Difference between revisions of "2012 USAJMO Problems"
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===Problem 3=== | ===Problem 3=== | ||
Let <math>a</math>, <math>b</math>, <math>c</math> be positive real numbers. Prove that | Let <math>a</math>, <math>b</math>, <math>c</math> be positive real numbers. Prove that | ||
− | <cmath>\frac{a^3 + 3b^3}{5a + b} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{ | + | <cmath>\frac{a^3 + 3b^3}{5a + b} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{3}{2} (a^2 + b^2 + c^2).</cmath> |
[[2012 USAJMO Problems/Problem 3|Solution]] | [[2012 USAJMO Problems/Problem 3|Solution]] | ||
+ | |||
==Day 2== | ==Day 2== | ||
===Problem 4=== | ===Problem 4=== |
Revision as of 05:03, 21 December 2019
Contents
Day 1
Problem 1
Given a triangle , let
and
be points on segments
and
, respectively, such that
. Let
and
be distinct points on segment
such that
lies between
and
,
, and
. Prove that
,
,
,
are concyclic (in other words, these four points lie on a circle).
Problem 2
Find all integers such that among any
positive real numbers
,
,
,
with
there exist three that are the side lengths of an acute triangle.
Problem 3
Let ,
,
be positive real numbers. Prove that
Day 2
Problem 4
Let be an irrational number with
, and draw a circle in the plane whose circumference has length 1. Given any integer
, define a sequence of points
,
,
,
as follows. First select any point
on the circle, and for
define
as the point on the circle for which the length of arc
is
, when travelling counterclockwise around the circle from
to
. Suppose that
and
are the nearest adjacent points on either side of
. Prove that
.
Problem 5
For distinct positive integers ,
, define
to be the number of integers
with
such that the remainder when
divided by 2012 is greater than that of
divided by 2012. Let
be the minimum value of
, where
and
range over all pairs of distinct positive integers less than 2012. Determine
.
Problem 6
Let be a point in the plane of triangle
, and
a line passing through
. Let
,
,
be the points where the reflections of lines
,
,
with respect to
intersect lines
,
,
, respectively. Prove that
,
,
are collinear.
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.