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- ...the roots of <math>Q(x) = 0,</math> find <math>P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).</math> ...g division to divide <math>P(x)</math> by <math>Q(x)</math>, the remainder is <math>x^2-x+1</math>.3 KB (475 words) - 21:53, 6 May 2024
- ...ath> such that <math>f(1)=1440</math>, <math>f(2)=1716</math>, and <math>f(3)=1848</math>. Plugging in the values for x gives us a system of three equat ...gives <math>a=-72, b=492,</math> and <math>c=1020</math>. Thus, the answer is <math>-72(8)^2+492\cdot8+1020= \boxed{348}.</math>5 KB (793 words) - 15:18, 14 July 2023
- ...tively, after a <math>180^\circ</math> rotation about <math>G.</math> What is the area of the union of the two regions enclosed by the triangles <math>AB Since a <math>13-14-15</math> triangle is a <math>5-12-13</math> triangle and a <math>9-12-15</math> triangle "glued"5 KB (787 words) - 17:38, 30 July 2022
- .... The ratio of the volume of the smaller tetrahedron to that of the larger is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively pr ...rac{1}{3}, \frac{1}{3})</math>,<math>(0,\frac{1}{3}, \frac{1}{3}, \frac{1}{3})</math>.3 KB (563 words) - 17:36, 30 July 2022
- .../math> is never immediately followed by <math>C</math>, and <math>C</math> is never immediately followed by <math>A</math>. How many seven-letter good wo ...s is restricted. Therefore, the number of seven-letter good words is <math>3*2^6=192</math>2 KB (336 words) - 17:29, 30 July 2022
- ...wo of whose digits are the same. What is the remainder when <math>N</math> is divided by 1000? ...<math>0,1,2</math> is also divisible by 8. The only arrangement that works is <math>120</math>.1,013 bytes (162 words) - 09:00, 11 July 2023
- ...eger]]s is <math>6</math> times their [[sum]], and one of the [[integer]]s is the sum of the other two. Find the sum of all possible values of <math>N</m ...<math>13, 8, 7</math> so the sum of all possible values of <math>N</math> is <math>12 \cdot (13 + 8 + 7) = 12(28) = \boxed{336}</math>.1 KB (174 words) - 08:56, 11 July 2023
- ..., <math>b</math>, and <math>c</math> are positive integers, <math>b</math> is not divisible by the square of any prime, and <math>a</math> and <math>c</m ...and <math>b</math> respectively. We know that the point <math>(9,6)</math> is a point on both circles, so we have that7 KB (1,188 words) - 17:46, 23 June 2024
- ...th center <math>O</math> on <math>\overline{AP}</math> is drawn so that it is [[Tangent (geometry)|tangent]] to <math>\overline{AM}</math> and <math>\ove ...Using the fact that the ratio of corresponding sides in similar triangles is equal to the ratio of their perimeters, we have4 KB (658 words) - 19:15, 19 December 2021
- ...area of triangle <math>PQR</math> to the area of triangle <math>ABC</math> is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively pr C=(1.9375,3.4994);6 KB (935 words) - 13:23, 3 September 2021
- ...h>a_n\le.4</math> for all <math>n</math> such that <math>1\le n\le9</math> is given to be <math>p^aq^br/\left(s^c\right)</math> where <math>p</math>, <ma ...h> to <math>(6,4)</math> that always stay below the line <math>y=\frac{2x}{3}</math>. We can find the number of such paths using a Pascal's Triangle typ7 KB (1,127 words) - 13:34, 19 June 2022
- ...math>n</math>, and <math>p</math> are positive integers and <math>m</math> is not divisible by the square of any prime. Find <math>100m+10n+p</math>. ...>. Then, the common ratio is <math>\frac{1}{8x}</math>, and the first term is <math>8x^2</math>.4 KB (710 words) - 16:06, 2 June 2024
- ...have no common elements.) Find the remainder obtained when <math>n</math> is divided by <math>1000</math>. ...</math>, <math>i \in B</math>, or <math>i \in C</math>. So there are <math>3^{10}</math> ways to organize the elements of <math>S</math> into disjoint <3 KB (404 words) - 23:07, 4 May 2024
- ...<math>n(n+1)<2002</math>, which is easy to solve by trial, as the solution is obviously <math>\simeq \sqrt{2002}</math>.) ...2002}{n}\right\rfloor=k</math> has no integer solutions for <math>n</math> is <math>\boxed{049}</math>.6 KB (908 words) - 14:22, 14 July 2023
- It is known that, for all positive integers <math>k</math>, <center><math>1^2+2^2+3^2+\ldots+k^{2}=\frac{k(k+1)(2k+1)}6</math>.</center>3 KB (403 words) - 12:10, 9 September 2023
- Find the integer that is closest to <math>1000\sum_{n=3}^{10000}\frac1{n^2-4}</math>. ...1 = (n-2)A + (n+2)B</math> or <math>1 = n(A+B)+ 2(B-A)</math>. Since there is no n term on the left hand side, <math> A+B=0</math> and by inspection <mat2 KB (330 words) - 05:56, 23 August 2022
- ...> and <math>m</math> are non-negative integers, for which <math>a^6</math> is not a divisor of <math>6^a</math>. ...l pairs of non-negative integers (n,m) for which <math>(2^n3^m)^{6}</math> is not a divisor of <math>6^{2^n3^m}</math>3 KB (515 words) - 14:46, 14 February 2021
- ...ath>m</math> is a positive integer. Find the remainder when <math>m</math> is divided by <math>1000</math>. ...Thus, the area of the garden enclosed by the path when <math>n=202</math> is2 KB (268 words) - 07:28, 13 September 2020
- ...r]]s that form an increasing [[geometric sequence]] and <math>b - a</math> is the [[Perfect square|square]] of an integer. Find <math>a + b + c.</math> ...the [[geometric mean]] of the [[product]] <math>abc</math>. <math>b=\sqrt[3]{abc}=6^2=36</math>.2 KB (263 words) - 22:50, 5 April 2024
- ...7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R=(11,3,9)</math>. What is the [[surface area]] of the cube? <math>PR=\sqrt{(11-7)^2+(3-12)^2+(9-10)^2}=\sqrt{98}</math>709 bytes (103 words) - 03:32, 6 December 2019
