15 January 2013

• 17:0817:08, 15 January 2013 diff hist +2,548 N Mock Geometry AIME 2011 Problems/Problem 12 Created page with "==Problem== A triangle has the property that its sides form an arithmetic progression, and that the angle opposite the longest side is three times the angle opposite the shortes..."

28 May 2012

• 14:0714:07, 28 May 2012 diff hist +3,793 N 2006 SMT/Geometry Problems/Problem 4 Created page with "==Problem== The distance <math> AB </math> is <math> l </math>. Find the area of the locus of points <math> X </math> such that <math> 15^\circ\le\angle AXB\le30^\circ </math> an..." current
• 11:2211:22, 28 May 2012 diff hist +1,363 N 2006 SMT/Geometry Problems/Problem 3 Created page with "==Problem== Circle <math> \gamma </math> is centered at <math> (0, 3) </math> with radius <math> 1 </math>. Circle <math> \delta </math> is externally tangent to circle <math> \g..." current
• 11:0011:00, 28 May 2012 diff hist +1,991 N 2006 SMT/Geometry Problems/Problem 2 Created page with "==Problem== Given square <math> ABCD </math> of side length <math> 1 </math>, with <math> E </math> on <math> \overline{CD} </math> and <math> F </math> in the interior of the sq..." current
• 10:3810:38, 28 May 2012 diff hist +1,860 N 2006 SMT/Geometry Problems/Problem 1 Created page with "==Problem== Given a cube, determine the ratio of the volume of the octahedron formed by connecting the centers of each face of the cube to the volume of the cube. ==Solution== ..." current
• 09:3509:35, 28 May 2012 diff hist +703 N 2006 SMT/Algebra Problems/Problem 10 Created page with "==Problem== Evaluate: <math> \sum_{k=1}^{\infty}\frac{k}{a^{k-1}} </math> for all <math> |a|<1 </math>. ==Solution== Let <math> S=\sum_{k=1}^{\infty}\frac{k}{a^{k-1}}=\frac{1}{a..." current
• 09:3009:30, 28 May 2012 diff hist +1,131 N 2006 SMT/Algebra Problems/Problem 9 Created page with "==Problem== Principal Skinner is thinking of two integers <math> m </math> and <math> n </math> and bets Superintendent Chalmers that he will not be able to determine these integ..." current
• 08:5208:52, 28 May 2012 diff hist +1,306 N 2006 SMT/Algebra Problems/Problem 8 Created page with "==Problem== Evaluate: <cmath> \sum_{x=2}^{10}\frac{2}{x(x^2-1)}. </cmath> ==Solution== Since the denominator of the fraction is factorable, we try partial fraction decompositio..." current
• 08:4008:40, 28 May 2012 diff hist +802 N 2006 SMT/Algebra Problems/Problem 7 Created page with "==Problem== Find all solutions to <math> aabb=n^4-6n^3 </math>, where <math> a </math> and <math> b </math> are nonzero digits, and <math> n </math> is an integer. (<math> a </ma..." current
• 08:3108:31, 28 May 2012 diff hist +1,360 N 2006 SMT/Algebra Problems/Problem 6 Created page with "==Problem== Let <math> a, b, c </math> be real numbers satisfying: <cmath> ab-a=b+119 </cmath> <cmath> bc-b=c+59 </cmath> <cmath> ca-c=a+71 </cmath> Determine all possible valu..."
• 08:0308:03, 28 May 2012 diff hist +1,778 N 2006 SMT/Algebra Problems/Problem 5 Created page with "==Problem== Jerry is bored one day, so he makes an array of Cocoa pebbles. He makes <math> 8 </math> equal rows with the pebbles remaining in a box. When Kramer drops by and eats..." current
• 07:5507:55, 28 May 2012 diff hist +1,250 N 2006 SMT/Algebra Problems/Problem 4 Created page with "==Problem== Simplify: <math> \frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)} </math> ==Solution== Firstly, we notice that the factors in the denominators ar..." current
• 07:4307:43, 28 May 2012 diff hist +1,313 N 2006 SMT/Algebra Problems/Problem 3 Created page with "==Problem== A Gaussian prime is a Gaussian integer <math> z=a+bi </math> (where <math> a </math> and <math> b </math> are integers) with no Guassian integer factors of smaller ab..." current
• 07:3107:31, 28 May 2012 diff hist +1,291 N 2006 SMT/Algebra Problems/Problem 2 Created page with "==Problem== Find the minimum value of <math> 2x^2+2y^2+5z^2-2xy-4yz-4x-2z+15 </math> for real numbers <math> x, y, z </math>. ==Solution== ===Solution 1=== Notice that <math> 2x..." current
• 07:0107:01, 28 May 2012 diff hist +1,399 N 2006 SMT/Algebra Problems/Problem 1 Created page with "==Problem== A finite sequence of positive integers <math> m_i </math> for <math> i=1, 2, \cdots, 2006 </math> are defined so that <math> m_1=1 </math> and <math> m_i=10m_{i-1}+1 ..." current

27 May 2012

• 20:3820:38, 27 May 2012 diff hist +1,713 N 2006 SMT/Advanced Topics Problems/Problem 7 Created page with "==Problem== A lattice point in the plane is a point whose coordinates are both integers. Given a set of <math> 100 </math> distinct lattice points in the plane, find the smallest..." current
• 20:1420:14, 27 May 2012 diff hist +1,334 N 2006 SMT/Advanced Topics Problems/Problem 6 Created page with "==Problem== Ten teams of five runners each compete in a cross-country race. A runner finishing in <math> n\text{th} </math> place contributes <math> n </math> points to his team,..." current
• 19:5819:58, 27 May 2012 diff hist +1,065 N 2006 SMT/Advanced Topics Problems/Problem 5 Created page with "==Problem== Evaluate: <math> \sum_{k=1}^{\infty}\frac{1}{k\sqrt{k+2}+(k+2)\sqrt{k}} </math> ==Solution== We try to rationalize the denominators. We have <math> \frac{1}{k\sqrt{k..." current
• 19:2019:20, 27 May 2012 diff hist +434 N 2006 SMT/Advanced Topics Problems/Problem 3 Created page with "==Problem== Simplify: <math> \sum_{k=10}^{2006}\binom{k}{10} </math> (Your answer should contain no summations but may still contain binomial coefficients/combinations). ==Solut..."
• 19:1219:12, 27 May 2012 diff hist −2 m 2006 SMT/Advanced Topics Problems/Problem 1 →Problem 1 current
• 19:0919:09, 27 May 2012 diff hist +1 m 2006 SMT/Advanced Topics Problems →Problem 2 current
• 19:0919:09, 27 May 2012 diff hist +2,090 N 2006 SMT/Advanced Topics Problems/Problem 2 Created page with "==Problem== Define <math> A=\left(\,\begin{matrix}0 & 1\\ 3 &0\end{matrix}\right) </math>. Find a vertical vector <math> v </math> such that <math> (A^8+A^6+A^4+A^2+I)v=\left(\be..." current
• 18:0718:07, 27 May 2012 diff hist +1,052 N 2006 SMT/Advanced Topics Problems/Problem 1 Created page with "==Problem 1== A college student is about to break up with her boyfriend, a mathematics major who is apparently more interested in math than her. Frustrated, she cries, ”You mat..."
• 16:3216:32, 27 May 2012 diff hist +5,805 N 2006 SMT/Team Problems Created page with "==Problem 1== Given <math> \triangle ABC </math>, where <math> A </math> is at <math> (0,0) </math>, <math> B </math> is at <math> (20,0) </math>, and <math> C </math> is on the ..." current
• 15:2215:22, 27 May 2012 diff hist +1,060 N 2006 SMT/Team Created page with "'''2006 SMT Team''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution. *[[2006 SMT/Te..." current
• 15:0215:02, 27 May 2012 diff hist +9,798 N 2006 SMT/General Problems Created page with "==Problem 1== After a cyclist has gone <math> \frac{2}{3} </math> of his route, he gets a flat tire. Finishing on foot, he spends twice as long walking as he did riding. How many..." current
• 13:1913:19, 27 May 2012 diff hist +1,641 N 2006 SMT/General Created page with "'''2006 SMT General''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution. *[[2006 SMT..." current
• 12:5912:59, 27 May 2012 diff hist +3,358 N 2006 SMT/Calculus Problems Created page with "==Problem 1== Evaluate: <cmath> \lim_{x\to0}\frac{\frac{d}{dx}\frac{\sin x}{x}}{x} </cmath> Solution ==Problem 2== Given the equation ..." current
• 11:3111:31, 27 May 2012 diff hist +858 N 2006 SMT/Calculus Created page with "'''2006 SMT Calculus''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution. *[[2006 SM..." current
• 11:2211:22, 27 May 2012 diff hist +5,385 N 2006 SMT/Geometry Problems Created page with "==Problem 1== Given a cube, determine the ratio of the volume of the octahedron formed by connecting the centers of each face of the cube to the volume of the cube. [[2006 SMT/G..." current
• 10:0910:09, 27 May 2012 diff hist +858 N 2006 SMT/Geometry Created page with "'''2006 SMT Geometry''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution. *[[2006 SM..." current
• 10:0210:02, 27 May 2012 diff hist +3,081 N 2006 SMT/Algebra Problems Created page with "==Problem 1== A finite sequence of positive integers <math> m_i </math> for <math> i=1, 2, \cdots, 2006 </math> are defined so that <math> m_1=1 </math> and <math> m_i=10m_{i-1}+..." current
• 09:2309:23, 27 May 2012 diff hist +845 N 2006 SMT/Algebra Created page with "'''2006 SMT Algebra''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution. *[[2006 SMT/..." current
• 09:1609:16, 27 May 2012 diff hist +4,073 N 2006 SMT/Advanced Topics Problems Created page with "==Problem 1== A college student is about to break up with her boyfriend, a mathematics major who is apparently more interested in math than her. Frustrated, she cries, ”You mat..."
• 08:4708:47, 27 May 2012 diff hist +941 N 2006 SMT/Advanced Topics Created page with "'''2006 SMT Advanced Topics''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution. *[[2..." current
• 08:3908:39, 27 May 2012 diff hist +404 N 2006 SMT Created page with "'''2006 SMT''' problems and solutions by category. *Advanced Topics *Algebra *[[2006 SMT/Ge..." current
• 08:3308:33, 27 May 2012 diff hist −28 Stanford Mathematics Tournament Problems No edit summary
• 08:3308:33, 27 May 2012 diff hist +29 Stanford Mathematics Tournament Problems No edit summary
• 08:1908:19, 27 May 2012 diff hist +117 m Stanford Mathematics Tournament Problems No edit summary
• 07:1207:12, 27 May 2012 diff hist +59 m Stanford Mathematics Tournament No edit summary
• 06:5606:56, 27 May 2012 diff hist +1,241 N 1974 AHSME Problems/Problem 30 Created page with "==Problem== A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If <math> R </math> is the ratio of the lesser part to t..."

26 May 2012

• 21:5821:58, 26 May 2012 diff hist +1,079 N 1974 AHSME Problems/Problem 29 Created page with "==Problem== For <math> p=1, 2, \cdots, 10 </math> let <math> S_p </math> be the sum of the first <math> 40 </math> terms of the arithmetic progression whose first term is <math> ..."
• 21:4121:41, 26 May 2012 diff hist +1,728 N 1974 AHSME Problems/Problem 28 Created page with "==Problem== Which of the following is satisfied by all numbers <math> x </math> of the form <cmath> x=\frac{a_1}{3}+\frac{a_2}{3^2}+\cdots+\frac{a_{25}}{3^{25}} </cmath> where ..."
• 21:3121:31, 26 May 2012 diff hist +770 N 1974 AHSME Problems/Problem 27 Created page with "==Problem== If <math> f(x)=3x+2 </math> for all real <math> x </math>, then the statement: "<math> |f(x)+4|<a </math> whenever <math> |x+2|<b </math> and <math> a>0 </math> and <..."
• 21:2721:27, 26 May 2012 diff hist +816 N 1974 AHSME Problems/Problem 26 Created page with "==Problem== The number of distinct positive integral divisors of <math> (30)^4 </math> excluding <math> 1 </math> and <math> (30)^4 </math> is <math> \mathrm{(A)\ } 100 \qquad ..."
• 21:2121:21, 26 May 2012 diff hist +1,510 N 1974 AHSME Problems/Problem 25 Created page with "==Problem== In parallelogram <math> ABCD </math> of the accompanying diagram, line <math> DP </math> is drawn bisecting <math> BC </math> at <math> N </math> and meeting <math> A..."
• 21:1221:12, 26 May 2012 diff hist +1,109 N 1974 AHSME Problems/Problem 24 Created page with "==Problem== A fair die is rolled six times. The probability of rolling at least a five at least five times is <math> \mathrm{(A)\ } \frac{13}{729} \qquad \mathrm{(B) \ }\frac{1..."
• 21:0321:03, 26 May 2012 diff hist +2,053 N 1974 AHSME Problems/Problem 23 Created page with "==Problem == In the adjoining figure <math> TP </math> and <math> T'Q </math> are parallel tangents to a circle of radius <math> r </math>, with <math> T </math> and <math> T' </..."
• 20:3220:32, 26 May 2012 diff hist +1,141 N 1974 AHSME Problems/Problem 22 Created page with "==Problem== The minimum of <math> \sin\frac{A}{2}-\sqrt{3}\cos\frac{A}{2} </math> is attained when <math> A </math> is <math> \mathrm{(A)\ } -180^\circ \qquad \mathrm{(B) \ }60..."
• 20:1420:14, 26 May 2012 diff hist +1,124 N 1974 AHSME Problems/Problem 21 Created page with "==Problem== In a geometric series of positive terms the difference between the fifth and fourth terms is <math> 576 </math>, and the difference between the second and first terms..."