Difference between revisions of "1992 AHSME Problems"
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== Problem 13 == | == Problem 13 == | ||
− | How many pairs of positive integers (a,b) with <math>a+b\le 100</math> satisfy the equation | + | How many pairs of positive integers <math>(a,b)</math> with <math>a+b\le 100</math> satisfy the equation |
<cmath>\frac{a+b^{-1}}{a^{-1}+b}=13?</cmath> | <cmath>\frac{a+b^{-1}}{a^{-1}+b}=13?</cmath> |
Revision as of 16:15, 16 July 2016
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 See also
Problem 1
If then
Problem 2
An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of objects in the urn are gold coins?
Problem 3
If and the points and lie on a line with slope , then
Problem 4
If and are positive integers and and are odd, then is
Problem 5
Problem 6
If , then
Problem 7
The ratio of to is , of to is and of to is . What is the ratio of to ?
Problem 8
A square floor is tiled with congruent square tiles. The tiles on the two diagonals of the floor are black. The rest of the tiles are white. If there are 101 black tiles, then the total number of tiles is
Problem 9
Five equilateral triangles, each with side , are arranged so they are all on the same side of a line containing one side of each vertex. Along this line, the midpoint of the base of one triangle is a vertex of the next. The area of the region of the plane that is covered by the union of the five triangular regions is
Problem 10
The number of positive integers for which the equation has an integer solution for is
Problem 11
The ratio of the radii of two concentric circles is . If is a diameter of the larger circle, is a chord of the larger circle that is tangent to the smaller circle, and , then the radius of the larger circle is
Problem 12
Let be the image when the line is reflected across the -axis. The value of is
Problem 13
How many pairs of positive integers with satisfy the equation
Problem 14
Which of the following equations have the same graph?
Problem 15
Let . Define a sequence of complex numbers by
In the complex plane, how far from the origin is ?
Problem 16
If for three positive numbers and , all different, then
Problem 17
The 2-digit integers from 19 to 92 are written consecutively to form the integer . Suppose that is the highest power of 3 that is a factor of . What is ?
Problem 18
The increasing sequence of positive integers has the property that
If , then is
Problem 19
For each vertex of a solid cube, consider the tetrahedron determined by the vertex and the midpoints of the three edges that meet at that vertex. The portion of the cube that remains when these eight tetrahedra are cut away is called a cubeoctahedron. The ratio of the volume of the cubeoctahedron to the volume of the original cube is closest to which of these?
Problem 20
Part of an "n-pointed regular star" is shown. It is a simple closed polygon in which all edges are congruent, angles are congruent, and angles are congruent. If the acute angle at is less than the acute angle at , then
Problem 21
For a finite sequence of numbers, the Cesáro sum of A is defined to be , where and . If the Cesáro sum of the 99-term sequence is 1000, what is the Cesáro sum of the 100-term sequence ?
Problem 22
Ten points are selected on the positive -axis,, and five points are selected on the positive -axis,. The fifty segments connecting the ten points on to the five points on are drawn. What is the maximum possible number of points of intersection of these fifty segments that could lie in the interior of the first quadrant?
Problem 23
Let be a subset of such that no pair of distinct elements in has a sum divisible by . What is the maximum number of elements in ?
Problem 24
Let be a parallelogram of area with and . Locate and on segments and , respectively, with . Let the line through parallel to intersect at . The area of quadrilateral is
Problem 25
In , and . If perpendiculars constructed to at and to at meet at , then
Problem 26
Semicircle has center and radius . Point is on and . Extend and to and , respectively, so that circular arcs and have and as their respective centers. Circular arc has center . The area of the shaded "smile" , is
Problem 27
A circle of radius has chords of length and of length 7. When and are extended through and , respectively, they intersect at , which is outside of the circle. If and , then
Problem 28
Let . The product of the real parts of the roots of is
Problem 29
An "unfair" coin has a probability of turning up heads. If this coin is tossed times, what is the probability that the total number of heads is even?
Problem 30
Let be an isosceles trapezoid with bases and . Suppose and a circle with center on is tangent to segments and . If is the smallest possible value of , then =
See also
1992 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1991 AHSME |
Followed by 1993 AHSME | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.