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| − | mathlete6453 was here
| + | '''2019 [[IMO]]''' problems and solutions. The 2019 IMO was held in Bath, United Kingdom. The first link contains the full set of test problems. The rest contain each individual problem and its solution |
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| − | ==Problem 1==
| + | *[[2019 IMO Problems|Entire Test]] |
| − | ''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
| + | **[[2019 IMO Problems/Problem 1 | Problem 1]] |
| − | ''integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>''
| + | **[[2019 IMO Problems/Problem 2 | Problem 2]] |
| | + | **[[2019 IMO Problems/Problem 3 | Problem 3]] |
| | + | **[[2019 IMO Problems/Problem 4 | Problem 4]] |
| | + | **[[2019 IMO Problems/Problem 5 | Problem 5]] |
| | + | **[[2019 IMO Problems/Problem 6 | Problem 6]] |
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| − | [[2019 IMO Problems/Problem 1|Solution]] | + | == See Also == |
| | + | * [[IMO Problems and Solutions, with authors]] |
| | + | * [[Mathematics competition resources]] |
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| − | ==Problem 2==
| + | {{IMO box|year=2019|before=[[2018 IMO]]|after=[[2020 IMO]]}} |
| − | 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>.
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| − | Prove that points <math>P,Q,P_1</math>, and <math>Q_1</math> are concyclic.
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| − | [[2019 IMO Problems/Problem 2|Solution]]
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| − | ==Problem 3==
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| − | 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:
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| − | 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.
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| − | 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.
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| − | [[2019 IMO Problems/Problem 3|Solution]]
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| − | ==Problem 4==
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| − | Find all pairs <math>(k,n)</math> of positive integers such that
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| − | <cmath>k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).</cmath>
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| − | [[2019 IMO Problems/Problem 4|Solution]]
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| − | ==Problem 5==
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| − | 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:
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| − | 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.
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| − | (a) Show that, for each initial configuration, Harry stops after a finite number of operations.
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| − | (b) For each initial configuration <math>C</math>, let <math>L(C)</math> be the number of operations before Harry stops. For
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| − | 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>
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| − | possible initial configurations <math>C</math>.
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| − | [[2019 IMO Problems/Problem 5|Solution]]
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| − | ==Problem 6==
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| − | 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>.
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| − | Prove that lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>.
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| − | [[2019 IMO Problems/Problem 6|Solution]]
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| − | {{IMO box|year=2019|before=[[2018 IMO Problems]]|after=[[2020 IMO Problems]]}} | |
