Difference between revisions of "2011 USAJMO Problems"
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− | =Day 1= | + | ==Day 1== |
− | ==Problem 1== | + | ===Problem 1=== |
Find, with proof, all positive integers <math>n</math> for which <math>2^n + 12^n + 2011^n</math> is a perfect square. | Find, with proof, all positive integers <math>n</math> for which <math>2^n + 12^n + 2011^n</math> is a perfect square. | ||
[[2011 USAJMO Problems/Problem 1|Solution]] | [[2011 USAJMO Problems/Problem 1|Solution]] | ||
− | ==Problem 2== | + | ===Problem 2=== |
Let <math>a</math>, <math>b</math>, <math>c</math> be positive real numbers such that <math>a^2 + b^2 + c^2 + (a + b + c)^2 \le 4</math>. Prove that | Let <math>a</math>, <math>b</math>, <math>c</math> be positive real numbers such that <math>a^2 + b^2 + c^2 + (a + b + c)^2 \le 4</math>. Prove that | ||
<cmath>\frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \ge 3.</cmath> | <cmath>\frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \ge 3.</cmath> | ||
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[[2011 USAMO Problems/Problem 1|Solution]] | [[2011 USAMO Problems/Problem 1|Solution]] | ||
− | ==Problem 3== | + | ===Problem 3=== |
For a point <math>P = (a,a^2)</math> in the coordinate plane, let <math>\ell(P)</math> denote the line passing through <math>P</math> with slope <math>2a</math>. Consider the set of triangles with vertices of the form <math>P_1 = (a_1, a_1^2)</math>, <math>P_2 = (a_2, a_2^2)</math>, <math>P_3 = (a_3, a_3^2)</math>, such that the intersections of the lines <math>\ell(P_1)</math>, <math>\ell(P_2)</math>, <math>\ell(P_3)</math> form an equilateral triangle <math>\Delta</math>. Find the locus of the center of <math>\Delta</math> as <math>P_1 P_2 P_3</math> ranges over all such triangles. | For a point <math>P = (a,a^2)</math> in the coordinate plane, let <math>\ell(P)</math> denote the line passing through <math>P</math> with slope <math>2a</math>. Consider the set of triangles with vertices of the form <math>P_1 = (a_1, a_1^2)</math>, <math>P_2 = (a_2, a_2^2)</math>, <math>P_3 = (a_3, a_3^2)</math>, such that the intersections of the lines <math>\ell(P_1)</math>, <math>\ell(P_2)</math>, <math>\ell(P_3)</math> form an equilateral triangle <math>\Delta</math>. Find the locus of the center of <math>\Delta</math> as <math>P_1 P_2 P_3</math> ranges over all such triangles. | ||
[[2011 USAJMO Problems/Problem 3|Solution]] | [[2011 USAJMO Problems/Problem 3|Solution]] | ||
− | =Day 2= | + | ==Day 2== |
− | ==Problem 4== | + | ===Problem 4=== |
A ''word'' is defined as any finite string of letters. A word is a ''palindrome'' if it reads the same backwards as forwards. Let a sequence of words <math>W_0</math>, <math>W_1</math>, <math>W_2</math>, <math>\dots</math> be defined as follows: <math>W_0 = a</math>, <math>W_1 = b</math>, and for <math>n \ge 2</math>, <math>W_n</math> is the word formed by writing <math>W_{n - 2}</math> followed by <math>W_{n - 1}</math>. Prove that for any <math>n \ge 1</math>, the word formed by writing <math>W_1</math>, <math>W_2</math>, <math>\dots</math>, <math>W_n</math> in succession is a palindrome. | A ''word'' is defined as any finite string of letters. A word is a ''palindrome'' if it reads the same backwards as forwards. Let a sequence of words <math>W_0</math>, <math>W_1</math>, <math>W_2</math>, <math>\dots</math> be defined as follows: <math>W_0 = a</math>, <math>W_1 = b</math>, and for <math>n \ge 2</math>, <math>W_n</math> is the word formed by writing <math>W_{n - 2}</math> followed by <math>W_{n - 1}</math>. Prove that for any <math>n \ge 1</math>, the word formed by writing <math>W_1</math>, <math>W_2</math>, <math>\dots</math>, <math>W_n</math> in succession is a palindrome. | ||
[[2011 USAJMO Problems/Problem 4|Solution]] | [[2011 USAJMO Problems/Problem 4|Solution]] | ||
− | ==Problem 5== | + | ===Problem 5=== |
Points <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, <math>E</math> lie on a circle <math>\omega</math> and point <math>P</math> lies outside the circle. The given points are such that (i) lines <math>PB</math> and <math>PD</math> are tangent to <math>\omega</math>, (ii) <math>P</math>, <math>A</math>, <math>C</math> are collinear, and (iii) <math>\overline{DE} \parallel \overline{AC}</math>. Prove that <math>\overline{BE}</math> bisects <math>\overline{AC}</math>. | Points <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, <math>E</math> lie on a circle <math>\omega</math> and point <math>P</math> lies outside the circle. The given points are such that (i) lines <math>PB</math> and <math>PD</math> are tangent to <math>\omega</math>, (ii) <math>P</math>, <math>A</math>, <math>C</math> are collinear, and (iii) <math>\overline{DE} \parallel \overline{AC}</math>. Prove that <math>\overline{BE}</math> bisects <math>\overline{AC}</math>. | ||
[[2011 USAJMO Problems/Problem 5|Solution]] | [[2011 USAJMO Problems/Problem 5|Solution]] | ||
− | ==Problem 6== | + | ===Problem 6=== |
Consider the assertion that for each positive integer <math>n \ge 2</math>, the remainder upon dividing <math>2^{2^n}</math> by <math>2^n - 1</math> is a power of 4. Either prove the assertion or find (with proof) a counterexample. | Consider the assertion that for each positive integer <math>n \ge 2</math>, the remainder upon dividing <math>2^{2^n}</math> by <math>2^n - 1</math> is a power of 4. Either prove the assertion or find (with proof) a counterexample. | ||
[[2011 USAMO Problems/Problem 4|Solution]] | [[2011 USAMO Problems/Problem 4|Solution]] | ||
− | = See | + | == See Also == |
*[[USAJMO Problems and Solutions]] | *[[USAJMO Problems and Solutions]] | ||
+ | |||
+ | {{USAJMO box|year=2011|before=[[2010 USAJMO Problems]]|after=[[2012 USAJMO Problems]]}} | ||
+ | {{MAA Notice}} |
Latest revision as of 16:40, 5 August 2023
Contents
Day 1
Problem 1
Find, with proof, all positive integers for which
is a perfect square.
Problem 2
Let ,
,
be positive real numbers such that
. Prove that
Problem 3
For a point in the coordinate plane, let
denote the line passing through
with slope
. Consider the set of triangles with vertices of the form
,
,
, such that the intersections of the lines
,
,
form an equilateral triangle
. Find the locus of the center of
as
ranges over all such triangles.
Day 2
Problem 4
A word is defined as any finite string of letters. A word is a palindrome if it reads the same backwards as forwards. Let a sequence of words ,
,
,
be defined as follows:
,
, and for
,
is the word formed by writing
followed by
. Prove that for any
, the word formed by writing
,
,
,
in succession is a palindrome.
Problem 5
Points ,
,
,
,
lie on a circle
and point
lies outside the circle. The given points are such that (i) lines
and
are tangent to
, (ii)
,
,
are collinear, and (iii)
. Prove that
bisects
.
Problem 6
Consider the assertion that for each positive integer , the remainder upon dividing
by
is a power of 4. Either prove the assertion or find (with proof) a counterexample.
See Also
2011 USAJMO (Problems • Resources) | ||
Preceded by 2010 USAJMO Problems |
Followed by 2012 USAJMO Problems | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.