Difference between revisions of "2015 IMO Problems"
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[[2015 IMO Problems/Problem 6|Solution]] | [[2015 IMO Problems/Problem 6|Solution]] | ||
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+ | ==Problem 1== | ||
+ | ''Let <math>\mathbb{Z}</math> be the set of integers. Determine all functions <math>f : \mathbb{Z} \to \mathbb{Z}</math> such that, for all | ||
+ | ''integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>'' | ||
+ | |||
+ | [[2019 IMO Problems/Problem 1|Solution]] | ||
+ | |||
+ | ==Problem 2== | ||
+ | In triangle <math>ABC</math>, point <math>A_1</math> lies on side <math>BC</math> and point <math>B_1</math> lies on side <math>AC</math>. Let <math>P</math> and <math>Q</math> be points on segments <math>AA_1</math> and <math>BB_1</math>, respectively, such that <math>PQ</math> is parallel to <math>AB</math>. Let <math>P_1</math> be a point on line <math>PB_1</math>, such that <math>B_1</math> lies strictly between <math>P</math> and <math>P_1</math>, and <math>\angle PP_1C=\angle BAC</math>. Similarly, let <math>Q_1</math> be the point on line <math>QA_1</math>, such that <math>A_1</math> lies strictly between <math>Q</math> and <math>Q_1</math>, and <math>\angle CQ_1Q=\angle CBA</math>. | ||
+ | |||
+ | Prove that points <math>P,Q,P_1</math>, and <math>Q_1</math> are concyclic. | ||
+ | |||
+ | [[2019 IMO Problems/Problem 2|Solution]] | ||
+ | |||
+ | ==Problem 3== | ||
+ | A social network has <math>2019</math> users, some pairs of whom are friends. Whenever user <math>A</math> is friends with user <math>B</math>, user <math>B</math> is also friends with user <math>A</math>. Events of the following kind may happen repeatedly, one at a time: | ||
+ | Three users <math>A</math>, <math>B</math>, and <math>C</math> such that <math>A</math> is friends with both <math>B</math> and <math>C</math>, but <math>B</math> and <math>C</math> are not friends, change their friendship statuses such that <math>B</math> and <math>C</math> are now friends, but <math>A</math> is no longer friends with <math>B</math>, and no longer friends with <math>C</math>. All other friendship statuses are unchanged. | ||
+ | Initially, <math>1010</math> users have <math>1009</math> friends each, and <math>1009</math> users have <math>1010</math> friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user. | ||
+ | |||
+ | [[2019 IMO Problems/Problem 3|Solution]] | ||
+ | |||
+ | ==Problem 4== | ||
+ | Find all pairs <math>(k,n)</math> of positive integers such that | ||
+ | |||
+ | <cmath>k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).</cmath> | ||
+ | |||
+ | [[2019 IMO Problems/Problem 4|Solution]] | ||
+ | |||
+ | ==Problem 5== | ||
+ | The Bank of Bath issues coins with an <math>H</math> on one side and a <math>T</math> on the other. Harry has <math>n</math> of these coins arranged in a line from left to right. He repeatedly performs the following operation: | ||
+ | |||
+ | If there are exactly <math>k > 0</math> coins showing <math>H</math>, then he turns over the <math>k^{th}</math> coin from the left; otherwise, all coins show <math>T</math> and he stops. For example, if <math>n = 3</math> the process starting with the configuration <math>THT</math> would be <math>THT \rightarrow HHT \rightarrow HTT \rightarrow TTT</math>, which stops after three operations. | ||
+ | |||
+ | (a) Show that, for each initial configuration, Harry stops after a finite number of operations. | ||
+ | |||
+ | (b) For each initial configuration <math>C</math>, let <math>L(C)</math> be the number of operations before Harry stops. For | ||
+ | example, <math>L(THT) = 3</math> and <math>L(TTT) = 0</math>. Determine the average value of <math>L(C)</math> over all <math>2^n</math> | ||
+ | possible initial configurations <math>C</math>. | ||
+ | |||
+ | [[2019 IMO Problems/Problem 5|Solution]] | ||
+ | |||
+ | ==Problem 6== | ||
+ | Let <math>I</math> be the incenter of acute triangle <math>ABC</math> with <math>AB \neq AC</math>. The incircle <math>\omega</math> of <math>ABC</math> is tangent to sides <math>BC</math>, <math>CA</math>, and <math>AB</math> at <math>D</math>, <math>E</math>, and <math>F</math>, respectively. The line through <math>D</math> perpendicular to <math>EF</math> meets ω again at <math>R</math>. Line <math>AR</math> meets ω again at <math>P</math>. The circumcircles of triangles <math>PCE</math> and <math>PBF</math> meet again at <math>Q</math>. | ||
+ | Prove that lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>. | ||
+ | |||
+ | [[2019 IMO Problems/Problem 6|Solution]] | ||
+ | |||
+ | {{IMO box|year=2015|before=[[2014 IMO Problems]]|after=[[2016 IMO Problems]]}} |
Revision as of 08:02, 10 September 2020
Contents
[hide]Problem 1
We say that a finite set in the plane is balanced if, for any two different points , in , there is a point in such that . We say that is centre-free if for any three points , , in , there is no point in such that .
- Show that for all integers , there exists a balanced set consisting of points.
- Determine all integers for which there exists a balanced centre-free set consisting of points.
Problem 2
Determine all triples of positive integers such that each of the numbers is a power of 2.
(A power of 2 is an integer of the form where is a non-negative integer ).
Problem 3
Let be an acute triangle with . Let be its circumcircle, its orthocenter, and the foot of the altitude from . Let be the midpoint of . Let be the point on such that . Assume that the points , , , , and are all different, and lie on in this order.
Prove that the circumcircles of triangles and are tangent to each other.
Problem 4
Triangle has circumcircle and circumcenter . A circle with center intersects the segment at points and , such that , , , and are all different and lie on line in this order. Let and be the points of intersection of and , such that , , , , and lie on in this order. Let be the second point of intersection of the circumcircle of triangle and the segment . Let be the second point of intersection of the circumcircle of triangle and the segment .
Suppose that the lines and are different and intersect at the point . Prove that lies on the line .
Problem 5
Let be the set of real numbers. Determine all functions : satisfying the equation
for all real numbers and .
Problem 6
The sequence of integers satisfies the conditions:
(i) for all ,
(ii) for all .
Prove that there exist two positive integers and for whichfor all integers and such that .
Problem 1
Let be the set of integers. Determine all functions such that, for all integers and ,
Problem 2
In triangle , point lies on side and point lies on side . Let and be points on segments and , respectively, such that is parallel to . Let be a point on line , such that lies strictly between and , and . Similarly, let be the point on line , such that lies strictly between and , and .
Prove that points , and are concyclic.
Problem 3
A social network has users, some pairs of whom are friends. Whenever user is friends with user , user is also friends with user . Events of the following kind may happen repeatedly, one at a time: Three users , , and such that is friends with both and , but and are not friends, change their friendship statuses such that and are now friends, but is no longer friends with , and no longer friends with . All other friendship statuses are unchanged. Initially, users have friends each, and users have friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.
Problem 4
Find all pairs of positive integers such that
Problem 5
The Bank of Bath issues coins with an on one side and a on the other. Harry has of these coins arranged in a line from left to right. He repeatedly performs the following operation:
If there are exactly coins showing , then he turns over the coin from the left; otherwise, all coins show and he stops. For example, if the process starting with the configuration would be , which stops after three operations.
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.
(b) For each initial configuration , let be the number of operations before Harry stops. For example, and . Determine the average value of over all possible initial configurations .
Problem 6
Let be the incenter of acute triangle with . The incircle of is tangent to sides , , and at , , and , respectively. The line through perpendicular to meets ω again at . Line meets ω again at . The circumcircles of triangles and meet again at . Prove that lines and meet on the line through perpendicular to .
2015 IMO (Problems) • Resources | ||
Preceded by 2014 IMO Problems |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2016 IMO Problems |
All IMO Problems and Solutions |