Difference between revisions of "1998 USAMO Problems"
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===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> |
Latest revision as of 06:11, 24 November 2020
Contents
Day 1
Problem 1
Suppose that the set has been partitioned into disjoint pairs () so that for all , equals or . Prove that the sum ends in the digit .
Problem 2
Let and be concentric circles, with in the interior of . From a point on one draws the tangent to (). Let be the second point of intersection of and , and let be the midpoint of . A line passing through intersects at and in such a way that the perpendicular bisectors of and intersect at a point on . Find, with proof, the ratio .
Problem 3
Let be numbers from the interval such that Prove that Solution
Day 2
Problem 4
A computer screen shows a 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.
Problem 5
Prove that for each , there is a set of integers such that divides for every distinct .
Problem 6
Let be an integer. Find the largest integer (as a function of ) such that there exists a convex -gon for which exactly of the quadrilaterals have an inscribed circle. (Here .)
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.