Difference between revisions of "2013 USAJMO"
Mrdavid445 (talk | contribs) (→Problem 6) |
|||
Line 10: | Line 10: | ||
(i) The difference between any two adjacent numbers is either <math>0</math> or <math>1</math>. | (i) The difference between any two adjacent numbers is either <math>0</math> or <math>1</math>. | ||
+ | |||
(ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to <math>0</math>. | (ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to <math>0</math>. | ||
Revision as of 19:31, 11 May 2013
Contents
Day 1
Problem 1
Are there integers and
such that
and
are both perfect cubes of integers?
Problem 2
Each cell of an board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a garden if it satisfies the following two conditions:
(i) The difference between any two adjacent numbers is either or
.
(ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to .
Determine the number of distinct gardens in terms of and
.
Problem 3
In triangle , points
lie on sides
respectively. Let
,
,
denote the circumcircles of triangles
,
,
, respectively. Given the fact that segment
intersects
,
,
again at
respectively, prove that
.
Day 2
Problem 4
Let be the number of ways to write
as a sum of powers of
, where we keep track of the order of the summation. For example,
because
can be written as
,
,
,
,
, and
. Find the smallest
greater than
for which
is odd.
Problem 5
Quadrilateral is inscribed in the semicircle
with diameter
. Segments
and
meet at
. Point
is the foot of the perpendicular from
to line
. Point
lies on
such that line
is perpendicular to line
. Let
be the intersection of segments
and
. Prove that
Problem 6
Find all real numbers satisfying
See Also
2013 USAJMO (Problems • Resources) | ||
Preceded by 2012 USAJMO |
Followed by 2014 USAJMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |