Algebra Puzzles
A car is traveling at 70 miles per hour. To the nearest tenth, how many seconds does it take to travel one mile?
51.43
An athlete has six trophies to place on an empty three-shelf display case. The six trophies are bowling trophies F, G, and H and tennis trophies J, K, and L. The three shelves of the display case are labelled 1 to 3 from top to bottom. Any of the shelves can remain empty. The athlete's placement of trophies must conform to the following conditions:
J and L cannot be on the same shelf
F must be on the shelf immediately above the shelf that L is on.
No single shelf can hold all three bowling trophies
K cannot be on Shelf 2
If L and G are on the same shelf, and if one of the shelves remains empty, which of the following must be true?
If H is on Shelf 3, then J is on Shelf 2.
In ΔABC , let the bisector of ∠A meet the circumcircle of ΔABC at point Q. Let QT be the line tangent to the circumcircle at Q. Prove that QT is parallel to BC.
Victor created a clock in three dimensional space in the following way. At t = 0 , the hour hand starts on the positive z-axis and rotates clockwise in the yz-plane when viewed from the positive x-axis. At t = 0 the minute hand starts on the positive y-axis and rotates clockwise in the xy-plane when viewed from the positive z-axis. At t = 0 the second hand starts on the positive x-axis and rotates clockwise in the xz-plane when viewed from the positive y-axis. Victor becomes melancholy when exactly two of his clock's hands coincide. How many times does he become melancholy in a 24 hour period?
What set of positive integers satisfies the equation, a squared + b squared = c squared?
A. trigonometric identities
B. Cartesian coordinates
C. Pythagorean triples
D. Fibonacci sequences
E. ordered pairs
C. Pythagorean triples
In the 2010 World Cup of Soccer the goalkeeper of the Kenyan team stopped 80% of the shots on goal prior to the game against South Africa. Against South Africa he did not stop any of the 15 shots the South Africans took and his percentage dropped to 50%. How many shots on goal did he stop before the game against South Africa?
20
A store sells large individual wooden letters for signs to put on houses. The letters are priced according to a logical rule. Your challenge is to find this rule. Use the information below to help you find the rule and answer the questions. Show your work.
The letters to make the name JONES cost $13.
The letters in SMITH cost $14.
The letters for ORTEGA cost $15.
The letters in VANG cost $11.
How much would the letters in the name GARDNER cost?
The rule was that vowels cost $2.00 apiece and consonants cost $3.00 apiece. Using this rule, the name GARDNER would cost $19.00 (5 consonants x $3.00 plus 2 vowels x $2.00).
Congruent circles C1 and C2 are externally tangent at P. Line Z is a common external tangent of C1 and C2. Circle C# is smaller than the other two circles and is tangent to C1, C2, and Z. Prove that ΔC1C3P is similar to a 3-4-5 triangle.
Solving, we have xx =R/4. So the legs are 3R/4 and R, and the hypotenuse is 5R/4. The result follows.
The five-digit number ABCDE has only even digits, such that A cannot be zero and the other digits may repeat. The four-digit number FGHJ has only odd digits, which may also repeat. How many ways can these numbers be configured so that ABCDE is twice FGHJ?
The largest four-digit number with odd digits is 9999. Since twice 9999 is 19,998, any five-digit number ABCDE must have A = 1.
The answer is 0.
Who deduced the heights of the pyramids from the lengths of their shadows and measured the distance to a ship from shore by the method of similar triangles?
A. Plato
B. Galen
C. Thales
D. Pericles
E. Euripides
C. Thales
A father and son drove out to California. The father drove 80% of the time and covered 60% of the distance. Assuming that each drove at a constant rate, determine the ratio of the father's speed to the son's speed.
3/8
5 pirates of different ages have a treasure of 100 gold coins. On their ship, they decide to split the coins using this scheme:
The oldest pirate proposes how to share the coins, and ALL pirates (including the oldest) vote for or against it.
If 50% or more of the pirates vote for it, then the coins will be shared that way. Otherwise, the pirate proposing the scheme will be thrown overboard, and the process is repeated with the pirates that remain.
As pirates tend to be a bloodthirsty bunch, if a pirate would get the same number of coins if he voted for or against a proposal, he will vote against so that the pirate who proposed the plan will be thrown overboard.
Assuming that all 5 pirates are intelligent, rational, greedy, and do not wish to die, (and are rather good at math for pirates) what will happen?
98 : 0 : 1 : 0 : 1.
Suppose that three circles of the same size are mutually tangent and snugly fit into an equilateral triangle of side 2a. What is the area covered by the three circles? Show the work that produced your answer.
A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least x times as large as the shorter piece (x ≤ 1) is
(A) 1/2 (B) 2/x (C) 1/(x+1) (D) 1/x (E) 2/(x+1)
2/(x+1)
In 200 B.C., Eratosthenes found the Sun's rays were vertical at Syene and 7 1/2 degrees from vertical at Alexandria. He knew the distance between the two cities. What was he able to calculate using these data?
Circumference of the earth
Find the number of distinct ways to rearrange the characters in KEEN225 such that there is a numeral at the end of the string.
540
Three men agree to share a cab from the airport into town. When they arrive, the meter reads $25. Each man gives the driver a $10 bill. She hands them five $1 bills as change. Each man takes one of the $1 bills. They give the driver the remaining two $1 bills as a tip.
Each man has now spent nine dollars, and the driver has two dollars, bringing the total to $29 (3 x 9 = 27, 27 + 2 = 29). What happened to the other dollar?
What needs to be added to this 27 dollars is the three dollars that the men keep, not the two dollar tip—the tip was accounted for in the cost of the ride.
Equilateral triangle ABC is inscribed in a circle. D is a point on minor arc BC? . The length of chord BD is 3 inches and the length of chord DC is 5 inches. How long is AD (in inches)?
There are twenty containers. One of them contains ten balls numbered 1 through 10. The other nineteen each contain 10,000 balls numbered 1 through 10,000. A container is selected at random and a ball is randomly selected from it. The ball is number 8. What is the probability that the container picked was the one with ten balls?
98%.
Which is not one of the five Platonic solids?
A. hexahedron
B. octahedron
C. tetrahedron
D. pentahedron
E. dodecahedron
D. pentahedron
If a single digit is removed from the decimal expansion of 8/11, resulting in a new decimal, determine the largest possible result.
7/22
In the following list of 4 people, if exactly one person is telling the truth and exactly one person did it, then who did it?
Al: I didn't do it.
Betty: Carl did it.
Carl: Debby did it.
Debby: I did it.
In ∆ABC , A is at (2, 4), B is at (8, 12), and C is at (16, 6). Find the coordinates (x, y) of point K on the segment AB so that the ratio of the area of ∆BKC to the area of ∆ABC is 1:5.
Five men agreed to meet on April 1st for lunch. Thereafter, each of them wanted to continue to have lunch together. Al said he would show up every second day. Bob said he would appear every third day. Charles said he would come every fourth day, Dan said he would come every fifth day, and Elton agreed to come every sixth day. During the next one hundred days, there were K days when only three men showed up. Find the value of K.
Who collaborated with Alfred North Whitehead in writing "Principia Mathematica" and later won the 1950 Nobel prize for literature?
A. M.C. Escher
B. Jean Wourier
C. Norbert Weiner
D. Leonhard Euler
E. Bertrand Russell
E. Bertrand Russell