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- ...will both be two-digit numbers and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let <math>d</math> be D <center><math>(1,2), (1,3), (2,3), (1,4), (2,4), (3,4), \dots , (8,9)</math></center>2 KB (246 words) - 17:02, 21 May 2023
- ...orean Theorem]], we find that the shorter dimension is <math>2r\left(\sqrt{3}+1\right)</math>. ...\cdot \left[\frac{\sqrt{3}-1}{\sqrt{3}-1}\right] = \frac{1}{2}\left(7\sqrt{3} - 7\right) = \frac{1}{2}\left(\sqrt{p}-q\right)</math>. Thus we have <math2 KB (287 words) - 19:54, 4 July 2013
- ...it arrangement that reads the same left-to-right as it does right-to-left) is <math>\dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are rela ...is <math>\frac{10 \times 10}{10^3} = \frac 1{10}</math>. Similarly, there is a <math>\frac 1{26}</math> probability of picking the three-letter palindro3 KB (369 words) - 23:36, 6 January 2024
- pair A=(0,0),C=(7.8,0),B=IP(CR(A,3.6),CR(C,5.07)), M=(A+C)/2, Da = bisectorpoint(A,B,C), D=IP(B--B+(Da-B)*10,A As <math>BD</math> is an angle bisector of both triangles <math>BAC</math> and <math>BF'F</math>,8 KB (1,382 words) - 14:23, 29 December 2022
- ...in that order. Find the smallest value of <math> n </math> for which this is possible. ...th>10^k m - nX</math> is an integer and <math>\frac{10^k m - nX}{n}</math> is a fraction between <math>0</math> and <math>1</math>, we can rewrite this a3 KB (477 words) - 14:23, 4 January 2024
- ...ath>'s than <math>0</math>'s. Find the [[remainder]] when <math> N </math> is divided by <math>1000</math>. ...are in the form <math>{2i \choose i}</math>, so the sum of these elements is <math>\sum_{i=0}^{5} {2i \choose i} = 1 + 2 +6 + 20 + 70 + 252 = 351</math>4 KB (651 words) - 19:42, 7 October 2023
- ...</math> and <math> \angle ACB = 106^\circ. </math> Point <math> M </math> is in the interior of the triangle so that <math> \angle MAC = 7^\circ </math> ...congruent (by ASA), <math>CM = CN</math>. Hence <math>\triangle CMN</math> is an [[equilateral triangle]], so <math>\angle CNM = 60^\circ</math>.7 KB (1,058 words) - 01:41, 6 December 2022
- An [[integer]] between <math>1000</math> and <math>9999</math>, inclusive, is called ''balanced'' if the sum of its two leftmost [[digit]]s equals the su ...balanced numbers. If the common sum of the first two and last two digits is <math>n</math>, such that <math>10 \leq n \leq 18</math>, there are <math>14 KB (696 words) - 11:55, 10 September 2023
- ...h <math> AB = 9 </math> and <math> BC = 21. </math> Point <math> D </math> is not on <math> \overline{AC} </math> so that <math> AD = CD, </math> and <ma ...<math>y^2 - 36 = x^2 - 225 \Longrightarrow x^2 - y^2 = 189</math>. The LHS is [[difference of squares]], so <math>(x + y)(x - y) = 189</math>. As both <m3 KB (490 words) - 18:13, 13 February 2021
- ...also vertices of a <math>1</math> by <math>1</math> by <math>1</math> cube is <math> m + \sqrt{n} + \sqrt{p}, </math> where <math> m, n, </math> and <mat ...x | vertices]] of a [[cube (geometry) | cube]], there are <math>{8 \choose 3} = 56</math> total [[triangle]]s to consider. They fall into three categor3 KB (477 words) - 18:35, 27 December 2021
- ...th>4</math> by <math>5</math> units. Given that the [[volume]] of this set is <math>\frac{m + n\pi}{p}, </math> where <math> m, n, </math> and <math> p < import three; currentprojection = perspective(5,4,3); defaultpen(linetype("8 8")+linewidth(0.6));2 KB (288 words) - 19:58, 4 July 2013
- ...tity, <math>\sin^2 x + \cos^2 x = 1</math> and <math>2\sin x \cos x</math> is simply <math>\frac{1}{5}</math>, via substitution. Thus, substituting these ==Solution 3==3 KB (516 words) - 12:32, 10 June 2024
- Let the [[set]] <math> \mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}. </math> Susan makes a list as follows: for each two-element subse ...the set from greatest to least to reduce error: <math>\{34, 21, 13, 8, 5, 3, 2, 1\}.</math>2 KB (317 words) - 00:09, 9 January 2024
- ...th>1</math> is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. T ...>, while the total area is given by <math>100^{2} \pi</math>, so the ratio is4 KB (523 words) - 15:49, 8 March 2021
- ...day. How many different dollar amounts could Taye have on Thursday, <math>3</math> days later? <math>\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 5\qquad\textbf{(D) } 6\qquad\textbf2 KB (384 words) - 22:57, 17 February 2024
- where <math>m, n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math> The five-element sum is just <math>\sin 30^\circ + \sin 60^\circ + \sin 90^\circ + \sin 120^\circ +4 KB (675 words) - 17:23, 30 July 2022
- .../math> where <math>m</math> and <math>n</math> are positive integers and n is not divisible by the square of any prime. Find <math>m + n.</math> ...rac{8}{\sqrt{3}}, 4\right)</math> and that <math>B = \left(\frac{10}{\sqrt{3}}, 2\right)</math>.9 KB (1,461 words) - 15:09, 18 August 2023
- ...he probability that the bug moves to its starting vertex on its tenth move is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively pr ...t the bug is at its starting vertex after <math>n</math> moves. If the bug is on its starting vertex after <math>n</math> moves, then it must be not on i15 KB (2,406 words) - 23:56, 23 November 2023
- ...ath>s=v_1+\cdots+v_{27}</math> be the total number of votes cast. Our goal is to determine the smallest possible <math>s</math>. ...is means "For all <math>i</math>, <math>\frac{100v_i}s + 1 \leq v_i</math> is true")4 KB (759 words) - 13:00, 11 December 2022
- ...<math>m</math> and <math>p</math> are relatively prime, and <math>n</math> is not divisible by the square of any prime, find <math>m + n + p.</math> Hence, the answer is <math>527+11+40=\boxed{578}.</math>5 KB (772 words) - 19:47, 1 August 2023
