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Difference between revisions of "2003 USAMO Problems"

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Problems of the [[2003 USAMO | 2003]] [[USAMO]].
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Problems of the [[2000 USAMO | 2000]] [[USAMO]].
  
 
== Day 1 ==
 
== Day 1 ==
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=== Problem 1 ===
 
=== Problem 1 ===
  
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Prove that for every positive integer <math> \displaystyle n </math> there exists an <math> \displaystyle n </math>-digit number divisible by <math> \displaystyle 5^n </math> all of whose digits are odd.
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Call a real-valued function <math>f</math> very convex if
  
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* [[2003 USAMO Problems/Problem 1 | Solution]]
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<cmath>\frac {f(x) + f(y)}{2} \ge f\left(\frac {x + y}{2}\right) + |x - y|</cmath>
 +
 
 +
holds for all real numbers <math>x</math> and <math>y</math>. Prove that no very convex function exists.
 +
 
 +
* [[2000 USAMO Problems/Problem 1 | Solution]]
  
 
=== Problem 2 ===
 
=== Problem 2 ===
  
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A convex polygon <math> \mathcal{P} </math> in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of the polygon <math> \mathcal{P} </math> are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.
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Let <math>S</math> be the set of all triangles <math>ABC</math> for which
 +
 
 +
<cmath>5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r},</cmath>
  
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* [[2003 USAMO Problems/Problem 2 | Solution]]
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where <math>r</math> is the inradius and <math>P, Q, R</math> are the points of tangency of the incircle with sides <math>AB, BC, CA,</math> respectively. Prove that all triangles in <math>S</math> are isosceles  and similar to one another.
 +
 
 +
* [[2000 USAMO Problems/Problem 2 | Solution]]
  
 
=== Problem 3 ===
 
=== Problem 3 ===
  
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Let <math> n \neq 0 </math>. For every sequence of integers
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A game of solitaire is played with <math>R</math> red cards, <math>W</math> white cards, and <math>B</math> 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 <math>R, W,</math> and <math>B,</math> the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
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<center>
 
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<math>
 
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A = a_0,a_1,a_2,\dots, a_n
 
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</math>
 
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</center>
 
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satisfying <math>0 \le a_i \le i</math>, for <math>i=0,\dots,n</math>, define another sequence
 
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<center>
 
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<math>
 
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t(A)= t(a_0), t(a_1), t(a_2), \dots, t(a_n)
 
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</math>
 
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</center>
 
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by setting <math> \displaystyle t(a_i) </math> to be the number of terms in the sequence <math> \displaystyle A </math> that precede the term <math> \displaystyle a_i </math> and are different from <math> \displaystyle a_i </math>. Show that, starting from any sequence <math> \displaystyle A </math> as above, fewer than <math> \displaystyle n </math> applications of the transformation <math> \displaystyle t </math> lead to a sequence <math> \displaystyle B </math> such that <math> \displaystyle t(B) = B </math>.
 
  
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* [[2003 USAMO Problems/Problem 3 | Solution]]
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* [[2000 USAMO Problems/Problem 3 | Solution]]
  
 
== Day 2 ==
 
== Day 2 ==
  
 
=== Problem 4 ===
 
=== Problem 4 ===
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Let <math>ABC</math> be a triangle. A circle passing through <math>A</math> and <math>B</math> intersects segments <math>AC</math> and <math>BC</math> at <math>D</math> and <math>E</math>, respectively. Lines <math>AB</math> and <math>DE</math> intersect at <math>F</math>, while lines <math>BD</math> and <math>CF</math> intersect at <math>M</math>. Prove that <math>MF = MC</math> if and only if <math>MB\cdot MD = MC^2</math>.
 
  
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* [[2003 USAMO Problems/Problem 4 | Solution]]
+
 
 +
* [[2000 USAMO Problems/Problem 4 | Solution]]
  
 
=== Problem 5 ===
 
=== Problem 5 ===
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Let <math>a</math>, <math>b</math>, <math>c</math> be positive real numbers. Prove that
 
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<center><math>\dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \dfrac{(2c + a + b)^2}{2c^2 + (a + b)^2} \le 8.</math></center>
 
  
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* [[2003 USAMO Problems/Problem 5 | Solution]]
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 +
* [[2000 USAMO Problems/Problem 5 | Solution]]
  
 
=== Problem 6 ===
 
=== Problem 6 ===
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At the vertices of a regular hexagon are written six nonnegative integers whose sum is 2003. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.
 
  
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* [[2003 USAMO Problems/Problem 6 | Solution]]
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 +
* [[2000 USAMO Problems/Problem 6 | Solution]]
  
 
== Resources ==
 
== Resources ==
  
 
* [[USAMO Problems and Solutions]]
 
* [[USAMO Problems and Solutions]]
−
* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2003-ua/03usamo-test.shtml 2003 USAMO Problems and Solutions]
+
* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2000-ua/03usamo-test.shtml 2000 USAMO Problems and Solutions]
−
* [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=2003 2003 USAMO Problems on the Resources page]
+
* [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=2000 2000 USAMO Problems on the Resources page]

Revision as of 15:41, 7 September 2008

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.

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.

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.

Day 2

Problem 4

Problem 5

Problem 6

Resources

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