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• 1979 IMO Problems/Problem 2
  == See Also == {{IMO box|year=1979|num-b=1|num-a=3}}
  2 KB (500 words) - 22:50, 29 January 2021
• 1979 IMO Problems/Problem 5
  Determine all real numbers a for which there exists non-negative reals <math>x_{1}, \ldots, x_{5}</math> which satisfy the relations ...1,2,3,4</math> From the latter and (2) we also have<cmath>\Sigma(n,n+1)=a(a-n^2)(a-(n+1)^2))\geq 0\implies a\notin (n^2,(n+1)^2)</cmath>So we have that<
  2 KB (391 words) - 10:55, 30 September 2022
• 1984 IMO Problems/Problem 3
  == See Also == {{IMO box|year=1984|num-b=2|num-a=4}}
  2 KB (388 words) - 23:49, 29 January 2021
• 1991 IMO Problems/Problem 2
  ...math> there is a prime <math>r</math> strictly between <math>a_2=p,\ a_3=2p-1</math>. <math>rq|n</math>, so, in particular, <math>pq<rq\le n</math>. On ...<math>q\not|p-1</math>, and all the numbers <math>1,1+p-1,\ldots,1+(q-1)(p-1)</math> are smaller than <math>n</math> (they are smaller than <math>pq</m
  11 KB (1,999 words) - 17:42, 30 January 2021
• 2021 AMC 12B Problems/Problem 23
  ...so the probability <math>x</math> is the middle bin is <math>6\cdot\frac{x-1}{8^x}</math>. Then, we want the sum 6\sum_{x=2}^{\infty}\frac{x-1}{8^x} &= \frac{6}{8}\left[\frac{1}{8} + \frac{2}{8^2} + \frac{3}{8^3}\cdot
  6 KB (862 words) - 03:25, 28 January 2023
• 2021 AMC 12B Problems/Problem 25
  ...n hopes of finding a pattern. I graphed the first couple positive integer x-coordinates, and found that the sum of the integers above the line is <math> ...-coordinate, a shift in the slope will increase the y-value of the higher x-coordinates. We turn our attention to <math>x=28, 29, 30</math> which the li
  17 KB (2,687 words) - 23:43, 7 November 2022
• 1956 AHSME Problems/Problem 28
  {{AHSME 50p box|year=1956|num-b=27|num-a=29}}
  861 bytes (123 words) - 22:26, 12 February 2021
• 2021 AIME II Problems/Problem 8
  ==Solution 1 (Four-Variable Recursion)== <ol style="margin-left: 1.5em;">
  17 KB (2,722 words) - 18:32, 23 January 2023
• 2021 AIME II Problems/Problem 11
  ...d to the class that the numbers in <math>S</math> were four consecutive two-digit positive integers, that some number in <math>S</math> was divisible by <ol style="margin-left: 1.5em;">
  15 KB (2,211 words) - 10:59, 4 February 2024
• Economics books
  332 bytes (42 words) - 23:03, 11 March 2021
• 2021 JMC 10 Problems/Problem 22
  Let <math>r_1,r_2,r_3,r_4</math> be the roots of <math>P(x)= x^4+4x^3-3x^2+2x-1.</math> Suppose <math>Q(x)</math> is the monic polynomial with all six roo ...s of each <math>(r_{i})^2</math> is produced, and there are <math>\tfrac{60-3\cdot 4}{6}=8</math> copies of <math>r_{i}r_{j}</math> by symmetry. By Viet
  1 KB (209 words) - 15:51, 1 April 2021
• 1994 IMO Problems/Problem 1
  <cmath>a_j+a_{m-j+1} \ge n+1</cmath> <cmath> a_j + a_{m-j+1} \le n</cmath>
  2 KB (370 words) - 01:28, 22 November 2023
• 2021 Fall MIMC 10
  ...nal of square <math>ABCD</math>. Square <math>IGED</math> has area <math>11-6\sqrt{2}</math>. Given that point <math>J</math> bisects line segment <math ...\pi+1}{280+100\pi} \qquad\textbf{(E)} ~\frac{50\pi^2+700\pi\sqrt{2}+3001\pi-70\sqrt{2}+60}{2\pi^2+240\pi+6920}\qquad</math>
  14 KB (2,226 words) - 23:39, 12 September 2021
• 2021 April MIMC 10 Problems/Problem 22
  ...nal of square <math>ABCD</math>. Square <math>IGED</math> has area <math>11-6\sqrt{2}</math>. Given that point <math>J</math> bisects line segment <math ...triangles, therefore, we can solve for length <math>AI</math>. <math>AI=AD-ID</math>. Use the technique of sum of squares and square root disintegratio
  3 KB (456 words) - 14:03, 26 April 2021
• 2021 GMC 12
  ...integers with <math>a_0=1</math> and <math>a_1=2</math> and <math>a_n=a_{n-1}\cdot a_{n+1}</math> for all integers <math>n</math> such that <math>n\geq ...quad\textbf{(D)} ~\frac{\pi+2\sqrt{3}-3}{4} \qquad\textbf{(E)} ~\frac{\pi+3-2\sqrt{2}}{4}</math>
  13 KB (2,097 words) - 17:38, 29 April 2021
• G285 2021 MC10B
  If <math>deg(Q(x))=3</math>, and <math>deg(K(x))=2</math>, and <math>Q(x)=(x-2)K(x)</math>, what is <math>deg(Q(x)-2K(x))</math>? A principal is pushing out an emergency COVID-19 alert to his school of <math>40</math> teachers and <math>500</math> stud
  4 KB (675 words) - 14:01, 28 May 2021
• G285 2021 MC10A
  ...k</math> is prime, for what value of <math>p</math> will <math>k_{2021} = k-2022p+1</math>? If a real number <math>k</math> is <math>happy</math> , <math>k^3+5k-3 \ge (k-1)^4</math>. If a real number <math>l</math> is <math>unhappy</math> , <ma
  8 KB (1,385 words) - 12:55, 23 June 2021
• 2022 AIME II Problems/Problem 2
  {{AIME box|year=2022|n=II|num-b=1|num-a=3}}
  4 KB (673 words) - 16:15, 24 January 2023
• 2022 AIME II Problems/Problem 8
  ...we can also say that if <math>60k+l</math> is a solution, then <math>60k-l-1</math> is a solution! Therefore, one doesn't have to go as far as determi ...and <math>20</math> possibilities for cases 2 and 4 each. However, we over-counted the cases where
  13 KB (2,136 words) - 22:39, 31 December 2023
• 2022 AIME II Problems/Problem 9
  ...mula of this recursion: <math>f(n+1)=f(n)+N_a+\frac{n\cdot (N_a) \cdot (N_a-1)}{2}</math>, where <math>N_a</math> is the number of points on <math>\ell_ ...>, it crosses <math>m*(n-2)</math> lines, thus making additional <math>m*(n-2)+1</math> bounded regions; etc. By simple algebra/recursion methods, we se
  12 KB (2,025 words) - 14:56, 25 January 2024

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