Difference between revisions of "1994 USAMO Problems"
(New page: Problems of the 1994 USAMO. ==Problem 1== Let <math>\, k_1 < k_2 < k_3 < \cdots \,</math> be positive integers, no two consecutive, and let <math>\, s_m = k_1 + k_2 ...) |
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[[1994 USAMO Problems/Problem 1|Solution]] | [[1994 USAMO Problems/Problem 1|Solution]] | ||
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+ | Induct on <math>n</math>. When <math>n = 1</math>, we are to show that the interval <math>\, [s_n, s_{n + 1}) \,</math> contains at least one perfect square. This interval is equivalent to <math>\, [k_0, k_0 + k_1) \,</math> when <math>n = 1</math>. Now for some <math>a , a^2 \le k_0^2 < (a+1)^2</math>.Then it suffices to show that the minimal "distance spanned" by the interval <math>\, [k_0, k_0 + k_1) \,</math> is greater than or equal to the maximum distance from <math>k_0</math> to the nearest perfect square. Since the smallest element that can follow <math>k_0</math> is <math>k_0 + 2</math>, we have to show the below. Note that we ignore the trivial case where <math>k_0 = a^2</math>, which should be mentioned. | ||
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+ | <math>2a + 1 \le k_0 + 2</math> | ||
+ | <math>2a \le k_0 + 1</math> | ||
+ | <math>2a \le k_0 + (q + 1)</math>, where <math>q</math> is a member of <math>\{1, 2, \ldots, 2\a}</math> | ||
+ | We now prove by contradiction. Assume that | ||
==Problem 2== | ==Problem 2== |
Revision as of 05:19, 11 July 2009
Problem 1
Let be positive integers, no two consecutive, and let for . Prove that, for each positive integer the interval contains at least one perfect square.
Induct on . When , we are to show that the interval contains at least one perfect square. This interval is equivalent to when . Now for some .Then it suffices to show that the minimal "distance spanned" by the interval is greater than or equal to the maximum distance from to the nearest perfect square. Since the smallest element that can follow is , we have to show the below. Note that we ignore the trivial case where , which should be mentioned.
, where is a member of $\{1, 2, \ldots, 2\a}$ (Error compiling LaTeX. Unknown error_msg)
We now prove by contradiction. Assume that
Problem 2
The sides of a 99-gon are initially colored so that consecutive sides are red, blue, red, blue, red, blue, yellow. We make a sequence of modifications in the coloring, changing the color of one side at a time to one of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in which consecutive sides are red, blue, red, blue, red, blue, red, yellow, blue?
Problem 3
A convex hexagon is inscribed in a circle such that and diagonals , , and are concurrent. Let be the intersection of and . Prove that .
Problem 4
Let be a sequence of positive real numbers satisfying for all . Prove that, for all
Problem 5
Let and denote the number of elements, the sum, and the product, respectively, of a finite set of positive integers. (If is the empty set, .) Let be a finite set of positive integers. As usual, let denote . Prove that
for all integers .
Resources
1994 USAMO (Problems • Resources) | ||
Preceded by 1993 USAMO |
Followed by 1995 USAMO | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |