Prove that the point symmetrical to <math>C</math> with respect to the midpoint of <math>BD</math> lies on line <math>A
...oint <math>BD \implies MF = FN, G</math> be the point symmetrical to <math>C</math> with respect <math>F \implies CF = FE \implies CMGN</math> is the pa
...
122 KB (21,381 words) - 13:47, 3 August 2025
...)</math> for which the sum of the roots is maximized. What is <math>\tilde{p}(1)</math>?
...textbf{(A) } \dfrac{5}{16} \qquad\textbf{(B) } \dfrac{1}{2} \qquad\textbf{(C) } \dfrac{5}{8} \qquad\textbf{(D) } 1 \qquad\textbf{(E) } \dfrac{9}{8}</mat
...
30 KB (5,245 words) - 03:26, 18 June 2025
.../math>, and the angle bisectors of <math>\angle{B}</math> and <math>\angle{C}</math> meet at <math>Q</math>. Find <math>PQ</math>.
pair A, B, C, D, A1, B1, C1, D1, P, Q;
...
25 KB (3,959 words) - 21:31, 28 September 2025
...ath>x^2</math> coefficient is <math>1.</math> Suppose the equation <math>P(P(x))=0</math> has four distinct solutions, <math>x=3,4,a,b.</math> Find the
...- 1}{2}\right) \implies a = \frac{7P(3) + 1}{8}</math>. Now, we note <math>P(4) = \frac{7}{2}</math> by plugging in again. Now, it's easy to find that <
...
11 KB (2,077 words) - 16:21, 31 August 2025
.... Barycentric coordinates were discovered by Möbius in 1827 (Coxeter 1969, p. 217; Fauvel et al. 1993).
...e lengths of <math>BC, AC, AB, \angle A = \alpha, \angle B = \beta, \angle C = \gamma.</math>
...
38 KB (7,569 words) - 02:35, 16 September 2025
...t square in the form <math>(p^2+1)(q^2+1)-((pq)^2-pq+1)</math> where <math>p</math> and <math>q</math> are prime. Find that perfect square.
...ath> and <math>s</math> are relatively prime positive integers. Find <math>r+s</math>.
...
76 KB (12,720 words) - 20:43, 1 November 2025
rt(a^2 - ab + b^2) rt(b^2 - bc + c^2) < rt(a^2 + ac + c^2)
cardinality of interval with R
...
55 KB (7,998 words) - 15:07, 16 September 2024
Find all triples <math>(a,b,p)</math> of positive integers with <math>p</math> prime and
<cmath>a^p = b! + p</cmath>
...
18 KB (3,366 words) - 12:49, 23 July 2025
...ength <math>s</math>, with the property that there is a unique point <math>P</math> inside the triangle such that <math>AP=1</math>, <math>BP=\sqrt{3}</
<math>\textbf{(A) } 1+\sqrt{2} \qquad \textbf{(B) } \sqrt{7} \qquad \textbf{(C) } \frac{8}{3} \qquad \textbf{(D) } \sqrt{5+\sqrt{5}} \qquad \textbf{(E) }
...
16 KB (2,518 words) - 18:41, 26 October 2025
...t]] to <math>\omega_{A},</math> <math>\omega_{B},</math> and <math>\omega_{C}</math>. If the sides of triangle <math>ABC</math> are <math>13,</math> <ma
pair A,B,C,X,Y,Z,P,Q,R;
...
12 KB (2,274 words) - 20:12, 8 November 2025
...be the isogonals with respect <math>\angle BAC.</math> Let <math>BD ||CE, P = BE \cap CD.</math>
Prove that <math>P</math> lies on bisector of <math>\angle BAC</math> and <math>BD||AP.</math>
...
50 KB (8,748 words) - 16:42, 20 October 2025
...isibility|divisible]] by <math>p</math>, then <math>a^{p-1} \equiv 1 \pmod{p}</math>.
...ed to restrict ourselves to integers <math>a</math> not divisible by <math>p</math>.
...
16 KB (2,683 words) - 09:54, 24 September 2025
...P</math> on the incircle of the rhombus such that the distances from <math>P</math> to the lines <math>DA,AB,</math> and <math>BC</math> are <math>9,</m
pair A, B, C, D, O, P, R, S, T;
...
19 KB (3,014 words) - 08:39, 26 October 2025
...the largest semicircle. What is the radius of the circle centered at <math>P</math>?
pair P = (-1,0)+(2+6/7)*dir(36.86989);
...
13 KB (1,982 words) - 16:12, 20 December 2022
<math>\text{(A) } \dfrac32 \quad \text{(B) } \dfrac{1+\sqrt5}2 \quad \text{(C) } \sqrt3 \quad \text{(D) } 2 \quad \text{(E) } \dfrac{3+\sqrt5}2</math>
real r=(3+sqrt(5))/2;
...
7 KB (1,284 words) - 11:52, 31 August 2025
real c=8.1,a=5*(c+sqrt(c^2-64))/6,b=5*(c-sqrt(c^2-64))/6;
pair B=(0,0),C=(c,0), D = (c/2-0.01, -2.26);
...
22 KB (3,772 words) - 00:13, 29 September 2025
...4)?</cmath><math>\textbf{(A) } 64 \qquad \textbf{(B) } 75 \qquad \textbf{(C) } 100 \qquad \textbf{(D) } 125 \qquad \textbf{(E) } 144</math>
...(p^2 + 4)(q^2 + 4)(r^2 + 4)</math> as <math>(p+2i)(p-2i)(q+2i)(q-2i)(r+2i)(r-2i)</math>.
...
5 KB (915 words) - 18:31, 1 November 2025
...gously show that <math>H'</math> also lies on the <math>B</math> and <math>C</math> altitudes, so <math>H'</math> is the orthocenter.
...hocenter is <math>H = A+B+C</math> and the centroid is <math>G = \frac{A+B+C}{3}</math>. Thus, <math>O, G, H</math> are collinear and <math>\frac{OG}{HG
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59 KB (10,203 words) - 03:47, 30 August 2023
...>BC</math>, <math>CD</math>, <math>DE</math>, <math>EA</math> at <math>P,Q,R,S,T</math> respectively. Then let <math>PB=x=BQ=RD=SD</math>, <math>ET=y=ES
...e resulting complex number. Thus, <math>(r + 2i)^2 \cdot (r + 4i)^2 \cdot (r + 3i)</math> is real. Expanding, we get:
...
14 KB (2,353 words) - 13:58, 13 September 2025
.... How many <math>\text{P}</math>s, <math>\text{Q}</math>s, and <math>\text{R}</math>s will appear in the completed table?
label(scale(.9)*"\textsf{P}", (.5,.5));
...
13 KB (1,932 words) - 17:38, 1 June 2025
