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- The plus or minus does not mean that there are two answers, but that the sign of the expression depend These are the identities that are not substantial enough to warrant a section of their own.8 KB (1,397 words) - 21:55, 20 January 2024
- ==Solution 2 (Trig)== ==Solution 4 (Trig)==11 KB (1,741 words) - 22:40, 23 November 2023
- ...sume that <math>R</math> lies on the line connecting the centers, which is not true in general. It is true here only because the perpendicular from <math> === Solution 3 (trig bash)===13 KB (2,151 words) - 17:48, 27 May 2024
- ...intersect the chord <math>BC</math> at two points, one point, or they may not have a point of intersection. By the problem condition, however, the circle ...ave found one possible case. However, in a full-solution contest, we could not assume that the answer is unique, but would need to prove that this is the20 KB (3,497 words) - 15:37, 27 May 2024
- == Solution 3 (less trig required, use of quadratic formula) == Remark: The quadratic need not be solved. The value of <math>ab</math> can be found through Vieta's.3 KB (545 words) - 23:44, 12 October 2023
- ...>q</math>, and <math>r</math> are positive integers, and <math>q</math> is not divisible by the square of any prime number. Find <math>p+q+r</math>. ==Solution 4 (Fast, no trig)==5 KB (811 words) - 21:39, 20 July 2024
- ...>b</math>, and <math>c</math> are positive integers, and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>. ==Solution 2 (no trig)==4 KB (694 words) - 17:16, 29 July 2024
- ...rs <math> \sin^2 x, \cos^2 x, </math> and <math> \sin x \cos x </math> are not the lengths of the sides of a triangle. Given that <math> p = d/n, </math> ...ath> is the largest of the three given expressions and those three lengths not forming a [[triangle]] is equivalent to a violation of the [[triangle inequ5 KB (895 words) - 15:24, 21 June 2024
- ...th>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime. Find <math>m + n.</math> ==Solution 4 (No Trig)==9 KB (1,463 words) - 18:08, 31 July 2024
- ==Solution 3 (trig)== Note: <math>BC=39</math> was not used in this problem.6 KB (892 words) - 01:50, 9 July 2024
- ...th> and <math> r</math> are [[relatively prime]], and <math> q </math> is not [[divisibility | divisible]] by the [[perfect square | square]] of any prim == Solution 3: Trig ==6 KB (1,033 words) - 02:36, 19 March 2022
- Note: Once <math>DY</math> is found, there is no need to do the trig. Notice that the hexagon consists of two trapezoids, <math>ABPQ</math> and ...= X, AP \cap CD = Y.</math> Then, angle chasing shows that <math>CQ</math> not only bisects <math>BX,</math> but is also perpendicular to it. This makes i12 KB (2,015 words) - 20:54, 9 October 2022
- ...very improbably due to the area being <math>\frac{7x}{2}</math>. This does not work. ...\frac{3}{x}</math>, and <math>\tan{90-\theta} = \frac{4}{x}</math>. Using trig identities, we can simplify the second one to <math>\cot{\theta} = \frac{4}5 KB (879 words) - 18:57, 30 April 2024
- ...irc</math> and <math>\angle ADC = 60^\circ</math> as you can see, let's do trig. Drop an altitude from <math>A</math> to <math>BC</math>; call this point < ...75^\circ</math> or <math>105^\circ</math>. Since <math>105^\circ</math> is not a choice, we know <math>\angle ACB = \boxed{75^\circ}</math>.8 KB (1,316 words) - 22:48, 7 March 2024
- ...th>\sin^{-1}(\sin(6x)) = 180 - 6x</math>. As <math>\frac{180}{7}</math> is not on the interval <math>30 \leq x \leq 45</math>, this yields no solution. .... Thus, <math>\sin^{-1}(\sin(6x)) = 6x - 360</math>. As <math>72</math> is not on the interval <math>45 \leq x \leq 60</math>, this yields no solution.7 KB (1,287 words) - 07:09, 22 December 2022
- ...k</math> do the graphs of <math>x^2+y^2=k^2</math> and <math>xy = k</math> not intersect? ...h of the circle. This should be enough to conclude that the hyperbola does not intersect the circle.9 KB (1,622 words) - 20:53, 11 September 2023
- ...egative. Continuing the pattern and accounting for doubled roots (which do not flip sign), we realize that there are <math>5</math> negative intervals fro ...count it as two disjoint intervals. Note that this will be important as to not undercounting disjoint intervals. )9 KB (1,434 words) - 17:54, 17 August 2022
- while word[0] not in vowels: ...erything from position 1, including position 1, all the way to position 3, not including position 3. This is called a slice.28 KB (4,762 words) - 21:20, 12 June 2023
- ==Solution 2 (with trig, not recommended)== ==Solution 3(Easier Trig)==5 KB (782 words) - 14:29, 1 April 2024
- == Solution 2 (trig) == ...e time in seconds. Since <math>x=0</math> is at the crest of the graph and not at the midline, we will use a cosine graph. Therefore, we will use the form3 KB (559 words) - 02:44, 8 February 2024