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| − | ==Problem 1==
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| − | ''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
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| − | ''integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>''
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| − | [[2019 IMO Problems/Problem 1|Solution]]
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| − | ==Problem 2==
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| − | 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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