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Difference between revisions of "1990 USAMO Problems"

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Problems from the '''1990 [[United States of America Mathematical Olympiad | USAMO]]'''.
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==Problem 1==
 
==Problem 1==
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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.
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A certain state issues license plates consisting of six digits (from 0 through 9). The state requires that any two plates differ in at least two places. (Thus the plates <math>\boxed{027592}</math> and <math>\boxed{020592}</math> cannot both be used.) Determine, with proof, the maximum number of distinct license plates that the state can use.
  
 
[[1990 USAMO Problems/Problem 1 | Solution]]
 
[[1990 USAMO Problems/Problem 1 | Solution]]
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==Problem 2==
 
==Problem 2==
 
A sequence of functions <math>\, \{f_n(x) \} \,</math> is defined recursively as follows:
 
A sequence of functions <math>\, \{f_n(x) \} \,</math> is defined recursively as follows:
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<cmath> \begin{align*}
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<math>
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f_1(x) &= \sqrt {x^2 + 48}, \quad \text{and} \\
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f_1(x) = \sqrt {x^2 + 48}, \quad \mbox{and} \\
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f_{n + 1}(x) &= \sqrt {x^2 + 6f_n(x)} \quad \text{for } n \geq 1.
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f_{n + 1}(x) = \sqrt {x^2 + 6f_n(x)} \quad \mbox{for } n \geq 1.
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\end{align*} </cmath>
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</math>
 
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(Recall that <math>\sqrt {\makebox[5mm]{}}</math> is understood to represent the positive square root.) For each positive integer <math>n</math>, find all real solutions of the equation <math>\, f_n(x) = 2x \,</math>.
 
(Recall that <math>\sqrt {\makebox[5mm]{}}</math> is understood to represent the positive square root.) For each positive integer <math>n</math>, find all real solutions of the equation <math>\, f_n(x) = 2x \,</math>.
  
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==Problem 3==
 
==Problem 3==
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Suppose that necklace <math>\, A \,</math> has 14 beads and necklace <math>\, B \,</math> has 19. Prove that for any odd integer <math>n \geq 1</math>, there is a way to number each of the 33 beads with an integer from the sequence
 
Suppose that necklace <math>\, A \,</math> has 14 beads and necklace <math>\, B \,</math> has 19. Prove that for any odd integer <math>n \geq 1</math>, there is a way to number each of the 33 beads with an integer from the sequence
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<cmath> \{ n, n + 1, n + 2, \dots, n + 32 \} </cmath>
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<math>
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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.)
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\{ n, n + 1, n + 2, \dots, n + 32 \}
 
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</math>
 
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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.)
 
  
 
[[1990 USAMO Problems/Problem 3 | Solution]]
 
[[1990 USAMO Problems/Problem 3 | Solution]]
  
 
==Problem 4==
 
==Problem 4==
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Find, with proof, the number of positive integers whose base-<math>n</math> representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by <math>\pm 1</math> from some digit further to the left. (Your answer should be an explicit function of <math>n</math> in simplest form.)
 
Find, with proof, the number of positive integers whose base-<math>n</math> representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by <math>\pm 1</math> from some digit further to the left. (Your answer should be an explicit function of <math>n</math> in simplest form.)
  
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==Problem 5==
 
==Problem 5==
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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
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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.
  
 
[[1990 USAMO Problems/Problem 5 | Solution]]
 
[[1990 USAMO Problems/Problem 5 | Solution]]
  
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== See also ==
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== See Also ==
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* [[1990 USAMO]]
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{{USAMO box|year=1990|before=[[1989 USAMO]]|after=[[1991 USAMO]]}}
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{{MAA Notice}}

Latest revision as of 19:47, 3 July 2013

Problems from the 1990 USAMO.

Problem 1

A certain state issues license plates consisting of six digits (from 0 through 9). The state requires that any two plates differ in at least two places. (Thus the plates $\boxed{027592}$ and $\boxed{020592}$ 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: \begin{align*} f_1(x) &= \sqrt {x^2 + 48}, \quad \text{and} \\ f_{n + 1}(x) &= \sqrt {x^2 + 6f_n(x)} \quad \text{for } n \geq 1. \end{align*} (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

1990 USAMO (Problems • Resources)
Preceded by
1989 USAMO 		Followed by
1991 USAMO
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

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