Difference between revisions of "2018 USAMO Problems"
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Note: For any geometry problem whose statement begins with an asterisk (<math>*</math>), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction. | Note: For any geometry problem whose statement begins with an asterisk (<math>*</math>), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction. | ||
− | + | ===Problem 1=== | |
− | ==Problem 1= | ||
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Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cmath> | Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cmath> | ||
− | Solution | + | [[2018 USAMO Problems/Problem 1|Solution]] |
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− | ==Problem 2== | + | ===Problem 2=== |
Find all functions <math>f:(0,\infty) \rightarrow (0,\infty)</math> such that | Find all functions <math>f:(0,\infty) \rightarrow (0,\infty)</math> such that | ||
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for all <math>x,y,z >0</math> with <math>xyz =1.</math> | for all <math>x,y,z >0</math> with <math>xyz =1.</math> | ||
− | Solution | + | [[2018 USAMO Problems/Problem 2|Solution]] |
− | + | ===Problem 3=== | |
− | ==Problem 3== | ||
For a given integer <math>n\ge 2,</math> let <math>\{a_1,a_2,…,a_m\}</math> be the set of positive integers less than <math>n</math> that are relatively prime to <math>n.</math> Prove that if every prime that divides <math>m</math> also divides <math>n,</math> then <math>a_1^k+a_2^k + \dots + a_m^k</math> is divisible by <math>m</math> for every positive integer <math>k.</math> | For a given integer <math>n\ge 2,</math> let <math>\{a_1,a_2,…,a_m\}</math> be the set of positive integers less than <math>n</math> that are relatively prime to <math>n.</math> Prove that if every prime that divides <math>m</math> also divides <math>n,</math> then <math>a_1^k+a_2^k + \dots + a_m^k</math> is divisible by <math>m</math> for every positive integer <math>k.</math> | ||
− | Solution | + | [[2018 USAMO Problems/Problem 3|Solution]] |
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==Day 2== | ==Day 2== | ||
Note: For any geometry problem whose statement begins with an asterisk (<math>*</math>), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction. | Note: For any geometry problem whose statement begins with an asterisk (<math>*</math>), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction. | ||
− | ==Problem 4== | + | ===Problem 4=== |
Let <math>p</math> be a prime, and let <math>a_1, \dots, a_p</math> be integers. Show that there exists an integer <math>k</math> such that the numbers <cmath>a_1 + k, a_2 + 2k, \dots, a_p + pk</cmath>produce at least <math>\tfrac{1}{2} p</math> distinct remainders upon division by <math>p</math>. | Let <math>p</math> be a prime, and let <math>a_1, \dots, a_p</math> be integers. Show that there exists an integer <math>k</math> such that the numbers <cmath>a_1 + k, a_2 + 2k, \dots, a_p + pk</cmath>produce at least <math>\tfrac{1}{2} p</math> distinct remainders upon division by <math>p</math>. | ||
− | Solution | + | [[2018 USAMO Problems/Problem 4|Solution]] |
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− | ==Problem 5== | + | ===Problem 5=== |
In convex cyclic quadrilateral <math>ABCD,</math> we know that lines <math>AC</math> and <math>BD</math> intersect at <math>E,</math> lines <math>AB</math> and <math>CD</math> intersect at <math>F,</math> and lines <math>BC</math> and <math>DA</math> intersect at <math>G.</math> Suppose that the circumcircle of <math>\triangle ABE</math> intersects line <math>CB</math> at <math>B</math> and <math>P</math>, and the circumcircle of <math>\triangle ADE</math> intersects line <math>CD</math> at <math>D</math> and <math>Q</math>, where <math>C,B,P,G</math> and <math>C,Q,D,F</math> are collinear in that order. Prove that if lines <math>FP</math> and <math>GQ</math> intersect at <math>M</math>, then <math>\angle MAC = 90^{\circ}.</math> | In convex cyclic quadrilateral <math>ABCD,</math> we know that lines <math>AC</math> and <math>BD</math> intersect at <math>E,</math> lines <math>AB</math> and <math>CD</math> intersect at <math>F,</math> and lines <math>BC</math> and <math>DA</math> intersect at <math>G.</math> Suppose that the circumcircle of <math>\triangle ABE</math> intersects line <math>CB</math> at <math>B</math> and <math>P</math>, and the circumcircle of <math>\triangle ADE</math> intersects line <math>CD</math> at <math>D</math> and <math>Q</math>, where <math>C,B,P,G</math> and <math>C,Q,D,F</math> are collinear in that order. Prove that if lines <math>FP</math> and <math>GQ</math> intersect at <math>M</math>, then <math>\angle MAC = 90^{\circ}.</math> | ||
− | Solution | + | [[2018 USAMO Problems/Problem 5|Solution]] |
+ | ===Problem 6=== | ||
+ | Let <math>a_n</math> be the number of permutations <math>(x_1, x_2, \dots, x_n)</math> of the numbers <math>(1,2,\dots, n)</math> such that the <math>n</math> ratios <math>\frac{x_k}{k}</math> for <math>1\le k\le n</math> are all distinct. Prove that <math>a_n</math> is odd for all <math>n\ge 1.</math> | ||
− | + | [[2018 USAMO Problems/Problem 6|Solution]] | |
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− | + | {{USAMO newbox|year=2018|before=[[2017 USAMO Problems]]|after=[[2019 USAMO Problems]]}} |
Latest revision as of 12:48, 22 November 2023
Contents
Day 1
Note: For any geometry problem whose statement begins with an asterisk (), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.
Problem 1
Let be positive real numbers such that . Prove that
Problem 2
Find all functions such that
for all with
Problem 3
For a given integer let be the set of positive integers less than that are relatively prime to Prove that if every prime that divides also divides then is divisible by for every positive integer
Day 2
Note: For any geometry problem whose statement begins with an asterisk (), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.
Problem 4
Let be a prime, and let be integers. Show that there exists an integer such that the numbers produce at least distinct remainders upon division by .
Problem 5
In convex cyclic quadrilateral we know that lines and intersect at lines and intersect at and lines and intersect at Suppose that the circumcircle of intersects line at and , and the circumcircle of intersects line at and , where and are collinear in that order. Prove that if lines and intersect at , then
Problem 6
Let be the number of permutations of the numbers such that the ratios for are all distinct. Prove that is odd for all
2018 USAMO (Problems • Resources) | ||
Preceded by 2017 USAMO Problems |
Followed by 2019 USAMO Problems | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |