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• 1977 Canadian MO Problems
  ....</math> Find the smallest positive integer <math>b</math> for which <math>N</math> is the fourth power of an integer. == Problem 4 ==
  3 KB (560 words) - 19:23, 10 March 2015
• 1959 IMO Problems/Problem 4
  <math> ab = \frac{c^2}{4}. </math> ...math> and a line parallel to <math>c</math> and of distance <math>\frac{c}{4}</math> from <math>c</math>, either point of intersection between the line
  6 KB (939 words) - 17:31, 15 July 2023
• 1959 IMO Problems/Problem 5
  ...t at <math>M </math> and also at another point <math>N </math>. Let <math>N' </math> denote the point of intersection of the straight lines <math>AF </ (a) Prove that the points <math>N </math> and <math>N' </math> coincide.
  4 KB (729 words) - 08:23, 23 May 2024
• 1970 IMO Problems/Problem 4
  ...th>n</math> with the property that the set <math>\{ n, n+1, n+2, n+3, n+4, n+5 \} </math> can be partitioned into two sets such that the product of the ...e factors. The only such number is 1, so the set must be <math>\{ 1, 2, 3, 4, 5, 6 \}</math>, but that does not work because only one of the numbers is
  1 KB (250 words) - 03:31, 2 January 2023
• 1970 IMO Problems/Problem 3
  <math>1 = a_{0} \leq a_{1} \leq \cdots \leq a_{n} \leq \cdots</math>. <math>b_n = \sum_{k=1}^{n} \frac{1 - \frac{a_{k-1}}{a_{k}} }{\sqrt{a_k}}</math>
  3 KB (577 words) - 16:10, 26 July 2009
• 2006 Romanian NMO Problems
  Let <math>ABC</math> be a triangle and the points <math>M</math> and <math>N</math> on the sides <math>AB</math> respectively <math>BC</math>, such that ...th> unit squares, each colored in red, yellow or green. Find minimal <math>n</math>, such that for each coloring, there exists a line and a column with
  11 KB (1,779 words) - 14:57, 7 May 2012
• 2006 Romanian NMO Problems/Grade 9/Problem 4
  ...h>(n \geq 5)</math> participated at table tennis contest, which took <math>4</math> days. Every day, every student played a match. (It is possible that * no student lost all <math>4</math> matches.
  2 KB (325 words) - 15:53, 11 December 2011
• 2005 Canadian MO Problems
  ...its a triangle. An example of one such path is illustrated below for <math>n=5</math>. Determine the value of <math>f(2005)</math>. ...ath>(a,b,c)</math> satisfying <math>\left(\frac{c}{a}+\frac{c}{b}\right)^2=n</math>.
  2 KB (436 words) - 11:45, 26 December 2018
• 2005 Canadian MO Problems/Problem 3
  Let <math>S</math> be a set of <math>n\ge 3</math> points in the interior of a circle. * Show that for no value of <math>n</math> can four such points in <math>S</math> (and corresponding points on
  2 KB (460 words) - 13:35, 9 June 2011
• 2005 Canadian MO Problems/Problem 5
  ...\le b \le c</math>, <math>\gcd(a,b,c) = 1</math>, and <math>a^n + b^n + c^n</math> is divisible by <math>a+b+c</math>. For example, <math>(1,2,2)</math ...l ordered triples (if any) which are <math>n</math>-powerful for all <math>n \ge 1</math>.
  984 bytes (177 words) - 18:21, 28 November 2023
• Midpoint
  draw((0,0)--(4,0)); label("A",(0,0),N);
  4 KB (596 words) - 17:09, 9 May 2024
• P versus NP
  ...\text{TIME}(f(n))</math> is the set of languages decidable by an <math>O(f(n))</math>-time deterministic Turing machine. *Given a list of <math>n</math> integers, is it sorted in non-decreasing order?
  6 KB (1,104 words) - 15:11, 25 October 2017
• 2005 AMC 10A Problems/Problem 10
  <math>4x^2 + ax + 8x + 9 = (mx + n)^2</math> <math>4x^2 + ax + 8x + 9 = m^2x^2 + 2mnx + n^2</math>
  3 KB (443 words) - 18:05, 29 March 2023
• 2005 AMC 10A Problems/Problem 11
  ...urth of the total number of faces of the unit cubes are red. What is <math>n</math>? <math> \textbf{(A) } 3\qquad \textbf{(B) } 4\qquad \textbf{(C) } 5\qquad \textbf{(D) } 6\qquad \textbf{(E) } 7 </math>
  883 bytes (144 words) - 12:40, 13 December 2021
• 2005 AMC 10A Problems/Problem 13
  How many positive integers <math>n</math> satisfy the following condition: <math> (130n)^{50} > n^{100} > 2^{200}\ ?</math>
  2 KB (223 words) - 07:18, 16 July 2022
• 2005 AMC 10A Problems/Problem 21
  For how many positive integers <math>n</math> does <math> 1+2+...+n </math> evenly divide from <math>6n</math>? ...n </math> evenly [[divide]]s <math>6n</math>, then <math>\frac{6n}{1+2+...+n}</math> is an [[integer]].
  1 KB (220 words) - 12:54, 14 December 2021
• 2005 AMC 10A Problems/Problem 25
  label("$A$", A, N); label("$A$", A, N);
  6 KB (854 words) - 20:25, 24 July 2022
• 2005 AMC 10A Problems/Problem 24
  ...t true that both <math> P(n) = \sqrt{n} </math> and <math> P(n+48) = \sqrt{n+48} </math>? ...bf{(A) } 0\qquad \textbf{(B) } 1\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5 </math>
  6 KB (895 words) - 13:52, 4 April 2024
• Volume
  ...> is not divisible by the square of any prime. Find <math> \lfloor m+\sqrt{n}\rfloor. </math> (The notation <math> \lfloor x\rfloor </math> denotes the
  3 KB (523 words) - 20:24, 17 August 2023
• Committee forming
  ...ee of size <math>k</math> from a set of size <math>n</math>. We use <math>{n\choose k}</math> which is called a [[binomial coefficient]]. From here on ...ath>{n\choose 0} + {n\choose 1} + {n\choose 2} + \cdots + {n\choose n} = 2^n.</math></center>
  3 KB (485 words) - 19:49, 16 July 2018

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