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]] |
− | + | ===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]] |
− | + | ===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 13:48, 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 |