Difference between revisions of "2018 USAMO Problems"
(→Problem 3) |
(→Problem 4) |
||
| Line 23: | Line 23: | ||
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 | Solution | ||
| − | |||
==Problem 5== | ==Problem 5== | ||
Revision as of 10:41, 21 April 2018
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 ![$a+b+c=4\sqrt[3]{abc}$](http://latex.artofproblemsolving.com/b/6/9/b69ff720e6d0801983832367327a018e7afce9ca.png)
. Prove that ![\[2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.\]](http://latex.artofproblemsolving.com/7/0/4/7043a170ac5535465a5a1b4daccb1f1de3043280.png)
Solution
Problem 2
Find all functions 
such that
![\[f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{z}\right) + f\left(z+\frac{1}{x}\right) = 1\]](http://latex.artofproblemsolving.com/8/9/4/8945fb40f421318fdb191894bc65db33f0253cd0.png)
for all 
with 
Solution
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 
Solution
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 ![\[a_1 + k, a_2 + 2k, \dots, a_p + pk\]](http://latex.artofproblemsolving.com/7/6/2/7624ce20fc9bfb798126ffacdb34110159ef0094.png)
produce at least 
distinct remainders upon division by .
Solution
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 
Solution
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 
Solution
