1. Into how many distinct regions do the angle trisectors of an
equilateral triangle divide the figure?
2. Chantal is rolling a die and wants to get all six different
outcomes. What is the probability that she succeeds in her first six
rolls?
3. Two counting numbers are personable if the product of the digits
of each number is greater than the sum of the digits of the other
number. Which of the following pairs of numbers are personable: 23
and 31; 205 and 42; 18 and 25; 8 and 111?
4. Two three-digit numbers 6A5 and 41A, with common digit A, are
personable. What is the smallest possible value of A?
5. A sequence of nine counting numbers begins and ends with 1's. Each
pair of adjacent numbers differs by exactly one. How many different
such lists are possible?
6. Esme is 60 m east and 80 m south of Rodrigue. Rodrigue cycles due
east at 10 m/s. Esme, who can bicycle at the same speed,
instantaneously calculates whether she can intercept Rodrigue. Can
she reach him by traveling in a straight line? If so, where will they
be when they collide? If not, why not?
7. At what speeds could Esme, from the previous problem, travel and
still reach Rodrigue at some point?
8. A palindromic number reads the same backward and forward. If all
palindromic counting numbers are listed in order, what are the
possible difference between consecutive terms in the list?
9. Consider the delayed Fibonacci sequence that starts with four I's
and is defined by F(n) = F(n-3) + F(n-4) : 1, 1, 1, 1, 2, 2, 2, 3, 4,
4, . . . In what position is the 1,000th even number in this sequence?
10. A rectangular piece of paper is folded so that one corner lands
on the opposite corner. The crease is as long as the longer side of
the rectangle. What is the ratio of the sides of the rectangle?
11. S(n) is defined to be the smallest number divisible by all
counting numbers from 1 to n. What is the smallest value for n such
that S(n) = S(n+3)?
Yip, I've spotted my mistake. ... first ... 'Twas me. It should read thus. Bo's result: x-y-z. Jo's result: x-(y-z) = x-y+z. Bo's result + 12 = Jo's result ...
1. Into how many distinct regions do the angle trisectors of an equilateral triangle divide the figure? 2. Chantal is rolling a die and wants to get all six...
... six ... Each ... due ... they ... I's ... 4, ... sequence? ... 1. 19 2. Prob(2nd roll diff from 1st)*Prob(3rd roll diff from 1st two) *...*Prob(last roll...
... Each ... 4. 30A > 5+A and 4A > 11+A 29A > 5 and 3A > 11 A > 5/29 and A > 11/3 A > 11/3 A >= 4 5. Contruct the 8 adjacent differences. They will each be...
... due ... they ... Let R be the point where Rodrigue started Let E be the point where Esme started. Let C be the point of collision. Let P be the point...
... n = 19 S(19) is divisible by 4 and 5, and therefore also by 20 S(19) is divisible by 3 and 7, and therefore also by 21 S(19) is divisible by 2 and 11, and...
... I started with 1 and went up incrementally (2,3,4,...) until I came to a number whose next three numbers' factors were already contained in numbers...
... I's ... 4, ... sequence? ... Note the following: F(1) is odd F(2) is odd F(3) is odd F(4) is odd F(5) is even F(6) is even F(7) is even F(8) is odd F(9) is...
... Since we're dealing with ratios, WLOG let the shorter side have length 1, and the longer side, length x Orient the rectangle so that the longer side is...
unless i'm mistaken slim, for which i extend an apology, your solution is elegant, concise, thorough and complete, but u should be (1+root(5))/2 in the...
You're right. I had solved the problem on paper, and had the right solution written down. When I typed it into my computer, I accidentally left out the '/2'...
... Given any palindrome x, how much must you add to it, to get the next higher palindrome? Let d = # digits in x. Case 1. x has an odd number of digits. ...
... all ... next ... non- ... be ... You defined two Cases 1c, so you have ruined your proof, haha. :P Were these problems all in the same post? Damn....
... due ... they ... Let R be the point where Rogrigue starts Let E be the point where Esme starts Let X be the intersection of E's longitude and R's latitude ...
Here's two math problems that were given to my sixth grade class in 1062 or 1963. To solve them correctly really requires a college education. Problem 1 ...
*A two digit number is 7 times the sum of its digits. If the digits are interchanged, the resultant becomes 27 less than the original number. Find the original...
... are interchanged, the resultant becomes 27 less than the original number. Find the original number. Hint: Don't forget that the two digit number, ab, has a...
... If not The answer to the first part is 63 The answer to the 2nd question can not be determined insufficient data all that is given is percehtages Best...
Hi All, For the second question, if you want the savings in percentage, here are my thoughts: Lets assume that: Income = X So, he spent 20% of his income in...
13. What is the radius of a circle in which a chord of length 10 is 5 units from the center? It's presumed that "5 units from the center" means prependicularly...
7. For what values of K is x – 1 a factor of x^2 – 6x + K? If x-1 is a factor of x^2 – 6x + K, then x = 1 satisfies x^2 – 6x + K = 0. (right?) 1^2 –...
10. The square of the complex number a + bi, where a and b are real numbers and i^2 = -1, is 2i. What are a and b? (a + bi)² = a² + 2abi + b²i² = a² +...
Please read b² = -1 (no-no cos b is real) or b² = 1 as b² = -1 (no-no because b is real) or b² = 1 Just in case anyone thought I was suddenly involving...
I thought I'd posted this before, but don't see it. Oh well; Clooneman's solution needs to be corrected: Let x = length of third side x > 5, because x + 6 >...
... Largest or not, the only prime satisfying the condition is 5 = 2 + 3 = 7 - 2. Suppose p1 = p2 + p3 = p4 - p5. Then p1 must be odd; otherwise we would have...
15. A rectangular picture has a frame that is 1 inch wide. The picture and the frame together form a larger rectangle. The area of the frame alone is 100...
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