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You are in the land of liars and truth-tellers, where each person you meet either always makes statements which are true, or always makes statements which are false. You come upon a group of four people, who make the following statements: A: C is a truth-teller B: D and C are opposite types of people C: B and A are opposite types of people D: At least one of A and B is a liarList whether each of A, B, C, and D is a liar, a truth-teller, or is impossible to determine.
Find the center of a sphere passing through the points (1, 1, 3), (2, 2, 1), (3, 4, 8), and (5, 6, 9).
Professor Moody gives two of his students, Hermione and Ron, the product and sum (respectively) of two (not necessarily distinct) non-zero digits (1 to 9). Hermione says, "I don't know the numbers." Ron says, "I don't know the numbers." Hermione says, "I don't know the numbers." Ron says, "I don't know the numbers." Hermione says, "I don't know the numbers." Ron says, "I don't know the numbers." Hermione says, "I know the numbers."What are the two numbers?
What is the smallest natural number that leaves remainders of 11, 2, and 44 when divided by 21, 37, and 51, respectively?
Julia is as old as John was when Julias age was one quarter of six less than the age that John will be when Julia is Johns current age. When John and Julias present ages are added together, and the digits of the result are reversed, it is twice Julias present age. Assuming John and Julia have the same birthday and thus the difference of their ages is always a constant, what is the absolute value of the difference between Johns age and Julias age?
A person sits down and begins flipping a coin repeatedly. What is the probability that he will get a string of at least four heads at some point before he first gets a string of at least two tails?
Some of the members of the math team decide to form the Calculator Club. In order to be exclusive, the requirements for being a member of the Calculator Club are that the number of calculators a member owns must be a perfect square. Also, each member must have a number of calculators which differs from another members number of calculators by 192. If this club eventually grows to have its maximum possible number of members, determine the difference between the greatest number of calculators owned by a member and the least number of calculators owned by a member.
An ant is on the surface of cube STUVWXYZ with edge length ten centimeters at point A, which is three centimeters from point Z along edge EMBED Equation.3 . Point B is another point on the cubes surface such that the minimum distance the ant must travel along the cubes surface to reach point B is a maximum. How far is point B from point U?
Given that A, B, and C are positive integers, and that GCF(A, B) = 300 GCF(B, C) = 150 GCF(C, A) = 6750 LCM(A, B) = 135000 LCM(B, C) = 81000 LCM(C, A) = 202500evaluate A + B + C.
There exists point A on circle O. Points E and F are chosen randomly in the interior of circle O. What is the probability that E lies within triangle OFA?
Tetrahedron ABCD has edges EMBED Equation.3 with lengths of EMBED Equation.3 , EMBED Equation.3 , 1, EMBED Equation.3 , EMBED Equation.3 , and EMBED Equation.3 centimeters, respectively. What is the volume of ABCD, in cubic centimeters?
What is the shortest distance between line A, which passes through the points (18, -2, -9) and (-6, -10, 7), and line B, which passes through the points (-31, -15, 12) and (-55, -18, 24)?
How many distinct five-card poker hands can Steph be dealt from a non-standard 48-card deck (all four 3s have been removed), which can be straight flushes if 2s are regarded as wild cards (they may act as any other card desired)? A straight flush is five cards that are in the same suit, which are in sequential order (4, 5, 6, 7, 8 of hearts would be a straight flush, as would 10, J, Q, K, A of spades). Note that Aces may only be the high card of a straight flush, not the low card. Because 2s are regarded as wild cards, hands such as 6 of diamonds, 2 of clubs, 8 of diamonds, 2 of hearts, 10 of diamonds will also count as straight flushes, because the wild 2s will become the 7 and 9 of diamonds.
Evaluate: EMBED Equation.3 .
The product of three daughters ages is 1200, while the sum of their ages is ten years less than their mothers age. A friend of the family, who knew the mothers age and was quite good at arithmetic, made two guesses at the age of the youngest daughter, but was wrong both times. What is the age of the mother?
Sample Collaborative Problem-Solving Contest
For information on this and other contests, visit our website, http://www.natassessment.com.
2002 National Assessment & Testing
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