tag:blogger.com,1999:blog-1974366466896776665.comments2022-03-27T05:59:34.842-05:00Math Problem of the DayDavidhttp://www.blogger.com/profile/09232747857608296294noreply@blogger.comBlogger35125tag:blogger.com,1999:blog-1974366466896776665.post-54026055965478218232010-12-16T22:36:25.683-06:002010-12-16T22:36:25.683-06:00no proof, but here's some prime numbers in a s...no proof, but here's some prime numbers in a spiral. http://www.flickr.com/photos/modern_carpentry/3896432339/in/set-72157622057420172/Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-83623843446938093032009-12-19T17:25:42.969-06:002009-12-19T17:25:42.969-06:00Correction: 5*7*9 - 4 = 311.Correction: 5*7*9 - 4 = 311.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-11277713616052798622009-10-24T14:09:29.844-05:002009-10-24T14:09:29.844-05:00How about f(x) = |x|^\sqrt{2}?
Since we take the a...How about f(x) = |x|^\sqrt{2}?<br />Since we take the absolute value before exponentiating to an irrational power, f is defined for each real number x.<br />Also, f(f(x)) = ||x|^\sqrt{2}|^\sqrt{2} = (|x|^\sqrt{2})^sqrt{2} = x^2.Japheth Woodhttps://www.blogger.com/profile/17314060401914172768noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-33744956949413233552009-10-24T06:23:09.677-05:002009-10-24T06:23:09.677-05:00The answer is (-6, 36).
The equation of the circl...The answer is (-6, 36).<br /><br />The equation of the circle has the form<br />x^2 + y^2 + ax + by + c = 0.<br /><br />After substituting y = x^2, we have<br />x^2 + x^4 + ax + bx^2 + c = 0.<br /><br />This is a fourth-degree equation, and the sum of the roots is zero, because the coefficient of x^3 is zero.<br /><br />It is given that 1, 2, and 3 are roots. So the fourth root must be -6, and so the fourth point of intersection is (-6, 36).Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-29420805661277723062009-10-16T21:52:32.577-05:002009-10-16T21:52:32.577-05:00The answer is 2 feet.
This is a special case of R...The answer is 2 feet.<br /><br />This is a special case of <a href="http://en.wikipedia.org/wiki/Regiomontanus'_angle_maximization_problem" rel="nofollow">Regiomontanus' angle maximization problem.</a>Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-20681123618901935562009-10-16T21:39:41.881-05:002009-10-16T21:39:41.881-05:00The answer is A = p^2 (2 - √3) / 4
A proof can be...The answer is A = p^2 (2 - √3) / 4<br /><br />A proof can be found <a href="http://answers.yahoo.com/question/index?qid=20081127220835AA641Uv" rel="nofollow">here</a>.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-28383873904133958792009-10-16T02:03:30.766-05:002009-10-16T02:03:30.766-05:00Answer: N = 10^24.Answer: N = 10^24.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-62894626213679028562009-10-16T01:57:39.270-05:002009-10-16T01:57:39.270-05:00Answer: 5*7*9 - 4 = 301.Answer: 5*7*9 - 4 = 301.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-59354033113783008242009-10-11T13:36:56.724-05:002009-10-11T13:36:56.724-05:00The answer is 1/4.
Once the student has answered ...The answer is 1/4.<br /><br />Once the student has answered the other questions, the parity of his score (odd or even) will depend on his response to the last question. Therefore, his score is even with probability 1/2.<br /><br />Suppose that the score S is even. If he had reversed all of his responses, writing true for false and false for true, then his score would have been (50 - S), which is also an even number. But note that exactly one of the two quantities S and (50 - S) is a multiple of 4. So, if S is even, then the probability is 1/2 that it is also divisible by 4.<br /><br />Therefore, the probability that the score is divisible by 4 is 1/2 * 1/2 = 1/4.<br /><br />REMARK: This argument is valid when the number of questions is congruent to 2 modulo 4. In other cases, the probability would be slightly different than 1/4. It is a challenge to find the exact probability in those cases.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-6485415750951972032009-10-10T11:59:12.272-05:002009-10-10T11:59:12.272-05:00The answer is 249,499,750,500.
Note that k(k+1)(k...The answer is 249,499,750,500.<br /><br />Note that k(k+1)(k+2) = k(k+1)(k+2)(k+3)/4 - (k-1)k(k+1)(k+2)/4. When this expression is summed from k=1 to 998, all of the terms cancel except for 998*999*1000*1001/4 = 249,499,750,500.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-63141990753641950202009-10-09T13:24:08.657-05:002009-10-09T13:24:08.657-05:00The answer is (3/4)^n.
The only way that A can fa...The answer is (3/4)^n.<br /><br />The only way that A can fail to be a subset of B is if there exists an element x that belongs to A but does not belong to B.<br /><br />For each x, the probability is 1/2 that x belongs to A, and 1/2 that x does not belong to B. Since the events are independent, the probability is 1/4 that x belongs to A but not B.<br /><br />The probability that this event does not occur for a given x is 1 - 1/4 = 3/4, and the probability that it never occurs for any x is (3/4)^n.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-5158406783364110762009-10-08T16:51:27.623-05:002009-10-08T16:51:27.623-05:00The answer is 161.
1/x + 1/y = 1/210
(x+y)/(xy) =...The answer is 161.<br /><br />1/x + 1/y = 1/210<br />(x+y)/(xy) = 1/210<br />xy = 210(x+y)<br />xy - 210x - 210y = 0<br />(x - 210)(y - 210) = 210^2<br /><br />Now, 210^2 has 81 positive divisors and 81 negative divisors. So there are 162 possible values for x - 210, and y - 210 is determined by x - 210. However, we must exclude the case where x - 210 = y - 210 = 210, because this leads to division by zero.<br /><br />Therefore the number of integer solutions is 2*81-1 = 161.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-25127711963852966612009-10-08T16:47:25.108-05:002009-10-08T16:47:25.108-05:00The answer is 97,038,694,279.
Let f(n) be the num...The answer is 97,038,694,279.<br /><br />Let f(n) be the number of n-digit numbers (with leading zeros allowed) that contain two consecutive nines. <br /><br />Let n>2 be given, and let S be the set of all n-digit numbers that contain two consecutive nines.<br /><br />If an n-digit number starts with 99 then it belongs to S, and there are 10^(n-2) such numbers.<br /><br />The number of elements in S that begin with a single 9 is 9*f(n-2), because there are nine choices for the second digit (0-8) and there are f(n-2) ways to complete the remaining n-2 digits.<br /><br />The number of elements in S that do not begin with 9 is 9*f(n-1), because there are nine choices for the first digit (0-8) and there are f(n-1) ways to complete the remaining n-1 digits.<br /><br />Therefore f(n) = 9*f(n-1) + 9*f(n-2) + 10^(n-2). We can use this recurrence to rapidly calculate f(12).<br /><br />f(1) = 0<br />f(2) = 1<br />f(3) = 9*1+0+10 = 19<br />f(4) = 9*19+9*1+100 = 280<br />...<br />f(12) = 97,038,694,279Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-43419785629173131442009-10-08T16:17:05.483-05:002009-10-08T16:17:05.483-05:00The answer is 02.
Note that (sqrt(3)+sqrt(2))^2 =...The answer is 02.<br /><br />Note that (sqrt(3)+sqrt(2))^2 = 5+sqrt(24).<br /><br />If n is a positive integer then (5+sqrt(24))^n + (5-sqrt(24))^n is an integer, because the terms involving radicals cancel in the binomial expansion. Since (5-sqrt(24))^n is less than 1/2, it follows that (5+sqrt(24))^n + (5-sqrt(24))^n is the integer nearest to (5+sqrt(24))^n.<br /><br />Let n = (10^100)/2. In the expansion of (5+sqrt(24))^n + (5-sqrt(24))^n, all terms are multiples of 100, except for the first term 2*5^n and the last term 2*24^(n/2)<br /><br />It is easy to check that 5^k = 25 (mod 100) for all k>1, and 24^k = 76 (mod 100) for all even k>1. Therefore, our integer is congruent to 2*25+76 modulo 100, so the last two digits are 02.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-75501528941927768052009-10-07T15:44:30.089-05:002009-10-07T15:44:30.089-05:00The answer is 3/4.
Observe that if A, B, and C ar...The answer is 3/4.<br /><br />Observe that if A, B, and C are three points on a circle, then the points lie on a common semicircle if and only if the center of the circle does not lie inside the triangle ABC.<br /><br />Let A, B, and C be three randomly selected points on the circle. Let a, b, and c be the points on the circle that are directly opposite to A, B, and C, respectively.<br /><br />Using these six points we form eight triangles: ABC, ABc, AbC, Abc, aBC, aBc, abC, and abc. Observe that exactly two of these triangles contain the center of the circle. But each triangle has the same likelihood of containing the center.<br /><br />Therefore, the probability that A, B, and C lie on a common semicircle is 6/8, or 3/4.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-81802637914711290782009-10-04T16:07:44.962-05:002009-10-04T16:07:44.962-05:00The answer is sin(1).
Let f(x)=cos(x)/(e^x+1). No...The answer is sin(1).<br /><br />Let f(x)=cos(x)/(e^x+1). Now, every real function is uniquely the sum of an even function e(x)=(f(x)+f(-x))/2 and an odd function o(x)=(f(x)-f(-x))/2. Since the integral of an odd function over the interval [-1,1] is 0, it suffices to integrate the even part.<br /><br />Now, it is easy to check that e(x)=cos(x)/2, and the integral from -1 to 1 of cos(x)/2 dx is sin(1).Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-91845273773123245352009-10-03T15:10:33.840-05:002009-10-03T15:10:33.840-05:00The answer is 5151.
Each term has the form A x^i ...The answer is 5151.<br /><br />Each term has the form A x^i y^j z^k where i, j, k are non-negative integers and i + j + k = 100.<br /><br />Note that i can take any value between 0 and 100.<br /><br />If i is given, then j can take any value between 0 and 100-i, and k is determined by i and j. So for each i, there are 101-i possibilities.<br /><br />Thus the total number of terms is the sum of (101-i) from i = 0 to 100, which is<br />101 + 100 + 99 + ... + 1 + 0 = 101*102/2 = 5151.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-2025417990909527342009-10-02T13:06:07.627-05:002009-10-02T13:06:07.627-05:00Answer: There must have been a point where the win...Answer: There must have been a point where the winning percentage was exactly 80%, but there need not have been a point where it was exactly 70%.<br /><br />Consider the first time that the team reached or exceeded 80%. Say that they have played N games, and won k of them. Thus k/N >= 4/5.<br /><br />Since this was the first time they reached 80%, the team must have won the Nth game. So, they had only k-1 wins after N-1 games, and (k-1)/(N-1) < 4/5.<br /><br />Now, (k-1)/(N-1) < 4/5 is equivalent to 4N > 5k-1, and k/N >= 4/5 is equivalent to 4N <= 5k.<br /><br />Since 4N is an integer, and there are no integers between 5k-1 and 5k, we conclude that 4N = 5k, or k = 4/5 N. Thus, the winning percentage was exactly 80% after N games.<br /><br />For the other part of the question, note that if the team loses the first game and wins the next nine games, then the winning percentage will never be exactly 70%.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-24246076343920098392009-10-01T08:20:07.816-05:002009-10-01T08:20:07.816-05:00The answer is 101/200
(3/4)(8/9)(15/16)...(9999/1...The answer is 101/200<br /><br />(3/4)(8/9)(15/16)...(9999/10000)<br />= (1-1/4)(1-1/9)(1-1/16)...(1-1/10000)<br />= (1-1/2)(1+1/2)(1-1/3)(1+1/3)...(1-1/100)(1+1/100)<br />= (1/2)(3/2)(2/3)(4/3)(3/4)...(100/99)(99/100)(101/100)<br />= (1/2)(101/100)<br />= 101/200Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-31472096993031440242009-09-30T13:48:43.084-05:002009-09-30T13:48:43.084-05:00The answer is x^2 - 1.
x^1000 = (x^4)(x^996 - 1) ...The answer is x^2 - 1.<br /><br />x^1000 = (x^4)(x^996 - 1) + x^4<br /> = (x^4)(x^996 - 1) + (x^4 - x^2 + 1) + (x^2 - 1).<br /><br />But x^996 - 1 is divisible by x^6 + 1, and<br />x^6 + 1 = (x^4 - x^2 + 1)(x^2 + 1).<br />Therefore the remainder is x^2 - 1.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-75025671238858763022009-09-29T13:26:49.512-05:002009-09-29T13:26:49.512-05:00The answer is c = f((a+b)/2).
Since f is continuo...The answer is c = f((a+b)/2).<br /><br />Since f is continuous and increasing, it has an inverse function g. By turning the graph sideways, we see that the area A(c) is given by the following equation.<br /><br />A(c) = int (f(a) to c) [g(x) - a] dx + int (c to f(b)) [b - g(x)] dx.<br /><br />Using the fundamental theorem of calculus,<br /><br />A'(c) = 2g(c) - a - b,<br /><br />so the minimum area occurs when A'(c) = 0, or when g(c) = (a+b)/2, which is equivalent to c = f((a+b)/2).Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-80817714859231114972009-09-28T10:39:25.849-05:002009-09-28T10:39:25.849-05:00Answer: 10/89
S = sum (1 to inf) Fib(n)/10^n
= 1/...Answer: 10/89<br /><br />S = sum (1 to inf) Fib(n)/10^n<br />= 1/10 + sum (2 to inf) Fib(n)/10^n<br />= 1/10 + sum (2 to inf) (Fib(n-1)+Fib(n-2))/10^n<br />= 1/10 + sum (2 to inf) Fib(n-1)/10^n + sum (2 to inf) Fib(n-2)/10^n<br />= 1/10 + sum (1 to inf) Fib(n)/10^(n+1) + sum (0 to inf) Fib(n)/10^(n+2)<br />= 1/10 + S/10 + S/100<br /><br />Solving S = 1/10 + S/10 + S/100 gives S = 10/89.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-86794722638287767192009-09-27T11:25:02.229-05:002009-09-27T11:25:02.229-05:00Answer: 1/8
The trick is to multiply by sin(20°) ...Answer: 1/8<br /><br />The trick is to multiply by sin(20°) and repeatedly apply the double-angle identity sin(2x)=2sin(x)cos(x).<br /><br />Let x = cos(20°)cos(40°)cos(80°).<br /><br />x sin(20°) = sin(20°)cos(20°)cos(40°)cos(80°)<br /> = 1/2 sin(40°)cos(40°)cos(80°)<br /> = 1/4 sin(80°)cos(80°)<br /> = 1/8 sin(160°)<br /> = 1/8 sin(20°).<br /><br />Therefore, x = 1/8.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-14050673293734899772009-09-26T22:44:19.378-05:002009-09-26T22:44:19.378-05:00Found this a slightly different way. Since the sl...Found this a slightly different way. Since the slope m, of the line y=mx+b, is tangent to f(x)=x^4 − 14x^3 + 69x^2, m must be the same as the first derivative of f(x) at these intersections. Since f(x)=f'(x)x-b, we can infer that b = f(x)-f'(x)*x.<br /><br />I then used a graphing calculator to graph the [m,b] values of the tangent line at each point along the graph: x(t) = [ f'(x), f(x)-f'(x)*x ]. This parametric line intersects itself at [140,-100], which gives the line y=140x-100.Unknownhttps://www.blogger.com/profile/05597332573842960982noreply@blogger.comtag:blogger.com,1999:blog-1974366466896776665.post-5715974862574878632009-09-26T22:35:37.541-05:002009-09-26T22:35:37.541-05:00;; Brute forced in scheme:
(define (potd x c) (if ...;; Brute forced in scheme:<br />(define (potd x c) (if (> 1001 (* 5 x)) (potd (+ 1 x) (+ c (floor (/ (- 1000 (* 5 x)) 4)))) c))<br />(potd 1 0)Unknownhttps://www.blogger.com/profile/05597332573842960982noreply@blogger.com