, Russia was represented by Ivan Bochkov - scored 25 points for tasks, Ivan Frolov - 22, Nikita Gladkov - 25, Alexander Kuznetsov - 21, Ruslan Salimov - 23, Alexander Zimin - 25.
Zhuo Qun (Alex) Song (Canada) - 42 points for the tasks, Chenjie Yu (China) - 41, Junghun Ju (Republic of Korea) - 40, Alexander Gunning (Australia) - 36, Jaehyung Kim (Republic of Korea) - 35, Allen Liu (USA) - 35, David Stoner (USA) - 35
Authorship of tasks:
Problem 1 proposed by Netherlands
Problem 2 proposed by Serbia
Problem 3 proposed by Ukraine
Problem 4 proposed by Silouanos Brazitikos and Vangelis Psychas, Greece
Problem 5 proposed by Dorlir Ahmeti, Albania
Problem 6 proposed by Ross Atkins and Ivan Guo, Australia
Original tasks in English here (PDF) .
Problem 1. A finite set of S points on the plane will be called balanced if for any different points A and B from the set S there is a point C from the set S such that AC = BC. A set S will be called eccentric if for any three different points A, B and C from the set S there is no point P from the set S such that PA = PB = PC.
(a) Prove that for any integer n ≥ 3 there exists a balanced set consisting of n points.
(b) Find all integers n ≥ 3 for which there exists a balanced eccentric set consisting of n points.
Task 2. Find all triples (a; b; c) of positive integers such that each of the numbers
ab - c, bc - a, ca - b is a power of two.
(The power of two is a number of the form 2 n , where n is a nonnegative integer.)
Problem 3. Let ABC be an acute triangle in which AB> AC. Let Γ be the circle circumscribed near it, H its orthocenter, and F the base of the height lowered from the vertex A. Let M be the midpoint of side BC. Let Q be a point on the circle Γ such that ∠HQA = 90 °, and K a point on the circle Γ such that ∠HKQ = 90 °. Let points A, B, C, K and Q are different and lie on the circle Γ in the indicated order.
Prove that the circles described near the triangles KQH and FKM are tangent to each other.
Time to work: 4 hours 30 minutes
Each task is estimated at 7 points.
(Boxers in the subject)
Problem 4. Let Ω be the circle described near the triangle ABC, and the point O its center. Circle Γ with center A intersects segment BC at points D and E so that points B, D, E and C are all different and lie on line BC in the indicated order. Let F and G be the intersection points of the circles G and Ω; moreover, the points A, F, B, C, and G lie on Ω in the indicated order. Let K be the second intersection point of the circle described near the triangle BDF and the segment AB. Let L be the second intersection point of the circle described near the triangle CGE and the segment CA. Let the lines FK and GL be different and intersect at the point X.
Prove that the point X lies on the line AO.
Problem 5. Let ℝ be the set of all real numbers. Find all functions f : ℝ → ℝ,
satisfying equality
f (x + f (x + y)) + f (xy) = x + f (x + y) + yf (x)
for all real numbers x and y .
Task 6. The sequence of a1, a2, ... integers satisfies the following conditions:
(i) 1 ≤ a j ≤ 2015 for all j ≥ 1;
(ii) k + a k ≠ L + a L for all 1 ≤ k < L .
Prove that there are two positive integers b and N such that:
For all integers m and n satisfying the condition n> m ≥ N.
Time to work: 4 hours 30 minutes:
Each task is estimated at 7 points.
P.S. All schoolchildren-mathematicians-Olympiad hello.
On July 15, the 56th International Mathematical Olympiad ended in Chiang Mai (Thailand). The first place was taken by the United States, the second - China, the third - South Korea, the fourth - North Korea, the fifth - Vietnam, the sixth - Australia, the seventh - Iran.
')
At the International Mathematical Olympiad no more than six students represent each country. They must solve six problems, for each of which you can get a maximum of seven points. About half of the participants receive medals; between them gold, silver and bronze awards are distributed in a ratio of 1: 2: 3. The last time Russian schoolchildren took first place at this Olympiad in 2007.
There is an opinion that task 6 "was the most difficult task in the entire history of the International Mathematical Olympiad, which was first held in 1959".
Can anyone decide anything?
Presentation video:
Arrival day:
The opening ceremony:
Lectures:
Russia
At the 2015 Olympics, Russia was represented by Ivan Bochkov - scored 25 points for tasks, Ivan Frolov - 22, Nikita Gladkov - 25, Alexander Kuznetsov - 21, Ruslan Salimov - 23, Alexander Zimin - 25.
Individual results
Zhuo Qun (Alex) Song (Canada) - 42 points for the tasks, Chenjie Yu (China) - 41, Junghun Ju (Republic of Korea) - 40, Alexander Gunning (Australia) - 36, Jaehyung Kim (Republic of Korea) - 35, Allen Liu (USA) - 35, David Stoner (USA) - 35
Tasks
Authorship of tasks:
Problem 1 proposed by Netherlands
Problem 2 proposed by Serbia
Problem 3 proposed by Ukraine
Problem 4 proposed by Silouanos Brazitikos and Vangelis Psychas, Greece
Problem 5 proposed by Dorlir Ahmeti, Albania
Problem 6 proposed by Ross Atkins and Ivan Guo, Australia
Original tasks in English here (PDF) .
Day 1
Problem 1. A finite set of S points on the plane will be called balanced if for any different points A and B from the set S there is a point C from the set S such that AC = BC. A set S will be called eccentric if for any three different points A, B and C from the set S there is no point P from the set S such that PA = PB = PC.
(a) Prove that for any integer n ≥ 3 there exists a balanced set consisting of n points.
(b) Find all integers n ≥ 3 for which there exists a balanced eccentric set consisting of n points.
Task 2. Find all triples (a; b; c) of positive integers such that each of the numbers
ab - c, bc - a, ca - b is a power of two.
(The power of two is a number of the form 2 n , where n is a nonnegative integer.)
Problem 3. Let ABC be an acute triangle in which AB> AC. Let Γ be the circle circumscribed near it, H its orthocenter, and F the base of the height lowered from the vertex A. Let M be the midpoint of side BC. Let Q be a point on the circle Γ such that ∠HQA = 90 °, and K a point on the circle Γ such that ∠HKQ = 90 °. Let points A, B, C, K and Q are different and lie on the circle Γ in the indicated order.
Prove that the circles described near the triangles KQH and FKM are tangent to each other.
Time to work: 4 hours 30 minutes
Each task is estimated at 7 points.
Day 2
(Boxers in the subject)
Problem 4. Let Ω be the circle described near the triangle ABC, and the point O its center. Circle Γ with center A intersects segment BC at points D and E so that points B, D, E and C are all different and lie on line BC in the indicated order. Let F and G be the intersection points of the circles G and Ω; moreover, the points A, F, B, C, and G lie on Ω in the indicated order. Let K be the second intersection point of the circle described near the triangle BDF and the segment AB. Let L be the second intersection point of the circle described near the triangle CGE and the segment CA. Let the lines FK and GL be different and intersect at the point X.
Prove that the point X lies on the line AO.
Problem 5. Let ℝ be the set of all real numbers. Find all functions f : ℝ → ℝ,
satisfying equality
f (x + f (x + y)) + f (xy) = x + f (x + y) + yf (x)
for all real numbers x and y .
Task 6. The sequence of a1, a2, ... integers satisfies the following conditions:
(i) 1 ≤ a j ≤ 2015 for all j ≥ 1;
(ii) k + a k ≠ L + a L for all 1 ≤ k < L .
Prove that there are two positive integers b and N such that:
For all integers m and n satisfying the condition n> m ≥ N.
Time to work: 4 hours 30 minutes:
Each task is estimated at 7 points.
Original tasks in Russian in the form of screenshots (to check all special characters)
Day 1:
Day 2:
Day 2:
P.S. All schoolchildren-mathematicians-Olympiad hello.
Source: https://habr.com/ru/post/263125/
All Articles