1990 AHSME Problems
1990 AHSME (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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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
Problem 3
The consecutive angles of a trapezoid form an arithmetic sequence. If the smallest angle is , then the largest angle is
Problem 4
Let be a parallelogram with and Extend through to so that If intersects at , then is closest to
Problem 5
Which of these numbers is largest?
Problem 6
Points and are units apart. How many lines in a given plane containing and are units from and units from ?
Problem 7
A triangle with integral sides has perimeter . The area of the triangle is
Problem 8
The number of real solutions of the equation is
Problem 9
Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest number possible of black edges is
Problem 10
An wooden cube is formed by gluing together unit cubes. What is the greatest number of unit cubes that can be seen from a single point?
Problem 11
How many positive integers less than have an odd number of positive integer divisors?
Problem 12
Let be the function defined by for some positive . If then
Problem 13
If the following instructions are carried out by a computer, which value of will be printed because of instruction ?
1. START AT AND AT . 2. INCREASE THE VALUE OF BY . 3. INCREASE THE VALUE OF BY THE VALUE OF . 4. IF IS AT LEAST , THEN GO TO INSTRUCTION ; OTHERWISE, GO TO INSTRUCTION . AND PROCEED FROM THERE. 5. PRINT THE VALUE OF . 6.STOP.
Problem 14
An acute isosceles triangle, , is inscribed in a circle. Through and , tangents to the circle are drawn, meeting at point . If and is the radian measure of , then
Problem 15
Four whole numbers, when added three at a time, give the sums and . What is the largest of the four numbers?
Problem 16
At one of George Washington's parties, each man shook hands with everyone except his spouse, and no handshakes took place between women. If married couples attended, how many handshakes were there among these people?
Problem 17
How many of the numbers, have three different digits in increasing order or in decreasing order?
Problem 18
First is chosen at random from the set , and then is chosen at random from the same set. The probability that the integer has units digit is
Problem 19
For how many integers between and is the improper fraction in lowest terms?
Problem 20
In the figure is a quadrilateral with right angles at and . Points and are on , and and are perpendicular to . If and , then
Problem 21
Consider a pyramid whose base is square and whose vertex is equidistant from and . If and , then the volume of the pyramid is
Problem 22
If the six solutions of are written in the form , where and are real, then the product of those solutions with is
Problem 23
If
Problem 24
All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined?
Problem 25
Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to the center sphere and to three faces of the cube. What is the radius of each sphere?
Problem 26
Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to them in the circle. Then each person computes and announces the average of the numbers of their two neighbors. The figure shows the average announced by each person (not the original number the person picked.) The number picked by the person who announced the average was
Problem 27
Which of these triples could be the lengths of the three altitudes of a triangle?
Problem 28
A quadrilateral that has consecutive sides of lengths and is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length and . Find .
Problem 29
A subset of the integers has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?
Problem 30
If where and , and then is an integer. Its units digit is
See also
1990 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1989 AHSME |
Followed by 1991 AHSME | |
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All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.