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2008 USAMO Problems/Problem 6
...mber of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form <math>2^k</math> for some positive integ ...e [[vector space]] of all such functions. Define the linear operator <math>A : V \to V</math> as

13 KB (2,414 words) - 14:37, 11 July 2016
2001 IMO Shortlist Problems/N6
...solving for the <math>99</math> integers <math>a_i</math>. This then each ai takes on the form, <math>j+b_1, j+b_2,..., j+b_{98}</math>. Then we must fi '''Lemma:''' We try to show that for a prime <math>p</math>, there are such <math>p</math> integers less than or e

2 KB (443 words) - 13:08, 17 August 2011
2008 iTest Problems/Problem 80
...1 - i\sqrt{3}}{2}</math>. Thus, <math>-b + a = 1</math> and <math>\tfrac12 a - \tfrac12 b + d = \tfrac12</math>, so a - b + 2d &= 1 \\

3 KB (560 words) - 19:49, 23 November 2018
1962 IMO Problems/Problem 6
...have circumcenter <math>O</math> and incenter <math>I</math>. Extend <math>AI</math> to meet the circumcircle again at <math>L</math>. Then extend <math> ...th>, or <math>\frac {ID}{BL} = \frac {AI} {ML} \iff ID \cdot ML = 2r\rho = AI \cdot BL</math>.

2 KB (308 words) - 06:29, 16 December 2023
2009 AIME I Problems
...s <math>3</math> distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest g ...s a complex number <math>z</math> with imaginary part <math>164</math> and a positive integer <math>n</math> such that

7 KB (1,152 words) - 02:24, 23 July 2021
2009 AIME I Problems/Problem 12
...h> be a point outside <math>\triangle ABC</math> such that <math>\overline{AI}</math> and <math>\overline{BI}</math> are both tangent to circle <math>\om ...h that <math>P</math> is on <math>BI</math> and <math>Q</math> is on <math>AI</math>. We know that <math>AD:CD = CD:BD = 12:35</math>.

12 KB (1,970 words) - 22:53, 22 January 2024
Radical axis
The theory of radical axis is a priceless geometric tool that can solve formidable geometric problems fairl ...rtains to that situation. I hope after you read this text, you will become a better math student, armed with another tool to solve difficult problems. B

12 KB (2,125 words) - 08:38, 23 May 2024
Mock AIME 2 2010 Answer Key
...an be easily proved through strong induction. Starting from 2010, which is a multiple of 15, we must first purge 1 lemming. We can then purge 4 lemmings ...atisfies <math>b_1+b_2+\ldots+b_{10}\equiv 0 \pmod{3}</math>, there exists a corresponding <math>(a_1, a_2, \ldots, a_{10})</math> such that <math>a_1+a

36 KB (6,214 words) - 20:22, 13 July 2023
2019 IMO Problems/Problem 6
...h>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>. We define <math>A' = PG \cap AI.</math>

5 KB (792 words) - 01:52, 19 November 2023
2010 IMO Problems/Problem 2
...Let <math>E</math> be a point on arc <math>BDC</math>, and <math>F</math> a point on the segment <math>BC</math>, such that <math>\angle BAF=\angle CAE ...e midpoint of arc <math>BDC</math> because it lies on angle bisector <math>AI</math>.

3 KB (525 words) - 14:52, 16 July 2023
2010 IMO Problems
...ft\lfloor a\right\rfloor</math> is greatest integer not greater than <math>a.</math> ...Let <math>E</math> be a point on arc <math>BDC</math>, and <math>F</math> a point on the segment <math>BC</math>, such that <math>\angle BAF=\angle CAE

4 KB (603 words) - 09:22, 10 September 2020
2015 IMO Problems/Problem 4
...tersection of <math>\Gamma</math> and <math>\Omega</math>, such that <math>A</math>, <math>F</math>, <math>B</math>, <math>C</math>, and <math>G</math> ...etric wrt <math>AO</math>, a diameter of <math>\Omega</math> through <math>A</math>. So are <math>F</math> and <math>G</math>, as <math>AF=AG</math>. Th

3 KB (502 words) - 23:58, 5 October 2015
2004 USAMO Problems/Problem 6
(''Zuming Feng'') A circle <math>\omega </math> is inscribed in a quadrilateral <math>ABCD </math>. Let <math>I </math> be the center of <ma (AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2

4 KB (700 words) - 23:18, 28 November 2014
2006 IMO Problems
Let <math>ABC</math> be a triangle with incentre <math>I.</math> A point <math>P</math> in the interior of the triangle satisfies <math>\angle Show that <math>AP \ge AI,</math> and that equality holds if and only if <math>P = I.</math>

2 KB (402 words) - 23:18, 12 April 2021
1974 AHSME Problems/Problem 16
...of radius <math> r </math> is inscribed in a right isosceles triangle, and a circle of radius <math> R </math> is circumscribed about the triangle. Then <math> \mathrm{(A)\ } 1+\sqrt{2} \qquad \mathrm{(B) \ }\frac{2+\sqrt{2}}{2} \qquad \mathrm{(C

2 KB (226 words) - 14:11, 3 March 2018
2013 AMC 12B Problems/Problem 25
...th>P(z)</math> has distinct roots of the form <math>a+ib</math> with <math>a</math> and <math>b</math> integers. How many polynomials are in <math>G</ma <math> \textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D)}\ 992

5 KB (750 words) - 01:10, 31 December 2022
Symmetry
A proof utilizes '''symmetry''' if the steps to prove one thing is identical ...nt <math>B</math> with respect to the lines containing the bisectors <math>AI</math> and <math>CI,</math> respectively.

19 KB (3,292 words) - 13:04, 13 May 2024
2014 USAJMO Problems/Problem 6
Let <math>ABC</math> be a triangle with incenter <math>I</math>, incircle <math>\gamma</math> and cir (a) Prove that <math>I</math> lies on ray <math>CV</math>.

7 KB (1,273 words) - 18:17, 28 August 2021
2006 IMO Problems/Problem 1
...gle PBA+\angle PCA = \angle PBC+\angle PCB</math>. Show that <math>AP \geq AI</math>, and that equality holds if and only if <math>P=I.</math> Let ray <math>AI</math> meet the circumcircle of <math>\triangle ABC\, </math> at point <ma

1 KB (247 words) - 01:02, 19 November 2023
1982 AHSME Problems
<math>\textbf{(A)} \ 2 \qquad If a number eight times as large as <math>x</math> is increased by two, then one

17 KB (2,500 words) - 19:05, 11 September 2023

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