1994 AHSME Problems
1994 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
Problem 2
A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle?
Problem 3
How many of the following are equal to for all ?
Problem 4
In the -plane, the segment with endpoints and is the diameter of a circle. If the point is on the circle, then
Problem 5
Pat intended to multiply a number by but instead divided by . Pat then meant to add but instead subtracted . After these mistakes, the result was . If the correct operations had been used, the value produced would have been
Problem 6
In the sequence each term is the sum of the two terms to its left. Find .
Problem 7
Squares and are congruent, , and is the center of square . The area of the region in the plane covered by these squares is
Problem 8
In the polygon shown, each side is perpendicular to its adjacent sides, and all 28 of the sides are congruent. The perimeter of the polygon is . The area of the region bounded by the polygon is
Problem 9
If is four times , and the complement of is four times the complement of , then
Problem 10
For distinct real numbers and , let be the larger of and and let be the smaller of and . If , then
Problem 11
Three cubes of volume and are glued together at their faces. The smallest possible surface area of the resulting configuration is
Problem 12
If , then
Problem 13
In triangle , . If there is a point strictly between and such that , then
Problem 14
Find the sum of the arithmetic series
Problem 15
For how many in is the tens digit of odd?
Problem 16
Some marbles in a bag are red and the rest are blue. If one red marble is removed, then one-seventh of the remaining marbles are red. If two blue marbles are removed instead of one red, then one-fifth of the remaining marbles are red. How many marbles were in the bag originally?
Problem 17
An by rectangle has the same center as a circle of radius . The area of the region common to both the rectangle and the circle is
Problem 18
Triangle is inscribed in a circle, and . If and are adjacent vertices of a regular polygon of sides inscribed in this circle, then
Problem 19
Label one disk "", two disks "", three disks "" fifty disks "". Put these labeled disks in a box. Disks are then drawn from the box at random without replacement. The minimum number of disks that must be drawn to guarantee drawing at least ten disks with the same label is
Problem 20
Suppose is a geometric sequence with common ratio and . If is an arithmetic sequence, then is
Problem 21
Find the number of counter examples to the statement:
Problem 22
Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta and Gamma. These three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can Professors Alpha, Beta and Gamma choose their chairs?
Problem 23
In the -plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at and . The slope of the line through the origin that divides the area of this region exactly in half is
Problem 24
A sample consisting of five observations has an arithmetic mean of and a median of . The smallest value that the range (largest observation minus smallest) can assume for such a sample is
Problem 25
If and are non-zero real numbers such that then the integer nearest to is
Problem 26
A regular polygon of sides is exactly enclosed (no overlaps, no gaps) by regular polygons of sides each. (Shown here for .) If , what is the value of ?
Problem 27
A bag of popping corn contains white kernels and yellow kernels. Only of the white kernels will pop, whereas of the yellow ones will pop. A kernel is selected at random from the bag, and pops when placed in the popper. What is the probability that the kernel selected was white?
Problem 28
In the -plane, how many lines whose -intercept is a positive prime number and whose -intercept is a positive integer pass through the point ?
Problem 29
Points and on a circle of radius are situated so that , and the length of minor arc is . If angles are measured in radians, then
Problem 30
When standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of . The smallest possible value of is
See also
1994 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1993 AHSME |
Followed by 1995 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.