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- A particle moves in the [[Cartesian plane]] according to the following rules: ...> the particle may only move to <math> (a+1,b), (a,b+1), </math> or <math>(a+1,b+1). </math>5 KB (897 words) - 00:21, 29 July 2022
- pair A = origin; pair C = rotate(15,A)*(A+dir(-50));13 KB (2,129 words) - 18:56, 1 January 2024
- ...es the rest equally between the other two. Given that each monkey receives a [[whole number]] of bananas whenever the bananas are divided, and the numbe ...fraction is equal to <math>8</math>, and the solution is <math>8(11 + 13 + 27) = \boxed{408}</math>.6 KB (950 words) - 14:18, 15 January 2024
- Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be dra ...ose <math>n_{}^{}</math> is a positive integer and <math>d_{}^{}</math> is a single digit in base 10. Find <math>n_{}^{}</math> if7 KB (1,045 words) - 20:47, 14 December 2023
- ...</math> rectangles, of which <math>s</math> are squares. The number <math>s/r</math> can be written in the form <math>m/n,</math> where <math>m</math> ...he two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah sho7 KB (1,098 words) - 17:08, 25 June 2020
- ...> <math>(10,114),</math> <math>(28,153),</math> and <math>(28,84).</math> A line through the origin cuts this figure into two congruent polygons. The ...all positive integers <math>n</math> for which <math>n^2-19n+99</math> is a perfect square.7 KB (1,094 words) - 13:39, 16 August 2020
- ...Let <math>A = (u,v)</math>, let <math>B</math> be the reflection of <math>A</math> across the line <math>y = x</math>, let <math>C</math> be the reflec ...icients of <math>x^{2}</math> and <math>x^{3}</math> are equal. Find <math>a + b</math>.7 KB (1,204 words) - 03:40, 4 January 2023
- ...than the mean of <math>\mathcal{S}</math>. Find the mean of <math>\mathcal{S}</math>. ...and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>.7 KB (1,212 words) - 22:16, 17 December 2023
- ...plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right- ...gram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the r8 KB (1,374 words) - 21:09, 27 July 2023
- ...ems and that one's score, <math>s</math>, is computed by the formula <math>s=30+4c-w</math>, where <math>c</math> is the number of correct answers and < Let Mary's score, number correct, and number wrong be <math>s,c,w</math> respectively. Then7 KB (1,163 words) - 23:53, 28 March 2022
- Clearly, if <math>x \ge 44</math>, it can be expressed as a sum of 2 odd composites. However, if <math>x = 42</math>, it can also be ex ...prime quintuplet is <math>5,11,17,23,</math> and <math>29</math>, yielding a maximal answer of 38. Since <math>38-25=13</math>, which is prime, the answ8 KB (1,346 words) - 01:16, 9 January 2024
- ...4,16,36,64</math>, we can express the difference of the two polynomials by a quartic polynomial that has roots at <math>t=4,16,36,64</math>, so ...+ 3^2 + 5^2 + 7^2 + x^2 + y^2 + z^2 + w^2)t^3 \dots = 0.</cmath> By Vieta's, we know that the sum of the roots of this equation is <cmath>1^2 + 3^2 + 56 KB (1,051 words) - 04:52, 8 May 2024
- ...= \frac{n}{729}</math> be the probability that the bug is at vertex <math>A</math> when it has crawled exactly <math>7</math> meters. Find the value of ...> let <math>P(k)</math> be the probability that the bug is at vertex <math>A</math> when it has crawled exactly <math>k</math> meters. We wish to find <17 KB (2,837 words) - 13:34, 4 April 2024
- ...rticipants to think of a three digit number <math>(abc)</math> where <math>a</math>, <math>b</math>, and <math>c</math> represent digits in base <math>1 ...math> be the number <math>100a+10b+c</math>. Observe that <math>3194+m=222(a+b+c)</math> so3 KB (565 words) - 16:51, 1 October 2023
- ...th>1,3,4,9,10,12,13\cdots</math> consists of all those positive [[integer]]s which are [[exponent|powers]] of 3 or sums of distinct powers of 3. Find th ...powers of 3, in base 3 each number is a sequence of 1s and 0s (if there is a 2, then it is no longer the sum of distinct powers of 3). Therefore, we can5 KB (866 words) - 00:00, 22 December 2022
- ...ments indicated in the figure. Find the product <math>abc</math> if <math>a + b + c = 43</math> and <math>d = 3</math>. Call the [[cevian]]s AD, BE, and CF. Using area ratios (<math>\triangle PBC</math> and <math>\tr4 KB (727 words) - 23:37, 7 March 2024
- Raise both as [[exponent]]s with base 8: ...ge of base formula]], which states <math>\log_a b = \frac{\log_k b}{\log_k a}</math> for arbitrary <math>k</math>.3 KB (481 words) - 21:52, 18 November 2020
- === Solution 1 (Ceva's Theorem, Stewart's Theorem) === ...th>RST</math>. We'll make use of the following fact: if <math>P</math> is a point in the interior of triangle <math>XYZ</math>, and line <math>XP</math13 KB (2,091 words) - 00:20, 26 October 2023
- ...e. Let <math>d</math> be the distance between the [[midpoint]]s of [[edge]]s <math>AB</math> and <math>CD</math>. Find <math>d^{2}</math>. pair A,B,C,D,M,P,Q;2 KB (376 words) - 13:49, 1 August 2022
- ...they showed there was a positive integer such that <cmath>133^5+110^5+84^5+27^5=n^{5}.</cmath> Find the value of <math>n</math>. By either <b>Fermat's Little Theorem (FLT)</b> or inspection, we get6 KB (874 words) - 15:50, 20 January 2024
