To Solve Or Not To Solve
Find all the solutions to the equation
sqrt{x+7}+x=13 .
x=9
k is a positive integer such that the least common multiple of {12,15,20,k} is 420. What is the second smallest possible value of k ?
14
The front face of a rectangular box has an area of 75 square inches, the top face has an area of 90 square inches, and the left face has an area of 13 1/3 square inches. What is the volume of the box, in cubic inches?
sqrt{75 times 90 times (40/3)}=300
Old McDonald wants to propose to his sweetheart Mrs. McBeth. He put signs with letters on the chests of seven chickens (one letter per chicken) and orders them in a line so that the letters read “MARRYME”. However, the moment Old McDonald looks away, the chickens change places, so that the letters are rearranged, every time in a totally new way. How many times does Old McDonald have to look away and back (and yell at the chickens), so that the naughty birds try all possible wrong permutations and finally spell it correctly?
{7!}/4 = 1260
The problem has a couple of different interpretations, so 1259 and 1261 are also acceptable.
Suppose your friend has a (0,0,0,6,6,6) die (that is, a die with 0 pips on three faces and 6 pips on the other three faces) and you have a (1,1,2,2,6,6) die (a die with 1 pips on two faces, 2 pips on two faces, and 6 pips on two faces). If both of you roll your dice at the same time, who is more likely to have the higher roll?
You are: half the rolls you win and sometimes you tie.
For what values of t does the quadratic equation
x^2+2(t-8)x+4t=0
have exactly 1 solution?
t in {4,16}
Find all the common divisors of 7854 and 2860
{1,2,11,22}
If a^4b^3=1080 and a^3b^4=18 , what is a^5b^2 ?
64,800
Old McDonald has a farm. One day he decided to host a “Miss Chicken Coop” competition. There are 10 chickens standing in a line on the stage: two gray ones, three red ones and five brown ones. What is the probability that the first one and the last one in the line have the same color?
5/10 * 4/9 + 2/10*1/9 + 3/10*2/9=28/90=14/45.
If the roots of the equation
x^2 +11x+c=0
are reciprocals of each other, compute all possible values of 𝑐.
c=1 , by Vieta.
Let a and b be two different numbers such that
a^2 + 3a + 1=0 and b^2+3b+1=0 .
Find the value of a/b + b/a .
a/b+b/a = {a^2+b^2}/{ab}=7
One bell rings every 4 hours, a second bell rings every 6 hours, and a third bell rings every 10 hours. If all three bells just rang together, how many times in the next 100 hours (including the 100th hour) will exactly two bells ring together?
7+4+2=13
Find the ordered pair (x,y) that satisfies the system
{4y-x}/{xy}=-1 ,
{y+3x}/{xy} =9.5
(2, 1/3)
In the finale of “Miss Chicken Coop” competition, four hens compete by testing how many eggs they can lay in a day. Four different boxes are placed on the stage in a row, and a hen is placed into each box. Each hen can lay between zero and three eggs. It is known that there was no tie for first place, and the winner laid as many eggs as the other three combined. In how many possible ways could the boxes with eggs look in the end of the day?
40
[(3,2,1,0) -> 24; (3,1,1,1) -> 4; (2, 1, 1, 0) -> 12]
Gil and Oleg each brought a lunchbox that contains one apricot, one banana, one kiwi, one peach, and one yuzu. Gil randomly selects one fruit from his lunchbox and puts it into Oleg’s box. Oleg then randomly selects one fruit from his lunchbox and puts it into Gil’s. What is the probability that after this process the contents of the two lunchboxes are the same?
2/6 = 1/3