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Difference between revisions of "1990 USAMO Problems"

(New page: ==Problem 1== A certain state issues license plates consisting of six digits (from 0 to 9). The state requires that any two license plates differ in at least two places. (For instance, t...)
 
(→‎Problem 5)
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==Problem 5==
 
==Problem 5==
 
An acute-angled triangle <math>ABC</math> is given in the plane. The circle with diameter <math>\, AB \,</math> intersects altitude <math>\, CC' \,</math> and its extension at points <math>\, M \,</math> and <math>\, N \,</math>, and the circle with diameter <math>\, AC \,</math> intersects altitude <math>\, BB' \,</math> and its extensions at <math>\, P \,</math> and <math>\, Q \,</math>. Prove that the points <math>\, M, N, P, Q \,</math> lie on a common circle
 
An acute-angled triangle <math>ABC</math> is given in the plane. The circle with diameter <math>\, AB \,</math> intersects altitude <math>\, CC' \,</math> and its extension at points <math>\, M \,</math> and <math>\, N \,</math>, and the circle with diameter <math>\, AC \,</math> intersects altitude <math>\, BB' \,</math> and its extensions at <math>\, P \,</math> and <math>\, Q \,</math>. Prove that the points <math>\, M, N, P, Q \,</math> lie on a common circle
 +
 
[[1989 USAMO Problems/Problem 5 | Solution]]
 
[[1989 USAMO Problems/Problem 5 | Solution]]
  
 
== See also ==
 
== See also ==
 
* [[1990 USAMO]]
 
* [[1990 USAMO]]

Revision as of 16:53, 16 October 2007

Problem 1

A certain state issues license plates consisting of six digits (from 0 to 9). The state requires that any two license plates differ in at least two places. (For instance, the numbers 027592 and 020592 cannot both be used.) Determine, with proof, the maximum number of distinct license plates that the state can use.

Solution

Problem 2

A sequence of functions $\, \{f_n(x) \} \,$ is defined recursively as follows:

$f_1(x) = \sqrt {x^2 + 48}, \quad \mbox{and} \\ f_{n + 1}(x) = \sqrt {x^2 + 6f_n(x)} \quad \mbox{for } n \geq 1.$

(Recall that $\sqrt {\makebox[5mm]{}}$ is understood to represent the positive square root.) For each positive integer $n$, find all real solutions of the equation $\, f_n(x) = 2x \,$.

Solution

Problem 3

Suppose that necklace $\, A \,$ has 14 beads and necklace $\, B \,$ has 19. Prove that for any odd integer $n \geq 1$, there is a way to number each of the 33 beads with an integer from the sequence

$\{ n, n + 1, n + 2, \dots, n + 32 \}$

so that each integer is used once, and adjacent beads correspond to relatively prime integers. (Here a ``necklace is viewed as a circle in which each bead is adjacent to two other beads.)

Solution

Problem 4

Find, with proof, the number of positive integers whose base-$n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\pm 1$ from some digit further to the left. (Your answer should be an explicit function of $n$ in simplest form.)

Solution

Problem 5

An acute-angled triangle $ABC$ is given in the plane. The circle with diameter $\, AB \,$ intersects altitude $\, CC' \,$ and its extension at points $\, M \,$ and $\, N \,$, and the circle with diameter $\, AC \,$ intersects altitude $\, BB' \,$ and its extensions at $\, P \,$ and $\, Q \,$. Prove that the points $\, M, N, P, Q \,$ lie on a common circle

Solution

See also

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