Difference between revisions of "2000 USAMO Problems"

Latest revision as of 18:02, 18 April 2014

Problems of the 2000 USAMO.

Day 1

Problem 1

Call a real-valued function $f$ very convex if

$\frac {f(x) + f(y)}{2} \ge f\left(\frac {x + y}{2}\right) + |x - y|$

holds for all real numbers $x$ and $y$ . Prove that no very convex function exists.

Solution

Problem 2

Let $S$ be the set of all triangles $ABC$ for which

$5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r},$

where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. Prove that all triangles in $S$ are isosceles and similar to one another.

Solution

Problem 3

A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.

Solution

Day 2

Problem 4

Find the smallest positive integer $n$ such that if $n$ squares of a $1000\times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with legs parallel to the edges of the board.

Solution

Problem 5

Let $A_1A_2A_3$ be a triangle and let $\omega_1$ be a circle in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\omega_2, \omega_3, \dots, \omega_7$ such that for $k = 2, 3, \dots, 7,$ $\omega_k$ is externally tangent to $\omega_{k - 1}$ and passes through $A_k$ and $A_{k + 1},$ where $A_{n + 3} = A_{n}$ for all $n \ge 1$ . Prove that $\omega_7 = \omega_1.$

Solution

Problem 6

Let $a_1, b_1, a_2, b_2, \dots , a_n, b_n$ be nonnegative real numbers. Prove that

$\sum_{i, j = 1}^{n} \min\{a_ia_j, b_ib_j\} \le \sum_{i, j = 1}^{n} \min\{a_ib_j, a_jb_i\}.$

Solution

@@ Line 32: / Line 32: @@
 === Problem 4 ===
+Find the smallest positive integer <math>n</math> such that if <math>n</math> squares of a <math>1000\times 1000</math> chessboard are colored, then there will exist three colored squares whose centers form a right triangle with legs parallel to the edges of the board.
 * [[2000 USAMO Problems/Problem 4 | Solution]]
 === Problem 5 ===
+Let <math>A_1A_2A_3</math> be a [[triangle]] and let <math>\omega_1</math> be a [[circle]] in its plane passing through <math>A_1</math> and <math>A_2.</math> Suppose there exist circles <math>\omega_2, \omega_3, \dots, \omega_7</math> such that for <math>k = 2, 3, \dots, 7,</math> <math>\omega_k</math> is externally [[tangent (geometry)|tangent]] to <math>\omega_{k - 1}</math> and passes through <math>A_k</math> and <math>A_{k + 1},</math>  where <math>A_{n + 3} = A_{n}</math> for all <math>n \ge 1</math>. Prove that <math>\omega_7 = \omega_1.</math>
 * [[2000 USAMO Problems/Problem 5 | Solution]]
 === Problem 6 ===
+Let <math>a_1, b_1, a_2, b_2, \dots , a_n, b_n</math> be nonnegative real numbers. Prove that
+<cmath>\sum_{i, j = 1}^{n} \min\{a_ia_j, b_ib_j\} \le \sum_{i, j = 1}^{n} \min\{a_ib_j, a_jb_i\}.</cmath>
 * [[2000 USAMO Problems/Problem 6 | Solution]]
-== Resources ==
+== See Also ==
-* [[USAMO Problems and Solutions]]
+{{USAMO newbox|year=2000|before=[[1999 USAMO]]|after=[[2001 USAMO]]}}
-* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2000-ua/03usamo-test.shtml 2000 USAMO Problems and Solutions]
+{{MAA Notice}}
-* [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=2000 2000 USAMO Problems on the Resources page]

2000 USAMO (Problems • Resources)
Preceded by 1999 USAMO	Followed by 2001 USAMO
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

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