Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1998 USAMO Problems"

m (→‎Resources)
(→‎Problem 3)
 
(9 intermediate revisions by 3 users not shown)
Line 1: Line 1:
 
Problems of the [[1998 USAMO | 1998]] [[USAMO]].
 
Problems of the [[1998 USAMO | 1998]] [[USAMO]].
  
−
==Problem 1==
+
==Day 1==
 +
===Problem 1===
 
Suppose that the set <math>\{1,2,\cdots, 1998\}</math> has been partitioned into disjoint pairs <math>\{a_i,b_i\}</math> (<math>1\leq i\leq 999</math>) so that for all <math>i</math>, <math>|a_i-b_i|</math> equals <math>1</math> or <math>6</math>. Prove that the sum
 
Suppose that the set <math>\{1,2,\cdots, 1998\}</math> has been partitioned into disjoint pairs <math>\{a_i,b_i\}</math> (<math>1\leq i\leq 999</math>) so that for all <math>i</math>, <math>|a_i-b_i|</math> equals <math>1</math> or <math>6</math>. Prove that the sum
 
<cmath> |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}|  </cmath>
 
<cmath> |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}|  </cmath>
Line 8: Line 9:
 
[[1998 USAMO Problems/Problem 1|Solution]]
 
[[1998 USAMO Problems/Problem 1|Solution]]
  
−
==Problem 2==
+
===Problem 2===
 
Let <math>{\cal C}_1</math> and <math>{\cal C}_2</math> be  concentric circles, with <math>{\cal C}_2</math> in the interior of  <math>{\cal C}_1</math>. From a point <math>A</math> on <math>{\cal C}_1</math> one draws the tangent <math>AB</math> to <math>{\cal C}_2</math> (<math>B\in {\cal C}_2</math>). Let <math>C</math> be the second point of intersection of <math>AB</math> and <math>{\cal C}_1</math>, and let <math>D</math> be the midpoint of <math>AB</math>. A line passing through <math>A</math> intersects <math>{\cal C}_2</math> at <math>E</math> and <math>F</math> in such a way that the perpendicular  bisectors of <math>DE</math> and <math>CF</math> intersect at a point <math>M</math> on <math>AB</math>. Find, with proof,  the ratio <math>AM/MC</math>.
 
Let <math>{\cal C}_1</math> and <math>{\cal C}_2</math> be  concentric circles, with <math>{\cal C}_2</math> in the interior of  <math>{\cal C}_1</math>. From a point <math>A</math> on <math>{\cal C}_1</math> one draws the tangent <math>AB</math> to <math>{\cal C}_2</math> (<math>B\in {\cal C}_2</math>). Let <math>C</math> be the second point of intersection of <math>AB</math> and <math>{\cal C}_1</math>, and let <math>D</math> be the midpoint of <math>AB</math>. A line passing through <math>A</math> intersects <math>{\cal C}_2</math> at <math>E</math> and <math>F</math> in such a way that the perpendicular  bisectors of <math>DE</math> and <math>CF</math> intersect at a point <math>M</math> on <math>AB</math>. Find, with proof,  the ratio <math>AM/MC</math>.
  
 
[[1998 USAMO Problems/Problem 2|Solution]]
 
[[1998 USAMO Problems/Problem 2|Solution]]
  
−
==Problem 3==
+
===Problem 3===
 
Let <math>a_0,a_1,\cdots ,a_n</math> be numbers from the interval <math>(0,\pi/2)</math> such that
 
Let <math>a_0,a_1,\cdots ,a_n</math> be numbers from the interval <math>(0,\pi/2)</math> such that
−
<cmath> \tan (a_0-\frac{\pi}{4})+ \tan (a_1-\frac{\pi}{4})+\cdots +\tan (a_n-\frac{\pi}{4})\geq n-1.  </cmath>
+
<cmath> \tan \left(a_0-\frac{\pi}{4}\right)+ \tan \left(a_1-\frac{\pi}{4}\right)+\cdots +\tan \left(a_n-\frac{\pi}{4}\right)\geq n-1.  </cmath>
 
Prove that
 
Prove that
 
<cmath> \tan a_0\tan a_1 \cdots \tan a_n\geq n^{n+1}.  </cmath>
 
<cmath> \tan a_0\tan a_1 \cdots \tan a_n\geq n^{n+1}.  </cmath>
 
[[1998 USAMO Problems/Problem 3|Solution]]
 
[[1998 USAMO Problems/Problem 3|Solution]]
  
−
==Problem 4==
+
==Day 2==
 +
===Problem 4===
 
A computer screen shows a <math>98 \times 98</math> chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result,  the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.
 
A computer screen shows a <math>98 \times 98</math> chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result,  the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.
  
 
[[1998 USAMO Problems/Problem 4|Solution]]
 
[[1998 USAMO Problems/Problem 4|Solution]]
  
−
==Problem 5==
+
===Problem 5===
 
Prove that for each <math>n\geq 2</math>, there is a set <math>S</math> of <math>n</math> integers such that <math>(a-b)^2</math> divides <math>ab</math> for every distinct <math>a,b\in S</math>.
 
Prove that for each <math>n\geq 2</math>, there is a set <math>S</math> of <math>n</math> integers such that <math>(a-b)^2</math> divides <math>ab</math> for every distinct <math>a,b\in S</math>.
  
 
[[1998 USAMO Problems/Problem 5|Solution]]
 
[[1998 USAMO Problems/Problem 5|Solution]]
  
−
==Problem 6==
+
===Problem 6===
 
Let <math>n \geq 5</math> be an integer. Find the largest integer <math>k</math> (as a function of <math>n</math>) such that there exists a convex <math>n</math>-gon <math>A_{1}A_{2}\dots A_{n}</math> for which exactly <math>k</math> of the quadrilaterals <math>A_{i}A_{i+1}A_{i+2}A_{i+3}</math> have an inscribed circle. (Here <math>A_{n+j} = A_{j}</math>.)
 
Let <math>n \geq 5</math> be an integer. Find the largest integer <math>k</math> (as a function of <math>n</math>) such that there exists a convex <math>n</math>-gon <math>A_{1}A_{2}\dots A_{n}</math> for which exactly <math>k</math> of the quadrilaterals <math>A_{i}A_{i+1}A_{i+2}A_{i+3}</math> have an inscribed circle. (Here <math>A_{n+j} = A_{j}</math>.)
  
 
[[1998 USAMO Problems/Problem 6|Solution]]
 
[[1998 USAMO Problems/Problem 6|Solution]]
  
−
== Resources ==
+
== See Also ==
−
 
 
 
{{USAMO newbox|year=1998|before=[[1997 USAMO]]|after=[[1999 USAMO]]}}
 
{{USAMO newbox|year=1998|before=[[1997 USAMO]]|after=[[1999 USAMO]]}}
−
 
+
{{MAA Notice}}
−
* [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=1998 1994 USAMO Problems on the resources page]
 

Latest revision as of 05:11, 24 November 2020

Problems of the 1998 USAMO.

Day 1

Problem 1

Suppose that the set $\{1,2,\cdots, 1998\}$ has been partitioned into disjoint pairs $\{a_i,b_i\}$ ($1\leq i\leq 999$) so that for all $i$, $|a_i-b_i|$ equals $1$ or $6$. Prove that the sum \[|a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}|\] ends in the digit $9$.

Solution

Problem 2

Let ${\cal C}_1$ and ${\cal C}_2$ be concentric circles, with ${\cal C}_2$ in the interior of ${\cal C}_1$. From a point $A$ on ${\cal C}_1$ one draws the tangent $AB$ to ${\cal C}_2$ ($B\in {\cal C}_2$). Let $C$ be the second point of intersection of $AB$ and ${\cal C}_1$, and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects ${\cal C}_2$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$.

Solution

Problem 3

Let $a_0,a_1,\cdots ,a_n$ be numbers from the interval $(0,\pi/2)$ such that \[\tan \left(a_0-\frac{\pi}{4}\right)+ \tan \left(a_1-\frac{\pi}{4}\right)+\cdots +\tan \left(a_n-\frac{\pi}{4}\right)\geq n-1.\] Prove that \[\tan a_0\tan a_1 \cdots \tan a_n\geq n^{n+1}.\] Solution

Day 2

Problem 4

A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.

Solution

Problem 5

Prove that for each $n\geq 2$, there is a set $S$ of $n$ integers such that $(a-b)^2$ divides $ab$ for every distinct $a,b\in S$.

Solution

Problem 6

Let $n \geq 5$ be an integer. Find the largest integer $k$ (as a function of $n$) such that there exists a convex $n$-gon $A_{1}A_{2}\dots A_{n}$ for which exactly $k$ of the quadrilaterals $A_{i}A_{i+1}A_{i+2}A_{i+3}$ have an inscribed circle. (Here $A_{n+j} = A_{j}$.)

Solution

See Also

1998 USAMO (Problems • Resources)
Preceded by
1997 USAMO 		Followed by
1999 USAMO
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1998_USAMO_Problems&oldid=138354"