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- Find the smallest prime that is the fifth term of an increasing [[arithmetic sequence]], all four preceding ...th>, and <math>29</math> form an [[arithmetic sequence]]. Thus, the answer is <math>029</math>.2 KB (332 words) - 13:22, 3 August 2020
- ...f <math>\mathcal{S}</math> divided by the area of <math>\mathcal{T}</math> is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relati ...of the above diagram, of <math>y \ge \frac{1}{3}, z \ge \frac{1}{6}</math> is the triangle at the right, and <math>x \ge \frac 12, z \ge \frac 16</math>3 KB (445 words) - 19:40, 4 July 2013
- ...ft to right, the labels on the cards are now in ascending order: <math>1,2,3,\ldots,1999,2000.</math> In the original stack of cards, how many cards wer ...4th card when there are 8 cards remaining. This pattern continues until it is the 512th card on the deck when there are 1024 cards remaining. Since there15 KB (2,673 words) - 19:16, 6 January 2024
- ...th>r</math> times as large as angle <math>APQ,</math> where <math>r</math> is a positive real number. Find <math>\lfloor 1000r \rfloor</math>. ...ath>which implies that <math>1000r = 571 + \tfrac 37</math>. So the answer is <math>\boxed{571}</math>.8 KB (1,275 words) - 03:04, 27 February 2022
- ...an be reached by the firetruck within six minutes. The area of this region is <math>m/n</math> square miles, where <math>m</math> and <math>n</math> are ...Arrows(4)); D((0,-6)--(0,6),Arrows(4)); truck((1,0)); truck((2,0)); truck((3,0)); truck((4,0));3 KB (571 words) - 00:38, 13 March 2014
- ...are [[relatively prime]] positive [[divisor]]s of <math>1000.</math> What is the [[floor function|greatest integer]] that does not exceed <math>S/10</ma ...pressed in the form of <math>2^{x}5^{y}</math>, where <math>-3 \le x,y \le 3</math>. Thus every number in the form of <math>a/b</math> will be expressed4 KB (667 words) - 13:58, 31 July 2020
- ...th>1</math> and <math>100,</math> inclusive, the number <math>x_{k}</math> is <math>k</math> less than the sum of the other <math>99</math> numbers. Give ...=\frac{75}{49} \Longrightarrow x_{50}=\frac{75}{98}</math>, and the answer is <math>75+98=\boxed{173}</math>.2 KB (319 words) - 22:26, 29 December 2022
- <cmath>\begin{eqnarray*} -\log x \log y + \log x + \log y - 1 &=& 3 - \log 2000\\ Small note from different author: <math>-(3 - \log 2000) = \log 2000 - 3 = \log 2000 - \log 1000 = \log 2.</math>4 KB (623 words) - 15:56, 8 May 2021
- ...math>n,</math> and <math>p</math> are positive integers and <math>p</math> is not divisible by the cube of any prime number. Find <math>m + n + p</math>. ...rac{3}{4}\right)^{3}\right)^{1/3}}{1}</math> of the height when the vertex is at the top.4 KB (677 words) - 16:33, 30 December 2023
- note: this is the type of problem that makes you think symmetry, but actually can be solv == Solution 3 ==5 KB (781 words) - 15:02, 20 April 2024
- ...ath> and that the [[arithmetic mean]] of <math>x</math> and <math>y</math> is exactly <math>2</math> more than the [[geometric mean]] of <math>x</math> a ...er of pairs of <math>(\sqrt{x},\sqrt{y})</math>, and so <math>(x,y)</math> is then <math>\boxed{997}</math>.6 KB (966 words) - 21:48, 29 January 2024
- .../math> and <math>n</math> are [[relatively prime]] positive integers. What is <math>m + n</math>? If we work with the problem for a little bit, we quickly see that there is no direct combinatorics way to calculate <math>m/n</math>. The [[Principle7 KB (1,011 words) - 20:09, 4 January 2024
- Expressing all terms 3 to 9 in terms of <math>a_1</math> and <math>a_2</math> and substituting the ...=69</math>. These numbers are relatively prime, as desired. The perimeter is <math>2(61)+2(69)=\boxed{260}</math>.3 KB (485 words) - 00:31, 19 January 2024
- ...th>D</math> across the y-axis. The area of [[pentagon]] <math>ABCDE</math> is <math>451</math>. Find <math>u + v</math>. ...)</math> or <math>(11,10)</math>. Since <math>v < u</math> the latter case is the answer, and <math>u+v = \boxed{021}</math>.3 KB (434 words) - 22:43, 16 May 2021
- ...ositive integer <math>n</math> such that no matter how <math>10^{n}</math> is expressed as the product of any two positive integers, at least one of thes <math>n = 3:</math> <cmath>2^3 = 8 , 5 ^3 = 125</cmath>1 KB (163 words) - 17:44, 16 December 2020
- ...are considered to be consecutive, are written on faces that share an edge is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively pr ...resent adjacent octahedral faces. Each assignment of the numbers <math>1,2,3,4,5,6,7</math>, and <math>8</math> to the faces of the octahedron correspon11 KB (1,837 words) - 18:53, 22 January 2024
- ...and <math>1</math> represent a house that does receive mail. This problem is now asking for the number of <math>19</math>-digit strings of <math>0</math n&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19\\\hline13 KB (2,298 words) - 19:46, 9 July 2020
- ...h>d < 120.</math> The length of the chord of a <math>3d</math>-degree arc is <math>- m + \sqrt {n}</math> centimeters, where <math>m</math> and <math>n< ...5}}{2},</math> which equals <math>-9 + \sqrt{165}.</math> Thus, the answer is <math>9 + 165 = \boxed{174}</math>.3 KB (561 words) - 19:25, 27 November 2022
- ...0,0,2),</math> and <math>D = (0,0,0).</math> The [[radius]] of the sphere is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively pr ...F = (0,2/3,0), G = (0,0,2/3), L = (0,2/3,2/3), M = (2/3,0,2/3), N = (2/3,2/3,0);6 KB (1,050 words) - 18:44, 27 September 2023
- ===Problem 3=== [[2014 USAJMO Problems/Problem 3|Solution]]3 KB (600 words) - 16:42, 5 August 2023
