30 Days of Problems for
30 Years of High School
Math Competitions
| 1 ~ November 13, 1971 | ||
| Two faulty clocks both read
twelve o’clock. One clock gains a minute each hour while the
Other loses a minute each hour. In how many hours will both clocks again show identical times? |
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| 2 ~ February 12, 1972 | ||
One glass contains 4 oz. of wine and another glass contains 4 oz. of water. From the first glass, one ounce of wine is removed and poured into the water. From the mixture, one ounce is removed and poured into the wine. What is the ratio of wine in the water glass to water in the wine glass? |
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| 3 ~ April 14, 1973 | ||
| A
plot of land consists of two equal square lots on opposite sides of a
larger square lot as in the figure below. The total area of the plot is
1752 square yards and the fence around the outside is 200 yards long.
Find the length of the side of each square if each length is an integer.
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| 4 ~ October 12, 1974 | ||
| A
man buys 100 chickens for $100. If roosters cost $10 each, hens cost $3
each and baby chicks are 2 for $1, how many of each did he buy? (Assume
he buys at least one of each.)
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| 5 ~ February 15, 1975 | ||
Find the sum of the 100 terms
(the log has a base of 10). .
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| 6 ~ February 21, 1976 | ||
| Arrange
the following in order from least to greatest.
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| 7 ~ November 12, 1977 | ||
| How many subsets of {1, 2, 3, 4, 5}
contain 5 as its largest number?
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| 8 ~ October 7, 1978 | ||
| Find the ratio of the area of a regular
hexagon to the area of an equilateral triangle if their perimeters are
equal.
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| 9 ~ November 10, 1979 | ||
In the circle below the
center is C and the distance from C to chord AB is half of the radius.
If the area of the shaded portion is 
square units, find the number of units in the radius of the circle.
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| 10 ~ November 8, 1980 | ||
Evaluate:  |
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| 11 ~ March 21, 1981 | ||
If 
and the inverse function of f,  ,
find a and b. |
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| 12 ~ February 6, 1982 | ||
What is the 63rd digit
to the right of the decimal point in the decimal representation of  ?
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| 13 ~ November 12, 1983 | ||
Find the ordered pair (a, b) such
that a and b are positive integers and  . |
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| 14 ~ January 14, 1984 | ||
| A palindromic number reads the same
forward and backward. (e.g., 61316 is a palindromic number.) How many
even five digit palindromic numbers are there?
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| 15 ~ February 16, 1985 | ||
| A person accidentally puts his
twelve-hour digital watch on upside-down and reads the time as 1:40 (hours:min)
later than it actually is. What time is it? |
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| 16 ~ November 8, 1986 | ||
If r is positive and the line whose
equation is x + y = r is tangent to the circle whose equation is  ,
what is the value of r?
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| 17 ~ January 17, 1987 | ||
| The arithmetic mean (average) of n
numbers is 38. If 1 is added to the first number, 2 to the second
number, and so on until n is added to the nth number, then the
arithmetic mean of the new set of numbers is 83. Find the value of n. |
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| 18 ~ January 16, 1988 | ||
| If Harry stands still on an escalator,
he can move from one floor of a building to another in 20 seconds. If
Harry walks at his standard pace on a parallel set of stairs, the trip
takes 15 seconds. How many seconds will the trip take if Harry walks on
the escalator at his standard pace in the direction of the escalator’s
movement?
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| 19 ~ November 11, 1989 | ||
| How many positive integers less than
1,000 are divisible by neither 5 nor 7?
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| 20 ~ January 20, 1990 | ||
In the diagram
assume that  is
a right angle,  AD=15and  . Find CE
(the length of segment CE).
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| 21 ~ November 17, 1990 | ||
| ABCD is a square where
points E and F are midpoints. Let x be the length of AE. Find the area
of the shaded region in terms of x.
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| 22 ~ October 31, 1992 | ||
| In a naval engagement one-third of the
fleet was captured, one-sixth sunk, and two ships were burned;
one-seventh of the remainder was lost in a storm after the battle.
Finally, 24 ships sailed home. How many ships were in the fleet before
the engagement? |
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| 23 ~ October 9, 1993 | ||
If 
and  ,
then t exceeds s by how much?
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| 24 ~ October 8, 1994 | ||
| If two numbers x and y are increased by
one and decreased by one, respectively, then their product is decreased
by four. If they are decreased by one and increased by two respectively,
their product is increased by sixteen. Find the sum of the two numbers x
and y. |
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| 25 ~ November 11, 1995 | ||
| A square with side-length
one is divided into five rectangles with equal areas by horizontal and
vertical lines as shown in the diagram. Find the perimeter of the shaded
rectangle.
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| 26 ~ January 20, 1996 | ||
| The vertices of quadrilateral ABCD are
A(8,1), B(3,5), C(-2,3), and D(2,0). Find the area of this
quadrilateral. |
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| 27 ~ March 8, 1997 | ||
| The product of two whole numbers is one
million, but neither ends in a zero. What is the sum of the two whole
numbers? |
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| 28 ~ January 10, 1998 | ||
| The sum of three consecutive terms of
an arithmetic progression is 24 and their product is 400. Find the
smallest of the three terms. |
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| 29 ~ January 9, 1999 | ||
| If five (5) turkeys eat two (2) pounds
of corn in three (3) days, how many pounds of corn will be needed to
feed one hundred (100) turkeys for thirty (30) days? |
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| 30 ~ March 11, 2000 | ||
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In the accompanying drawing of three right triangles, AE = BC, BE = CD, AB = 3, and DE = 9. Find the exact length of AE.
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