World of Math
———————-CONGRATS————————
NATIONAL NEWS: Calvin wins fourth place in written, Bobby Shen (3rd written) wins countdown, and David Yang (who was top USAMO finisher too) won written. Maximilian Schindler got 2nd and finished first in the masters round. The Texas team finished first, with Missouri behind and Washington State third. North Carolina finished eighth. New Mexico took most improved team.
NCSSM NEWS: The first MaCh 8 took place as of today, 5/9/09! I won third in the general section and Roy won third in discreet math.
————————–DIRECTIONS—————————–
Every math problem has an answer that will be posted the next month. If you want to figure a problem out, just put the solution on your comment! Every month, the first person to get a problem right will be a winner. I’ll post his/her name on the page; if you request anonymity, please say so in your comment.
There are 4 rounds: the MathCounts/AMC 8 Round, the MathCounts Countdown Round (the easiest level), the AMC 10/AMC 12 (AHSME) round, and the AIME Round (which has problems worth 40+ pts). Problems within each round are ordered from least difficult to most difficult. Solve the problems you can solve; this page will prove if you need extra preparation or if you have sufficient knowledge in your level.
POINTS: MathCounts Countdown (1-2) MathCounts/AMC 8 (2-3) AMC 10/12 (4-6) AIME (25+)
Any problem not solved by the end of a month will go on to the next month, therefore adding points to its value.
———-PENALTIES/REWARDS———-
The first rule that you must always remember is, NO CHEATING!!! I am only going to name the test the question was from, so that you will have to look up more questions if you want to cheat. (It’s almost impossible to cheat on the AoPS FTW questions, anyways, as they pop up randomly on games.) Cheating is becoming a major problem on my math page, Roy’s contemporary music page, and Samuel’s unscrambling page (well, not anymore, because Roy isn’t doing questions).
The second rule you should know is that if your comment contains ANY chat terms besides “lol” and “omg”, your comment will be ruled as spam. And this IS automatic, so BEWARE!!!
And of course, there are rewards. If you detect a spelling/grammatical error, you will receive one point on this page. If you detect a math error, you will receive 2 pts. on this page.
—————————–MATHCOUNTS/AMC 8 ROUND (7th/8th grade)——————————-
Note: Questions are not necessarily from the MathCounts Competition or AMC 8.
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Math Problem of June No. 7 (From AMC 8/Mist Academy; points: 2) A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are 6.1, 8.2, and 9.7 centimeters. Find the area of the square in square centimeters.
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Math Problem of June No. 6 (From MATHCOUNTS/Mist Academy; points: 2) Roy did a number trick with Amy G. Roy told Amy G to pick any even number, double it, add 48, divide by 4, subtract 7, multiply by 2, and subtract her original number. What is the result Amy G should have attained? Find how to get the result WITHOUT substituting a number in. (1 point)
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Math Problem of June No. 8 (From AoPS FTW; points: 2) The third term of an arithmetic sequence is 5 and the sixth term is -1. Find the twelfth term.
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Math Problem of June No. 9 (From MATHCOUNTS Foundation; points: 2) Define a & b as a²-b²+ab when a and b are real numbers. Calculate the value of (3 & 4) & 5.
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Math Problem of May No. 4 (Made up; points: 2) What is the sum of all numbers whose digits add up to 9?
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Math Problem of June No. 15 (From MATHCOUNTS; points: 2) How many two-digit numbers greater than 50 have the property that the sum of the digits is more than double the ones digit?
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Math Problem of May No. 20 (Made up; points: 2) If x+1/x = 5, then find x²+1/x².
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Math Problem of May No. 43 (From AoPS FTW; points: 2) Given that both p and p+1 are prime numbers, find the least integer n that is divisible by neither p nor p+1.
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Math Problem of May No. 50 (From AHSME 1956; points: 2) How many scalene triangles have all sides of integral lengths and perimeter less than 13?
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Math Problem of June No. 16 (Based on MATHCOUNTS; points: 2) Amy G has a piece of land whose area is 30 square feet, all for her pet turtle. If she doubles the plot’s dimensions, how many square yards will the area of the new plot be? Express your answer as a common decimal reduced to the simplest form.
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Math Problem of May No. 2 (Made up; points: 2) How many squares are in a 3×4 array?
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Math Problem of May No. 50 (From AoPS; points: 2) Find the units digit of 1! + 2! + 3! + … + 2009! [Hint: n! is equal to n x (n-1) x (n-2) x .... x 1.]
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Math Problem of June No. 3 (From AoPS FTW; points: 2) Find the sum of the prime factors of 85,085.
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Math Problem of May No. 44 (Based on AoPS; points: 3) How many even factors does 96 have?
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Math Problem of July No. 1 (Made up; points: 3) How many different arrangements of there of 8 distinct keys in a circle? (An arrangement flipped over or rotated is the same one.)
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Math Problem of May No. 33 (From 2002 USA AMC 12 a.k.a. AHSME 2002; points: 3) Both roots of the quadratic equation x² – 63x + k = 0 are prime numbers. The number of possible values of k is…
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Math Problem of June No. 10 (Based on AGMath.com/Mr. Jason Batterson; points: 3) In Carnivoresville, there is a restaurant which offers two packs of Chicken McNuggets. One pack offers 7 in a box. The other has 10 in a box. What is the largest number of whole nuggets that you can NOT purchase?
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Math Problem of May No. 18 (Made up; points: 3) On the Altered AMC General Tests, there are 15 multiple choice questions with choices A through D (AMC 8, 10, and 12), 5 bubble-ins (AIME) with integer answers 0-999, and 5 proving questions. Assuming that mathemagician Tom Liang can do all the proving questions correctly but is too tired to do the rest and just guesses them all, what is the probability that he can get a perfect score? Express your answer in simplest exponential form.
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Math Problem of May No. 27 (From NCSSM Math Competition General Test #2; points: 3) How many five-digit numbers can be formed from the digits {0, 1, 2, 3, 4, 5, 6} ?
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Math Problem of May No. 48 (From AoPS FTW; points: 3) How many numbers between 100 and 1,000 are divisible by 13?
- Math Problem of February No. 9 (Based on AoPS; points: 4): If x + y = 10 and xy = 3, find x cubed – y cubed.
————-MATHCOUNTS COUNTDOWN ROUND (6th/7th grade)———–
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Math Problem of June No. 22 (From MATHCOUNTS Chapter Countdown) What is the value of 6 + (8/2)
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Math Problem of June No. 21 (From MATHCOUNTS Chapter Countdown) For what value of x is the expression (x-3)/4x equal to 0?
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Math Problem of June No. 23 (From MATHCOUNTS Chapter Countdown) A tree grew from 6 feet to 15 feet. By how much did its height increase?
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Math Problem of June No. 24 (From MATHCOUNTS Chapter Countdown) If f(x) = |x+3| and g(x) = x^2 – 6, what is the value of f(g(2))?
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Math Problem of June No. 25 (From MATHCOUNTS Chapter Countdown) The side of a square is increased by 50%. By how much is the area increased?
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Math Problem of May No. 24 (From AoPS FTW; points: 1) How many segments are determined by 4 points on a line?
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Math Problem of June No. 2 (From AoPS FTW; points: 1) If I designed a logo 2 inches wide and 1.5 inches tall, but the logo’s width needs to be 0.5 feet, how tall will the logo be?
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Math Problem of May No. 15 (From MathCounts National Countdown Round; points: 2) The Bulls are playing the Pistons in a playoff match, and the Bulls are winning the series 3-2. In basketball, you need 4 games to win the playoff match. Assume that the probability of the Bulls beating the Pistons is 3/5. Find the probability that the Pistons will win the playoff match.
—————AMC 10/AMC 12 Round (High schoolers)————-
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Math Problem of June No. 11 (Made up; points: 2) Given that sin x = 2009 and cos x = 287, find tan x.
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Math Problem of March No. 40 (Based on AoPS FTW; points: 4) There is a function rule that says a§b = (a³+a)[b²+90b+2009]. Based on this function, find [(0§7) § (41§49)] § [(7§287) § (√2009 § √2010)].
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Math Problem of May No. 46 (Made up; points: 4) What is the probability that a randomly chosen factor of 216 is divisible by 2? Express your answer as a fraction reduced to the lowest terms.
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Math Problem of April No. 1 (From MathCounts 1997; points: 5) In the Land of Eternal Darkness, people cannot see anything. The currency is made by cutting 1 x 1 triangles from 2.5 x 5 rectangles. How many types of currency are there, assuming a bill with no cuts is possible?
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Math Problem of May No. 29 (Based on NCSSM Math Competition General Test #2; points: 5) A school orders 99 textbooks, all costing the same. Unfortunately, the managers of the textbooks at the school weren’t very smart, and they forgot two digits in the total price, so it appeared as _381. _5. What is the cost of one textbook?
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Math Problem of May No. 30 (Made up; points: 5) Find log 10 in terms of α and β if log 2 = α and log 5 = β.
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Math Problem of May No. 31 (From MathCounts; rating: points: 5) Somebody is shooting darts at a target made up of concentric circles of radius 3 and 4. Every dart that lands in the radius 4 circle counts as 7 points, while every dart that lands in the radius 3 circle counts as 15 points. Find the greatest unattainable score. BONUS: Find the theorem that shows you a shortcut through this problem. (FREEBIE for Roy, 5 pts for everyone else)
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Math Problem of May No. 34 (From 2002 USA AMC 12 a.k.a. AHSME 2002; points: 5) Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages 40 miles per hour, he arrives at his workplace three minutes late. When he averages 60 miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace right on time?
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Math Problem of June No. 20 (From AMC 12; points: 5) A subset of the integers 1,2,…..100 has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?
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Math Problem of June No. 8 (From MATHCOUNTS; points: 5) What is the probability that a randomly chosen factor of 15^90 is a multiple of 15^65? Expres your answer in simplest fraction form.
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Math Problem of February No. 4 (Made up; points: 6): If a+b=10, b+c=20, c+d=30, and so on and so forth until z+a, what is the sum of all the letters of the alphabet excluding a and z?
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Math Problem of March No. 4 (From 2009 MathCounts Chapter Target Round; points: 6) [ANYBODY] A 3 by x array board is created. It has 70 different squares of 3 different sizes. Find x.
————-AIME ROUND (Juniors and seniors in high school)———-
Note: There will not be proving questions like the questions on the USAMO and IMO because they are too advanced, they would be too hard for most of you anyways, and you can find sample problems on Roy’s page.
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Math Problem of May No. 38 (From AIME; 35 pts) How many positive integers have exactly three proper divisors excluding itself, each of which is less than 50?
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Math Problem of June No. 1 (From Russian Olympiad; 35 points) The denominators of two irreducible fractions are 600 and 700. Find the minimum value of the denominator of their sum. Express your answer in simplest form.
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Math Problem of May No. 39 (From AIME; 40 pts) A box has 10 polka-dotted candies and 10 striped candies. Roy picks two candies at random, Amy G picks two of the remaining candies at random, and Samuel picks two of the remaining candies at random. Given the probability that Roy and Amy G get the same color combination, regardless of order, is m/n, where m and n are prime, positive integers, find |m-n|.
———————————————————————-
WINNER OF MATHCOUNTS/AMC 8 ROUND: Amy G. (3)
WINNER OF MATHCOUNTS COUNTDOWN ROUND: Samuel C. (2) and Amy G. (2)
WINNER OF AMC 10/AMC 12 ROUND: Samuel C. (5)
WINNER OF AIME ROUND: Amy G. (70)
WINNER OF FREE POINTS ROUND: Samuel C. (1.5)
——————–RATINGS:——————-
These ratings are similar to AoPS’s FTW.
Amy G-1290 Samuel-1235 Roy-1140
HALL OF FAME
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Winner Number of times Place
Roy ** 57.5 3
Amy G* ** 124 LALALA 1
Samuel 31 2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
* indicates the top winner(s).
** indicates winner of quarter.
SCOREBOARD (6/22/09):
Amy G: 6 Samuel: 0 Everybody else: 0
Entries (RSS)
March 13th, 2009 at 10:19 pm
Proving Math Problem of February No. 1 (From Adv. Class, beyond geometry; rating: very hard)
Prove that two numbers on the Pascal’s Triangle always add up to the number below them in the next row. Hint: You’ll have to set up a factorials equation before solving.
GAH THIS IS GOING TO BE SO ANNOYING TO TYPE WITHOUT FRACTION BARS. STUFF IN { } are fractions.
First we must define each item we are using. If we write Pascal’s triangle with 0th row as 0c0, 1st row as 1c0 and 1c1, 2nd row as 2c0 2c1 2c2, etc, we see that the two values above a certain value are (n-1)c(k-1) + (n-1)c(k), where n is the first value of the combination and k is the 2nd. Now, we just show that those two are equal.
(n-1)c(k-1) + (n-1)c(k) = {(n-1)!/((k-1)!(n-k)!} + {(n-1)!/(k!(n-k-1)!)}
Using n! = n(n-1)!,
(n-1)c(k-1) + (n-1)c(k) = (n-1)! * [{k/(k!(n-k)!)} + {(n-k)/(k!(n-k)!)}]
Now, just simplify.
(n-1)c(k-1) + (n-1)c(k) = (n-1)! * {n/(k!(n-k)!)}
(n-1)c(k-1) + (n-1)c(k) = {n!/(k!(n-k)!)}
(n-1)c(k-1) + (n-1)c(k) = nck
Haha that looks funny. nck. It’s n combination k.
March 13th, 2009 at 10:22 pm
Math Problem of February No. 7 (Made up; rating: very hard)
John J. Goodwill writes the letters I LOVE MATH on a piece of paper. How many combinations of these letters can he make?
One. n combination n is always 1.
I hope you meant ORDERINGS. Then, it’s 9! or 362880.
March 13th, 2009 at 10:25 pm
Math Problem of February No. 8 (From Adv. Math Class, beyond geometry; rating: hard)
An octopus who had three legs bitten off by a shark has 5 black and 5 white socks. How many ways can he wear 3 black and 2 white socks?
Well, it’s not a probability, so we can ignore 5 of each. He has five feet, and we can choose 3 of them to have black socks. 5c3 = 10.
March 13th, 2009 at 10:29 pm
Math Problem of March No. 2 (From 2009 AMC 10A question 24; rating: hard) [FOR 94QM AND OTHERS]
Three vertices are randomly selected from a regular cube. What is the probability that, when connected, form a shape inside the cube?
The cube has 8c3, or 56, ways to choose 3 vertices. Let’s count the ways that they can’t be a shape in the cube. The only time that occurs is when all three points are on the same face. That can occur in 4 different ways per face, so (56-24)/56 = 4/7.
March 13th, 2009 at 10:35 pm
Math Problem of March No. 3 (From 2009 AMC 10A question 19; rating: medium) [FOR ALL VIEWERS]
Two circles are internally tangent to one another. One is Circle A and has radius 100. The other is Circle B and has radius r < 100. Circle B rolls around Circle A’s circumference 360 degrees. They have the same point of tangency as they did before Circle B rolled around Circle A’s circumference. How many possible values can r have? a) 4 b) 8 c)9 d) 10 e) 99 f) other
The circumference of the big circle is 200pi. It must be divisible by 2pi to make the little circle have an integer radius r. We must now look for factors of 100. You can have 0, 1, 2 quantities of 2’s and same for 5’s. 3*3=9. However, one of them is 100 but r<100. So 9-1 =8.
March 13th, 2009 at 10:36 pm
OK I’m done for now… This is a lot more time consuming than real math competitions–I have to think of a good way to word my solution.
March 14th, 2009 at 3:23 pm
21 in one day, wow.
March 15th, 2009 at 4:47 pm
How did you prove the pascal’s thing in math?
March 16th, 2009 at 8:04 pm
Oh, we had this factorial equation (which was ezpz) and we had to solve it to make it true.
March 17th, 2009 at 6:17 pm
haha a whole class can’t beat my proof
March 17th, 2009 at 6:18 pm
i realize how to do like all the aime problems now…
i fail
March 28th, 2009 at 6:47 pm
don’t brag please
March 29th, 2009 at 9:46 am
that whole class thing was a joke…
of course they can beat me…
March 29th, 2009 at 5:35 pm
Now do the problems, if you please, instead of ranting on about the class thing.
April 28th, 2009 at 7:27 pm
Math Problem of April No. 2 (From AoPS FTW; rating: easy)
(the numbers are 12 and 13) *feels oddly happy*
Two counting numbers’ squares’ difference is 25. What is the sum of those two numbers?
25!
BTW, in the archives, for Math Problem of March No. 5, is the difference 8? *hopefully*
April 28th, 2009 at 8:26 pm
Math Problem of April–correct
Archive–wrong, but very close
April 28th, 2009 at 10:15 pm
** the difference is 7?
April 29th, 2009 at 4:39 pm
April 1. 5 types… ME GOT LOTS AND LOTS OF MONIES!!!!
March 1. 10
April 29th, 2009 at 6:06 pm
agchan correct
roy: Apr. 1 wrong
March 1 duh
April 29th, 2009 at 7:37 pm
Math Problem of February No. 4 (Made up; rating: very hard plus):
If a+b=10, b+c=20, c+d=30, and so on and so forth until z+a, what is the sum of all the letters of the alphabet excluding a and z?
1445?
Math Problem of May No. 2 (Made up; rating: medium)
How many squares are in a 3×4 array?
20?
Math Problem of May No. 5 (Made up; rating: easy)
Two numbers whose product is 221 are added, and the sum is divided by their difference. Then, the quotient is added to the greater number and the little number is subtracted from this sum. Find the final answer.
11.5?
April 29th, 2009 at 8:12 pm
Wait…there’s an error in my answer for no. 4, you get 0.25 pts
Math Problem No. 2 how can there be so many?
Math Problem No. 5 correct..no calculator? It’s doable in your mind
April 29th, 2009 at 9:07 pm
3×4 array of royness
****
****
****
Two 3×3s
6 2×2s
12 1×1s
BUT I’m gonna stupid this one.
There are infinite squares because the question didn’t specify anything.
April 29th, 2009 at 9:13 pm
2009/2^3271 for the weird exponent one
April 29th, 2009 at 9:21 pm
49/100 for the repetitive fractions
April 29th, 2009 at 9:21 pm
LALALALA
SING, JUSTIN
SING, ROY
IT IS TIME FOR MUSIC ♪ ♫
April 29th, 2009 at 9:24 pm
1 digit number: 0 (negative factors) if only positive, 1.
2 digit number: none b/c the number is always a factor of itself.
April 29th, 2009 at 9:25 pm
also for 2 digit: if there are negatives, it’s always 0
April 30th, 2009 at 7:35 pm
23 and 24 correct
26 I didn’t word clearly; you get +0.5. The rules are: if you do a problem with an error, you get +0.25. If you actually diagnose the error, then you get 0.5, but this was way too blatant, so I added a bonus.
May 3rd, 2009 at 6:18 pm
Math Problem of May No. 6 (Made up; rating: easy) There is one one-digit number whose factors (excluding itself) add up to itself. What is it?
6
May 3rd, 2009 at 8:04 pm
Yes, that’s correct.
May 4th, 2009 at 7:42 pm
Math Problem of May No. 11 (From Advanced Math Class; rating; v easy for Roy and Amy G, hard for everyone else) Find the fifteenth term of the Fibonacci sequence.
610?
May 4th, 2009 at 7:44 pm
That’s correct. But how? Did you use a calculator? (2pts) Or did you use your mind, pencil, and paper? (4pts) Or did you use just your mind? (6pts)
May 4th, 2009 at 8:26 pm
Math Problem of May No. 7 (From North Carolina State Algebra I Test 2000; rating: hard) How many digits are in the product (4^5) x (5^13)?
13 digits
Math Problem of May No. 8 (From North Carolina State Geometry Test; rating: hard) The hypotenuse of a right triangle is 7, and the perimeter of the same triangle is 16. Find the area.
8 sq. units
Math Problem of May No. 10 (Made up; rating: v easy for Roy, easy for everyone else) There is a function a
b such that a 
b is equal to [(5a+12b)^2] / [8a-3b]. Assume that x = 3 and y = 8. Find x 
y.
Undefined… because 8(3)-3(8)=0 and division by 0 is undefined (right?)
Math Problem of May No. 12 (From North Carolina State Geometry Test; rating: easy for Roy, hard for everyone else) A quadrilateral has sides 12, 16, 15, and 25. There is a right angle where the side measuring 12 and the side measuring 16 meet. Find the area of the quadrilateral.
246 sq. units
May 5th, 2009 at 3:38 pm
Math Problem No. 7 correct
Math Problem No. 8 correct
Math Problem No. 10 correct
Math Problem Nol 12 correct
+14, agchan, that almost triples your score
May 6th, 2009 at 5:53 pm
Math Problem of May No. 9 (From North Carolina State Algebra I Test 2000; rating: hard for Roy, v. hard for everyone else) There is a sequence as shown: 1, 2, 5, 12, x, 70, 169,… Find x.
x=29
May 6th, 2009 at 6:45 pm
That’s correct, +5
But how?
May 7th, 2009 at 5:13 pm
i’m amazing
double the previous number and then add that number with the one before that
May 8th, 2009 at 2:50 pm
Yeah, nice. I’m the expert at those sequences. Did you use your head? That was my way.
May 9th, 2009 at 9:12 am
Math Problem of May No. 17 (From Advanced Math Class; rating: easy for Roy, medium for Amy G, hard for everyone else) A lock-cracker is trying to break through a lock to steal jewels, an MP3 player, and gold and cash valued at $2,000,000. Unfortunately for him, the numbers are from 1-40, there are three numbers which make up a locker combination, and no number can be used twice in the combination. The robber is quite dumb, so he needs your assistance for how many combinations there are. Can you help him? (Note: If you do get it correct, the robber will give you $500,000.)
59, 280 combinations
or is it 9,880?
May 9th, 2009 at 3:29 pm
Just tell me how to do it: like if you have 5600002 x 15625600, you don’t need to multiply it out.
59280 is correct, I used calculator.
May 10th, 2009 at 10:50 am
Oh justin. My score vs. everyone else should be 57.5
i have enough competition now
May 10th, 2009 at 2:58 pm
calculator!!
i just did permutations: 40 P 3
May 10th, 2009 at 3:43 pm
Roy–wrong. If you read carefully it says,”4/29/09″. By that, I mean the points scored since that date.
Okay, acceptable, Samuel.
May 11th, 2009 at 3:03 pm
Math Problem of May No. 28 (From AoPS FTW; rating: v easy for Roy, medium for everyone else) At the mall, Crystal wants to buy one entree, one drink, and one desert. The entrees are pizza, chicken teriyaki, corn dog, and fish and chips; the drinks are lemonade and root beer; and the desserts are frozen yogurt and chocolate chip cookies. How many distinct possible meals can she buy?
FIRST OF ALL, YOU BLAME ME FOR INCORRECT SPELLING AND GRAMMAR ALL THE TIME, YET I FIND MORE ERRORS ON YOUR BLOG THAN MINE.
16 possible meals
May 11th, 2009 at 3:08 pm
I guess that’s correct.
Oh, and guess what? Good news! If you find an error you get 1 point, not 0.5 points. And this is general speaking, not just limited to this page.
So where’s my error?
May 11th, 2009 at 5:32 pm
desert=dessert
May 11th, 2009 at 5:38 pm
Math Problem of May No. 29 (From AoPS FTW; rating: easy) For a world record, George Adrian picked 15832 lbs of apples in 8 hours. Assuming he maintained a constant rate, how many pounds of apples did he pick in 3 hours?
5937 lbs of apples??
May 11th, 2009 at 6:00 pm
47 correct
Error check validated
Therefore, +3
May 12th, 2009 at 3:25 pm
Math Problem of May No. 13 (Made up; rating: v. easy for Roy, easy for Amy G, medium for everyone else) Find the midpoint of (2,7) and (8,6).
(5, 6.5)
Math Problem of May No. 16 (Made up; rating: v easy for Roy, easy for everyone else) What is {1/[1/(½)]}² ? (Apparently, you don’t need LaTeX.)
1/4
Math Problem of May No. 19 (Made up; rating: easy) Find three different numbers that multiply up to 2009.
7 x 7 x 41
Math Problem of May No. 20 (Made up; rating: v easy for Roy, medium for everyone else) If x+1/x = 5, then find x²+1/x².
4 1/16
Math Problem of May No. 21 (From MathCounts Competition; rating: medium) Darts are thrown randomly at a board showing three concentric circles of radii 2, 3, and 4. If a dart hits inside one of the circles, what is the probability, expressed as a common fraction, that it is in the interior of the circle of radius 3 but not in the interior of the circle of radius 2?
5/16
Math Problem of May No. 22 (From AoPS FTW; rating: easy for Roy, medium for everyone else) Julie is preparing a speech for her class. Her speech must last between 30 minutes and 45 minutes. The ideal rate of speech is 150 words per minute. If Julie speaks at this rate, which of the following would be the appropriate length of her speech? A. 2250 B. 3000 C. 4200 D. 4350 E. 5650
E. 5650 (it must be between 4500 and 6750 words)
Math Problem of May No. 23 (From AoPS FTW; rating: medium) A wheel has a diameter of 9/∏ (pi) feet. How many revolutions does the wheel make when rolled for 60 yards?
20 revolutions (and that was the coolest pi ever)
Math Problem of May No. 25 (From AoPS FTW; rating: medium) Every other time that Roy Li saw a friend today, he gave away half of his flowers. At the end of the day, he had five flowers left. How many flowers did he start the day with if he saw 8 friends?
80 flowers (and I recieved none D:)
Math Problem of May No. 26 (From AoPS FTW; rating: easy) If the reciprocal of (x-2) is (x+2), what is the greatest possible value of x? Express your answer as a common decimal and in simplest radical form.
erm… sqrt(5)?
May 12th, 2009 at 3:55 pm
Wow, Amy, I guess Roy will be relying on you to win the group competition. And you might win the individual competition.
13 correct
16 correct
19 correct
20 hmm, maybe not this time. How did you do it?
21 wrong
22 correct
23 correct. Did you copy/paste that? Do you want to learn how to make such things?
25 correct. That was supposed to be a joke! Maybe if you see him and he has flowers, he’ll give you some.
26 correct–The “common decimal” was supposed to be a trick
GRAND TOTAL FOR AMY G: 17
Won’t Samuel be surprised!
May 12th, 2009 at 5:04 pm
Oh gee, I’m losing. Shucks
May 12th, 2009 at 8:06 pm
20 – If x+1/x = 5, then find x²+1/x².
x+1/x=5
x+1=5x
1=4x
1/4=x
(1/4)²+1/(1/4)²
=1/16+1/(1/16)
=1/16+16
=16 1/16?
*blinks* Is it… 16 1/16? I think I forgot to square the second x² the first time I did that problem… o__o Then again, I could be totally on the wrong track and 16 1/16 could be wrong too. xD
21 – Darts are thrown randomly at a board showing three concentric circles of radii 2, 3, and 4. If a dart hits inside one of the circles, what is the probability, expressed as a common fraction, that it is in the interior of the circle of radius 3 but not in the interior of the circle of radius 2?
probability of landing in radius-3 circle but not in radius-2 circle: (pi)(3)²-(pi)(2)²
total area of board: (pi)(4)²
probability of landing in radius 3 but not in radius 2
=
(pi)(3)²-(pi)(2)²
_____________
(pi)(4)²
= (3²-2²)/4²
=(9-4)/16
=5/16
what’s wrong? D:
Hehe. Yeah. I copy-pasted. Do you do the pi symbol the same way you can do symbols like ♪, ♫, ☺☻♦ on the keypad?
lol no.
May 13th, 2009 at 3:02 pm
Oh, you misread the problem. It’s not (x+1) / x, it’s x + (1/x).
And I miscalculated! Nice error catch. So +4.
May 14th, 2009 at 7:46 pm
Math Problem of May No. 6 (Made up; rating: easy) There is one one-digit number whose factors (excluding itself) add up to itself. What is it? Find a two-digit number which also satisfies this property. (2 pts each)
28
May 15th, 2009 at 2:25 pm
So, +2
May 16th, 2009 at 10:41 pm
yawn
good thing you put amy on my team
i can just lounge around
May 16th, 2009 at 10:43 pm
btw pythag was A winner
not THE winner
May 16th, 2009 at 10:46 pm
put an ULTRA ULTRA HARD problem worth 90 points kk?
May 17th, 2009 at 9:54 am
Ugh!!! Laziness! Maybe I should put you alone again…
So pythag is David Yang? *trait: Most of the past National MathCounts winners are users on AoPS FTW. And what does “btw” mean?
Ultra hard problem. Hmm…I’m not sure if my abilities range that high. If you want I’ll extract some AIME/USAMO problems with answer keys. But there’s no way they’re going to be that high-rated; the maximum number of points will be 75. And I guess my planet problem will count as 45.
May 18th, 2009 at 4:41 pm
Math Problem of May No. 35 (Made up; rating: very easy) If John bikes 2 miles in 30 minutes, find his average speed.
15 minutes/mile
btw means By the Way
and gtg means Got to go NOT Good to go
everyone knows that even if they don’t text or IM
May 18th, 2009 at 5:17 pm
Okay, +1. Shouldn’t provide such easy questions.
NEXT TIME, PLEASE LIST THE CATEGORY (e.g. MathCounts Countdown Round, etc.)
And how do they know? Did you know when you were first chatting? Did Roy know?
Please no chat-talk next time, or your comment may count as spam and you will not be credited.
May 21st, 2009 at 3:22 pm
Chat-free world, especially on my top page, World of Math (your tagline)
hmm…so no chat-talk??
May 21st, 2009 at 3:26 pm
Yes, no chat-talk.
I was surprised you’d actually notice it! +1
Free points
May 22nd, 2009 at 3:20 pm
Math Problem of May No. 42 (From 2009 MathCounts Chapter Countdown Round; points: 1) Find 3/(1/3).
9
May 22nd, 2009 at 3:27 pm
That’s correct.
And you’re winning countdown by 2 points! With only 12 days of school left–if you do problems in the other rounds then you’ll win the sections!
May 23rd, 2009 at 3:42 pm
take the last digit of the number and double it (2009–double the 9 to a 18) (1001–double the 1 to a 2)
then take the remaining number and subtract it from the doubled last digit (18-200 and 2-100)
then divide the difference by 7 (-182/7=-26) (-98/7=-14)
May 23rd, 2009 at 4:05 pm
CHEATER^10000000000000!
How?
May 24th, 2009 at 2:27 pm
1. you are not the only person that knows a bunch of random facts
2. take a pencil and a piece of paper…guess and check…blah
May 30th, 2009 at 8:47 am
MC/AMC8:
Math Problem of May No. 49 (From Algebra Workbook; points: 3) Parallelogram ABCD has vertices A(2,4) B (4,1) C (8,5) and D (6,x). Find x.
x=8
Countdown:
Math Problem of May No. 32 (From 2002 USA AMC 12 a.k.a. AHSME 2002; points: 2) Find the degree measure of an angle whose complement is 25% of its supplement.
50 degrees
AIME round:
Math Problem of May No. 37 (From AIME; 30 pts) In quadrilateral ABCD, angle B is right, diagonal AC is perpendicular to CD, AB = 18, BC = 21, and CD = 14. Find the perimeter of ABCD.
84 units
Math Problem of May No. 36 (Made up; 45 points) Cephalo, Rhododron, Xenei, and Titanic are four planets. Cephalo’s year is 36 Earth years, Rhododron’s year is 1525 Earth days, Xenei’s year is 1080 Earth months, and Titanic’s year is 0.52 Earth years. They all revolve around a sun. On September 2009, they are all in a line. When is the next year they will be in a line? (Solution: 20 pts; Method: 25 pts)
(oh, geezums, here we go)
First, convert all the planets’ years to Earth years.
Cephalo’s year is 36 Earth years.
Rhododron’s year is 1525/365 Earth years, or about 4.2 Earth years.
Xenei’s year is 1080/12 Earth years, or 90 Earth years.
Titanic’s year is .52 Earth years.
Find the least common multiple of 36, 4.2, 90, and .52. It is …whoa! Uh… 142740. (I think. o__O I’m not sure. Actually, now that I look at that number, I’m REALLY not sure. But whatever.) They will all be in a line in 142740 Earth years after 2009, or in the year… 144749?
Fudge…. I don’t think that’s right. *blinks* Help.
May 30th, 2009 at 8:51 am
Oh lord, that’s a lot.
MC/AMC 8: correct (so you beat Samuel)
Countdown-incorrect, sorry
AIME-dang, that’s correct! Roy will have difficulty beating you!
I’ll give you partial credit for that (rounding). 40 points
It was Roy who advised me to put those questions up
May 30th, 2009 at 8:01 pm
Oh, fail. -.- 60 degrees.
May 30th, 2009 at 8:16 pm
Good job. +2
June 19th, 2009 at 2:23 pm
i can do these questions so trivially, so i stopped doing this page
June 22nd, 2009 at 12:41 pm
Math Problem of June No. 5 (From MATHCOUNTS/Mist Academy; points: 1) If x=3, what is the value of 2x+3?
9
Math Problem of May No. 24 (From AoPS FTW; points: 1) How many segments are determined by 4 points on a line?
3
Math Problem of May No. 47 (Made up; points: 1) Find the tens digit of 5^2008.
2
Math Problem of June No. 2 (From AoPS FTW; points: 1) If I designed a logo 2 inches wide and 1.5 inches tall, but the logo’s width needs to be 0.5 feet, how tall will the logo be?
4.5 in
Math Problem of June No. 4 (From AoPS FTW; points: 2) One day, scientists capture and mark 45 frogs. The next day, they catch 40 frogs, 10 of which are marked. Find the best estimate for the total number of frogs.
180 frogs
Math Problem of May No. 44 (Made up; points: 2) What are the last three digits of 5^2009? 5^2010?
125 and 625
June 22nd, 2009 at 4:36 pm
Errors: there is no Fun and Misc. page anymore so how can I get a point on that page if I find an error like this one
“Introduction to this blog”-same problem
June 22nd, 2009 at 7:41 pm
AGCHAN:
June No. 5 correct
May No. 24 sorry
May No. 47 correct
June No. 2 incorrect (watch your units)
June No. 4 correct
May No. 44 correct
TOTAL: 6 pts
SAMUEL:
BWAHAHAHAHAHAHAHAHAHAHAHAHAHAHA!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Maybe you forgot to look at the page widget.
MWAHAHAHAHAHAHAHAHAHAHAHAHAHAHA!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
But it’s true, I do need to fix that for new visitors.
June 23rd, 2009 at 7:01 pm
AAAAAAAAHHHHH
EXPLAIN.
June 24th, 2009 at 11:04 am
Explain what? Math, my laughter,…?
June 25th, 2009 at 3:54 pm
*after considering for a minute or two* Both, please.
June 25th, 2009 at 6:33 pm
May #24–the lines don’t all have to be 1 unit long
June #2–watch your units
My cacophonious laughter was because Samuel didn’t realize that my page widget still gave visitors access to all pages I had.
June 27th, 2009 at 6:41 pm
Hmm… so the segments can overlap? …six, then?
And what’s wrong with my units? 1.5 inches over 2 inches is equal to 4.5 inches over 6 inches.
Fantastic. *nods solemnly*
July 10th, 2009 at 2:41 pm
$1 + 1 = 2$
July 18th, 2009 at 6:20 pm
Hi, nice site you have here!
How do we submit answers/solutions to the problems?
July 18th, 2009 at 6:33 pm
Hullo! Thanks for the compliment.
You just copy-paste a question, then write out your answer via comment. You don’t have to include your solution, but if you want to you can.
Thanks for commenting!