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• 1993 AIME Problems/Problem 1
  The thousands digit is <math>\in \{4,5,6\}</math>. Case <math>1</math>: Thousands digit is even
  3 KB (440 words) - 21:20, 22 July 2021
• 1994 AIME Problems/Problem 15
  ...rs and <math>s\,</math> is not divisible by the square of any prime. What is <math>q+r+s\,</math>? ...th>; the [[locus]] of each of the respective conditions for <math>P</math> is the region inside the (semi)circles with diameters <math>\overline{AB}, \ov
  4 KB (717 words) - 22:20, 3 June 2021
• 1994 AIME Problems/Problem 13
  ...\text{cis}\left(\frac {(2k + 1)\pi}{10}\right)</math> where <math>k</math> is an integer between <math>0</math> and <math>9</math>. The expression to find is <math>\sum t\bar{t} = 850 - 26\sum_{k = 0}^4 \cos \frac {(2k + 1)\pi}{10}</
  3 KB (375 words) - 23:46, 6 August 2021
• 1994 AIME Problems/Problem 12
  ...the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plots into which the field can be partiti ...ac {13}6n</math> squares in every row. Then <math>6|n</math>, and our goal is to maximize the value of <math>n</math>.
  3 KB (473 words) - 17:06, 1 January 2024
• 1994 AIME Problems/Problem 11
  ...multiples of <math>3</math>, the change will always be a multiple of <math>3</math>, so we just need to find the number of changes we can get from <math ...r <math>m,n</math> being [[positive integer]]s is <math>5 \times 2 - 5 - 2=3</math>.
  4 KB (645 words) - 15:12, 15 July 2019
• 1994 AIME Problems/Problem 10
  ...ths of the sides of <math>\triangle ABC\,</math> are integers, <math>BD=29^3,\,</math> and <math>\cos B=m/n\,</math>, where <math>m\,</math> and <math>n ...he form <math>29^2 x</math> and <math>29 x^2</math>, respectively, where x is an integer.
  3 KB (534 words) - 16:23, 26 August 2018
• 1994 AIME Problems/Problem 9
  ...nues until the bag is empty. The probability that the bag will be emptied is <math>p/q,\,</math> where <math>p\,</math> and <math>q\,</math> are relativ *Case 1. We draw a pair on the first two cards. The second card is the same as the first with probability <math>\frac {1}{2k - 1}</math>, then
  3 KB (589 words) - 14:18, 21 July 2019
• 1994 AIME Problems/Problem 8
  Consider the points on the [[complex plane]]. The point <math>b+37i</math> is then a rotation of <math>60</math> degrees of <math>a+11i</math> about the ...)\left(\mathrm{cis}\,60^{\circ}\right) = (a+11i)\left(\frac 12+\frac{\sqrt{3}i}2\right)=b+37i.</cmath>
  5 KB (788 words) - 13:53, 8 July 2023
• 1994 AIME Problems/Problem 7
  has at least one solution, and each solution is an ordered pair <math>(x,y)\,</math> of integers. How many such ordered pa ...h> where the signs are all independent of each other, for a total of <math>3\cdot 2\cdot 2=12</math> lattice points. They are indicated by the blue dots
  3 KB (442 words) - 19:51, 8 January 2024
• 1994 AIME Problems/Problem 6
  <center><math>y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k,</math></center> ...the plane into equilateral triangles of side length <math>\tfrac{2}{\sqrt{3}}.\,</math> How many such triangles are formed?
  4 KB (721 words) - 16:14, 8 March 2021
• 1994 AIME Problems/Problem 5
  ...math>. (If <math>n\,</math> has only one digits, then <math>p(n)\,</math> is equal to that digit.) Let <center><math>S=p(1)+p(2)+p(3)+\cdots+p(999)</math></center>.
  2 KB (275 words) - 19:27, 4 July 2013
• 1994 AIME Problems/Problem 4
  \lfloor\log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994 (For real <math>x\,</math>, <math>\lfloor x\rfloor\,</math> is the greatest integer <math>\le x.\,</math>)
  2 KB (264 words) - 13:33, 11 August 2018
• 1994 AIME Problems/Problem 2
  ...ngent at <math>P</math> to a circle of radius 20. Square <math>ABCD</math> is constructed with <math>A</math> and <math>B</math> on the larger circle, <m ...that <math>OE = AB - OQ = AB - 10</math>. The other leg, <math>AE</math>, is just <math>\frac 12 AB</math>.
  2 KB (272 words) - 03:53, 23 January 2023
• 1994 AIME Problems/Problem 1
  ...square]]. What is the [[remainder]] when the 1994th term of the sequence is divided by 1000? ...= 2992</math>. The value of <math>n^2 - 1 = 2992^2 - 1 \pmod{1000}</math> is <math>\boxed{063}</math>.
  946 bytes (139 words) - 21:05, 1 September 2023
• 1995 AIME Problems/Problem 15
  ...ld start with a block of <tt>H</tt>'s, the total probability is that <math>3/2</math> of it has to start with an <tt>H</tt>. ...ose sum is <math>\frac{3/64}{1-(15/32)}=\frac{3}{34}</math>, so the answer is <math>\boxed{037}</math>.
  6 KB (979 words) - 13:20, 11 April 2022
• 1995 AIME Problems/Problem 14
  ...,</math> and <math>d_{}</math> are positive integers and <math>d_{}</math> is not divisible by the square of any prime number. Find <math>m+n+d.</math> ...math>, and <math>EF = \sqrt{OE^2 - OF^2} = 9</math>. Then <math>OEF</math> is a <math>30-60-90</math> [[right triangle]], so <math>\angle OEB = \angle OE
  3 KB (484 words) - 13:11, 14 January 2023
• 1995 AIME Problems/Problem 13
  ...- \left(k^4 - 2k^3 + \frac 32k^2 - \frac 12k + \frac 1{16}\right)\\ &= 4k^3 + k. \end{align*}</cmath> ...^3 + k</math> times, and the sum for each <math>k</math> is then <math>(4k^3 + k) \cdot \frac{1}{k} = 4k^2 + 1</math>. From <math>k = 1</math> to <math>
  2 KB (287 words) - 01:25, 12 December 2019
• 1995 AIME Problems/Problem 12
  ...ity]], let <math>AP = 1.</math> It follows that <math>\triangle OPA</math> is a <math>45-45-90</math> [[right triangle]], so <math>OP = AP = 1,</math> <m ...= 1 + 1 - 2\cos \theta \Longrightarrow \cos \theta = - 3 + 2\sqrt {2} = - 3 + \sqrt{8}.</cmath>
  8 KB (1,172 words) - 21:57, 22 September 2022
• 1995 AIME Problems/Problem 11
  ...of <math>P_{}</math> cuts <math>P_{}</math> into two prisms, one of which is [[similar]] to <math>P_{},</math> and both of which have nonzero volume. G ...nce <math>x < a, y < b, z < c</math>, it follows that the only possibility is <math>y=a,z=b=1995</math>. Then,
  2 KB (292 words) - 19:30, 4 July 2013
• 1995 AIME Problems/Problem 10
  What is the largest positive integer that is not the sum of a positive integral multiple of <math>42</math> and a positi ...h>\mod {42}</math> must be a [[composite]] number. Also, every number that is a multiple of <math>42</math> greater than that prime number must also be p
  3 KB (436 words) - 19:26, 2 September 2023
• 1995 AIME Problems/Problem 9
  ...on <math>\overline{AM}</math> with <math>AD=10</math> and <math>\angle BDC=3\angle BAC.</math> Then the perimeter of <math>\triangle ABC</math> may be ...defaultpen(dps); pen ds=black; real xmin=-1.55,xmax=7.95,ymin=-4.41,ymax=5.3;
  7 KB (1,181 words) - 13:47, 3 February 2023
• 1995 AIME Problems/Problem 8
  ...tory positive integers for all integers <math>y \le 100</math>. The answer is ...1}^{99} \left\lfloor\frac{100-y}{y(y+1)} \right\rfloor = 49 + 16 + 8 + 4 + 3 + 2 + 1 + 1 + 1 = \boxed{085}.</cmath>
  4 KB (646 words) - 17:37, 1 January 2024
• 1995 AIME Problems/Problem 7
  ...s t - \sin t - \cos t + 1 = \frac{13}{4} - \sqrt{10}</math>, so the answer is <math>13 + 4 + 10 = \boxed{027}</math>. ...ath>. Therefore, <math>x = \frac{13}{4} - \sqrt{10}</math>, and the answer is <math> 13 + 4 + 10 = \boxed{027}</math>.
  3 KB (427 words) - 09:23, 13 December 2023
• 1995 AIME Problems/Problem 6
  Let <math>n=2^{31}3^{19}.</math> How many positive [[integer]] [[divisor]]s of <math>n^2</math ...h>) into pairs that multiply to <math>n^2</math>, then one factor per pair is less than <math>n</math>, and so there are <math>\frac{63\times 39-1}{2} =
  2 KB (407 words) - 08:14, 4 November 2022
• 1995 AIME Problems/Problem 5
  ...ese roots is <math>13+i</math> and the sum of the other two roots is <math>3+4i,</math> where <math>i=\sqrt{-1}.</math> Find <math>b.</math> ...pairs. Let the first two roots be <math>m,n</math>. Since <math>m+n</math> is not real, <math>m,n</math> are not conjugates, so the other pair of roots m
  3 KB (451 words) - 15:02, 6 September 2021
• 1995 AIME Problems/Problem 4
  ...adius <math>9</math>. The circle of radius <math>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the ...i(acos(1/3)), F=B+3*expi(acos(1/3)), P=IP(F--F+3*(D-F),CR(A,9)), Q=IP(F--F+3*(F-D),CR(A,9));
  3 KB (605 words) - 11:30, 5 May 2024
• 1995 AIME Problems/Problem 3
  ...coordinate plane]] via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let <math>p</math> be t ...reach <math>(2,2)</math>, so the number of steps the object may have taken is either <math>4</math> or <math>6</math>.
  3 KB (602 words) - 23:15, 16 June 2019
• 1995 AIME Problems/Problem 2
  ...2}</math>. Thus, the product of the two roots (both of which are positive) is <math>1995^{1+\sqrt{2}/2} \cdot 1995^{1 - \sqrt{2}/2} = 1995^2</math>, maki ...od{1000}\equiv (-5)^2\pmod{1000}\equiv 25\pmod{1000}</math>, so our answer is <math>\boxed{025}</math>.
  2 KB (362 words) - 00:40, 29 January 2021
• 1995 AIME Problems/Problem 1
  ...</math> The total area enclosed by at least one of <math>S_{1}, S_{2}, S_{3}, S_{4}, S_{5}</math> can be written in the form <math>m/n,</math> where <m The sum of the areas of the [[square]]s if they were not interconnected is a [[geometric sequence]]:
  2 KB (302 words) - 19:29, 4 July 2013
• 1996 AIME Problems/Problem 15
  ...each twice as large as angle <math>DBA</math>, and angle <math>ACB</math> is <math>r</math> times as large as angle <math>AOB</math>. Find <math>\lfloor pair B=(0,0), A=expi(pi/4), C=IP(A--A + 2*expi(17*pi/12), B--(3,0)), D=A+C, O=IP(A--C,B--D);
  5 KB (710 words) - 21:04, 14 September 2020
• 1996 AIME Problems/Problem 14
  A <math>150\times 324\times 375</math> [[rectangle|rectangular]] [[solid]] is made by gluing together <math>1\times 1\times 1</math> cubes. An internal [ ...point on the diagonal with coordinates <math>(ma, mb, mc)</math>. We have 3 key observations as this point moves from <math>(0,0,0)</math> towards <mat
  5 KB (923 words) - 21:21, 22 September 2023
• 1996 AIME Problems/Problem 13
  .../math> [[bisect]]s <math>\overline{BC}</math>, and <math>\angle ADB</math> is a right angle. The ratio <math>\frac{[ADB]}{[ABC]}</math> can be written in ...he problem asks for a ratio, we can divide each side length by <math>\sqrt{3}</math> to make things simpler. We now have a triangle with sides <math>\sq
  3 KB (521 words) - 01:18, 25 February 2016
• 1996 AIME Problems/Problem 12
  ...utation]] <math>a_1,a_2,a_3,\cdots,a_{10}</math> of the integers <math>1,2,3,\cdots,10</math>, form the sum + |2 - 10| + \ldots + |2 - 3| + |2 - 1|\\
  5 KB (879 words) - 11:23, 5 September 2021
• 1996 AIME Problems/Problem 11
  ...math>\mathrm {P}</math> be the product of the [[root]]s of <math>z^6+z^4+z^3+z^2+1=0</math> that have a positive [[imaginary]] part, and suppose that <m 0 &=& z^6 - z + z^4 + z^3 + z^2 + z + 1 = z(z^5 - 1) + \frac{z^5-1}{z-1}\\
  6 KB (1,022 words) - 20:23, 17 April 2021
• 1996 AIME Problems/Problem 10
  ...ent|tangent]] function is <math>180^\circ</math>, and the tangent function is [[one-to-one]] over each period of its domain. Therefore, the smallest positive solution is <math>x = \boxed{159}</math>.
  4 KB (503 words) - 15:46, 3 August 2022
• 1996 AIME Problems/Problem 9
  ...s wandering back and forth in this manner until every locker is open. What is the number of the last locker he opens? ...>, leaving lockers <math>86, 342, 598</math>, and <math>854</math>, and he is at where he started again. He then opens <math>86</math> and <math>598</mat
  3 KB (525 words) - 23:51, 6 September 2023
• 1996 AIME Problems/Problem 8
  ...rdered pairs of positive integers <math>(x,y)</math> with <math>x<y</math> is the harmonic mean of <math>x</math> and <math>y</math> equal to <math>6^{20 ...h>x<y</math>, the answer is half of the remaining number of factors, which is <math>\frac{1599-1}{2}= \boxed{799}</math>.
  1 KB (155 words) - 19:32, 4 July 2013
• 1996 AIME Problems/Problem 7
  pathpen = black; pair O = (3.5,3.5); D(O); fill(shift(4,3)*unitsquare,rgb(1,1,.4));fill(shift(4,5)*unitsquare,rgb(1,1,.4));
  4 KB (551 words) - 11:44, 26 June 2020
• 1996 AIME Problems/Problem 6
  The probability that one team wins all games is <math>5\cdot \left(\frac{1}{2}\right)^4=\frac{5}{16}</math>. Similarity, the probability that one team loses all games is <math>\frac{5}{16}</math>.
  3 KB (461 words) - 01:00, 19 June 2019
• 1996 AIME Problems/Problem 5
  ...</math>, <math>b</math>, and <math>c</math>, and that the roots of <math>x^3+rx^2+sx+t=0</math> are <math>a+b</math>, <math>b+c</math>, and <math>c+a</m ...^3+3x^2+4x-11 = (x-a)(x-b)(x-c) = 0</math>, we have <math>a + b + c = s = -3</math>, <math>ab + bc + ca = 4</math>, and <math>abc = 11</math>. Then
  3 KB (585 words) - 22:08, 19 November 2022
• 1996 AIME Problems/Problem 4
  .... The area of the shadow, which does not include the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed <math (Figure not to scale) The area of the square shadow base is <math>48 + 1 = 49</math>, and so the sides of the shadow are <math>7</math>
  2 KB (257 words) - 17:50, 4 January 2016
• 1996 AIME Problems/Problem 3
  ...square after <math>1996</math> is <math>2025 = 45^2</math>, so our answer is <math>45 - 1 = \boxed{044}</math>. ...</math> terms. Therefore, we need to find the smallest perfect square that is greater than <math>1996</math>. From trial and error, we get <math>44^2 = 1
  3 KB (515 words) - 04:29, 27 November 2023
• 1996 AIME Problems/Problem 2
  ...that <math>n<1000</math> and that <math>\lfloor \log_{2} n \rfloor</math> is a positive even integer? ...4</math> for the third, and <math>256</math> for the fourth, so the answer is <math>4+16+64+256=\boxed{340}</math>.
  1 KB (163 words) - 19:31, 4 July 2013
• 1996 AIME Problems/Problem 1
  ...c [[square]], the sum of the three entries in any row, column, or diagonal is the same value. The figure shows four of the entries of a magic square. Fin \multicolumn{3}{c}{\text{Table}}\\\hline
  2 KB (332 words) - 11:28, 4 August 2021
• 1997 AIME Problems/Problem 15
  ...ath>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>. ...\left(\frac {11}{2} - \frac {a\sqrt {3}}{2}\right) + \left(\frac {11\sqrt {3}}{2} + \frac {a}{2}\right)i = b + 10i</math>.
  4 KB (609 words) - 22:49, 17 July 2023
• 1997 AIME Problems/Problem 14
  ...t <math>\frac{m}{n}</math> be the [[probability]] that <math>\sqrt{2+\sqrt{3}}\le\left|v+w\right|</math>, where <math>m</math> and <math>n</math> are [[ ...rt{3}</math>, which simplifies to <cmath>\cos((m-n)\theta) \ge \frac{\sqrt{3}}{2}</cmath>Thus, <cmath>|m - n| \le \frac{\pi}{6} \cdot \frac{1997}{2 \pi}
  5 KB (874 words) - 22:30, 1 April 2022
• 1997 AIME Problems/Problem 13
  ...<math>a</math> and <math>b</math> are positive integers and <math>b</math> is not divisible by the square of any prime number. Find <math>a+b</math>. | <math>f(3) = 0</math> || <math>f(1.1) = 0.1</math>
  7 KB (1,225 words) - 19:56, 4 August 2021
• 1997 AIME Problems/Problem 12
  ...r all values except <math>\frac{-d}{c}</math>. Find the unique number that is not in the range of <math>f</math>. The only value that is not in the range of this function is <math>\frac {a}{c}</math>. To find <math>\frac {a}{c}</math>, we use the tw
  11 KB (2,063 words) - 22:59, 21 October 2023
• 1997 AIME Problems/Problem 11
  ...{n=1}^{44} \cos n^\circ}{\sum\limits_{n=1}^{44} \sin n^\circ}</math>. What is the greatest integer that does not exceed <math>100x</math>? We want to pair up <math>[1, 44]</math>, <math>[2, 43]</math>, <math>[3, 42]</math>, etc. from the numerator and <math>[46, 89]</math>, <math>[47,
  10 KB (1,514 words) - 14:35, 29 March 2024
• 1997 AIME Problems/Problem 10
  ...pe-color-shade combination represented. A set of three cards from the deck is called complementary if all of the following statements are true: *'''Case 1''': All three attributes are the same. This is impossible since sets contain distinct cards.
  3 KB (585 words) - 19:37, 25 April 2022
• 1997 AIME Problems/Problem 9
  ...ve, <math>\langle a^{-1}\rangle=\langle a^2\rangle</math>, and <math>2<a^2<3</math>. Find the value of <math>a^{12}-144a^{-1}</math>. ...+\sqrt{5}}2</math> (the [[phi|golden ratio]]) is the answer. The following is the way to derive that:
  4 KB (586 words) - 21:53, 30 December 2023
• 1997 AIME Problems/Problem 8
  ...of the entries in each row is 0 and the sum of the entries in each column is 0? The problem is asking us for all configurations of <math>4\times 4</math> grids with 2 1's
  4 KB (638 words) - 16:41, 22 January 2024
• 1997 AIME Problems/Problem 7
  ...</math> mile per minute. At time <math>t=0</math>, the center of the storm is <math>110</math> miles due north of the car. At time <math>t=t_1</math> min ...ar is at <math>\left(\frac 23t,0\right)</math> and the center of the storm is at <math>\left(\frac{t}{2}, 110 - \frac{t}{2}\right)</math>. Using the dist
  4 KB (617 words) - 18:47, 17 July 2022
• 1997 AIME Problems/Problem 6
  ...A_n</math>, and <math>A_1A_2B</math> is an [[equilateral triangle]]. What is the largest value of <math>n</math> for which <math>A_1</math>, <math>A_n</ Clearly <math>n</math> is maximized when <math>m = 7, n = \boxed{042}</math>.
  3 KB (497 words) - 00:39, 22 December 2018
• 1997 AIME Problems/Problem 4
  ...<math>5, x, 5+r</math> and <math>5, 8+r+x, 13</math>, wher <math>x</math> is the distance between the center of the circle in question and the segment c NOTE: It can be seen that there is no apparent need to use the variable x as a 5,12,13 right triangle has been
  2 KB (354 words) - 22:33, 2 February 2021
• 1997 AIME Problems/Problem 3
  ...number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number? ...and <math>y=112</math>, which satisifies the conditions. Hence the answer is <math>112 + 14 = \boxed{126}</math>.
  2 KB (375 words) - 19:34, 4 August 2021
• 1997 AIME Problems/Problem 2
  ...is pattern can be easily generalized and we see that the number of squares is just <math>\sum^8_{i=1}{i^2}</math>. This can be simplified by using the we ...{i=1}{i}}</math>. This gets us <math>{(\frac{9\cdot8}{2})}^2,</math> which is just <math>1296.</math>
  3 KB (416 words) - 21:09, 27 October 2022
• 1998 AIME Problems/Problem 15
  ...n ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after ...the domino a set of 40 points (labeled 1 through 40) in which every point is connected with every other point. The connections represent the dominoes.
  9 KB (1,671 words) - 22:10, 15 March 2024
• 1998 AIME Problems/Problem 14
  ...math> and <math>p</math> are integers, and <math>m\le n\le p.</math> What is the largest possible value of <math>p</math>? ...the second pair gives <math>98</math>. We now check that <math>130</math> is optimal, setting <math>a=m-2</math>, <math>b=n-2</math> in order to simplif
  2 KB (390 words) - 21:05, 29 May 2023
• 1998 AIME Problems/Problem 13
  ...n,</math> its complex power sum is defined to be <math>a_1i + a_2i^2+ a_3i^3 + \cdots + a_ni^n,</math> where <math>i^2 = - 1.</math> Let <math>S_n</mat ...we will just define to have a power sum of zero) with <math>9</math> in it is equal to the number of subsets without a <math>9</math>. To easily see this
  2 KB (384 words) - 19:02, 20 October 2023
• 1998 AIME Problems/Problem 12
  ...and <math>c</math> is not divisible by the square of any [[prime]]. What is <math>a^{2} + b^{2} + c^{2}</math>? ...math> gives <math>x = \frac {\sqrt {5} - 1}{2}</math> and <math>y = \frac {3 - \sqrt {5}}{2}</math>.
  5 KB (876 words) - 20:27, 9 June 2022
• 1998 AIME Problems/Problem 9
  ...math> and <math>c</math> are [[positive]] [[integer]]s, and <math>c</math> is not divisible by the square of any [[prime]]. Find <math>a + b + c.</math> ...e the area of the unshaded region over the area of the total region, which is the probability that the mathematicians do not meet:
  4 KB (624 words) - 18:34, 18 February 2018
• 1998 AIME Problems/Problem 8
  ...he preceding term from the one before that. The last term of the sequence is the first [[negative]] term encounted. What positive integer <math>x</math The best way to start is to just write out some terms.
  2 KB (354 words) - 19:37, 24 September 2023
• 1998 AIME Problems/Problem 7
  Note that this is an algebraic bijection, we have simplified the problem and essentially remo ...hrough 50; thus the answer is <math>n = {50\choose3} = \frac{50 * 49 * 48}{3 * 2} = 19600</math>, and <math>\frac n{100} = \boxed{196}</math>.
  5 KB (684 words) - 11:41, 13 August 2023
• 1998 AIME Problems/Problem 4
  ...values. The [[probability]] that all three players obtain an [[odd]] sum is <math>m/n,</math> where <math>m</math> and <math>n</math> are [[relatively ...that it matters in what order the people pick the tiles; the final answer is the same if we assume the opposite, that order doesn't matter.)
  5 KB (917 words) - 02:37, 12 December 2022
• 1998 AIME Problems/Problem 2
  ...ac 12</math> of the 30 by 30 [[square]] it is in. A simple way to see this is to note that the two triangles outside of the quadrilateral form half of th ...positive integers, so <math>(0,0)</math> doesn't count; hence, the answer is <math>480</math>.
  6 KB (913 words) - 16:34, 6 August 2020
• 1998 AIME Problems/Problem 1
  For how many values of <math>k</math> is <math>12^{12}</math> the [[least common multiple]] of the positive integers It is evident that <math>k</math> has only 2s and 3s in its prime factorization,
  2 KB (289 words) - 22:50, 23 April 2024
• 1999 AIME Problems/Problem 15
  ...ed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid? ...om <math>P</math> to <math>\triangle ABC</math>. The crux of this problem is the following lemma.
  7 KB (1,107 words) - 20:34, 27 January 2023
• 1999 AIME Problems/Problem 14
  ...ath> and <math>CA=15,</math> and the [[tangent]] of angle <math>PAB</math> is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relativ /* constructing P, C is there as check */
  7 KB (1,184 words) - 13:25, 22 December 2022
• 1999 AIME Problems/Problem 12
  ...]] to <math>\overline{AB}</math> at <math>P_{},</math> and its [[radius]] is <math>21</math>. Given that <math>AP=23</math> and <math>PB=27,</math> fin We want the perimeter, which is <math>2s = 2\left(50 + \frac{245}{2}\right) = \boxed{345}</math>.
  3 KB (472 words) - 15:59, 25 February 2022
• 1999 AIME Problems/Problem 11
  ...s equal to the second half), but it quickly becomes apparent that this way is difficult to pull off. Instead, we look to [[telescope]] the sum. Using the ...}{\sin 175} \Longrightarrow s = \tan \frac{175}{2},</cmath> and our answer is <math>\boxed{177}</math>.
  4 KB (614 words) - 04:38, 8 December 2023
• 1999 AIME Problems/Problem 10
  ...segments form a [[triangle]] whose vertices are among the ten given points is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are [[relat ...e picked. Since the triangle accounts for 3 segments, there are <math>45 - 3 = 42</math> segments remaining.
  3 KB (524 words) - 17:25, 17 July 2023
• 1999 AIME Problems/Problem 9
  ...ction]] has the property that the image of each point in the complex plane is [[equidistant]] from that point and the [[origin]]. Given that <math>|a+bi ...h passes through <math>\left(\frac 12, \frac12\right)</math>, so its graph is <math>x + y = 1</math>. Substituting <math>x = (a-b)</math> and <math>y = (
  6 KB (1,010 words) - 19:01, 24 May 2023
• 1999 AIME Problems/Problem 7
  ...9</math>. At step i of a 1000-step process, the <math>i</math>-th switch is advanced one step, and so are all the other switches whose labels divide th ...f <math>\frac{N}{d}</math> must be a multiple of 4 to ensure that a switch is in position A:
  3 KB (475 words) - 13:33, 4 July 2016
• 1999 AIME Problems/Problem 6
  ...er manipulation <math>y = \frac {x}{\sqrt {3}}</math> and <math>y = \sqrt {3}x</math>, respectively, which are still linear functions. Basically the squ ...2\pi) = \frac {1}{12}(2400\pi - 1200\pi) = 100\pi</math>. Hence the answer is <math>\boxed{314}</math>.
  2 KB (354 words) - 16:42, 20 July 2021
• 1999 AIME Problems/Problem 4
  ...h> is <math>43/99</math> and the [[area]] of octagon <math>ABCDEFGH</math> is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are [[relat ...of all <math>8</math> of them is <math>\frac{86}{99}</math> and the answer is <math>\boxed{185}</math>.
  3 KB (398 words) - 13:27, 12 December 2020
• 1999 AIME Problems/Problem 3
  ...all [[positive integer]]s <math>n</math> for which <math>n^2-19n+99</math> is a [[perfect square]]. ...math>. This gives <math>n=1, 9, 10,</math> or <math>18</math>, and the sum is <math>1+9+10+18=\boxed{38}</math>.
  2 KB (296 words) - 01:18, 29 January 2021
• 1999 AIME Problems/Problem 2
  ...his figure into two [[congruent]] [[polygon]]s. The [[slope]] of the line is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are [[relat ...th>\frac{45 + \frac{135}{19}}{10} = \frac{99}{19}</math>, and the solution is <math>m + n = \boxed{118}</math>.
  3 KB (423 words) - 11:06, 27 April 2023
• 1999 AIME Problems/Problem 1
  Find the smallest prime that is the fifth term of an increasing [[arithmetic sequence]], all four preceding ...th>, and <math>29</math> form an [[arithmetic sequence]]. Thus, the answer is <math>029</math>.
  2 KB (332 words) - 13:22, 3 August 2020
• 1999 AIME Problems/Problem 8
  ...f <math>\mathcal{S}</math> divided by the area of <math>\mathcal{T}</math> is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relati ...of the above diagram, of <math>y \ge \frac{1}{3}, z \ge \frac{1}{6}</math> is the triangle at the right, and <math>x \ge \frac 12, z \ge \frac 16</math>
  3 KB (445 words) - 19:40, 4 July 2013
• 2000 AIME I Problems/Problem 15
  ...ft to right, the labels on the cards are now in ascending order: <math>1,2,3,\ldots,1999,2000.</math> In the original stack of cards, how many cards wer ...4th card when there are 8 cards remaining. This pattern continues until it is the 512th card on the deck when there are 1024 cards remaining. Since there
  15 KB (2,673 words) - 19:16, 6 January 2024
• 2000 AIME I Problems/Problem 14
  ...th>r</math> times as large as angle <math>APQ,</math> where <math>r</math> is a positive real number. Find <math>\lfloor 1000r \rfloor</math>. ...ath>which implies that <math>1000r = 571 + \tfrac 37</math>. So the answer is <math>\boxed{571}</math>.
  8 KB (1,275 words) - 03:04, 27 February 2022
• 2000 AIME I Problems/Problem 13
  ...an be reached by the firetruck within six minutes. The area of this region is <math>m/n</math> square miles, where <math>m</math> and <math>n</math> are ...Arrows(4)); D((0,-6)--(0,6),Arrows(4)); truck((1,0)); truck((2,0)); truck((3,0)); truck((4,0));
  3 KB (571 words) - 00:38, 13 March 2014
• 2000 AIME I Problems/Problem 11
  ...are [[relatively prime]] positive [[divisor]]s of <math>1000.</math> What is the [[floor function|greatest integer]] that does not exceed <math>S/10</ma ...pressed in the form of <math>2^{x}5^{y}</math>, where <math>-3 \le x,y \le 3</math>. Thus every number in the form of <math>a/b</math> will be expressed
  4 KB (667 words) - 13:58, 31 July 2020
• 2000 AIME I Problems/Problem 10
  ...th>1</math> and <math>100,</math> inclusive, the number <math>x_{k}</math> is <math>k</math> less than the sum of the other <math>99</math> numbers. Give ...=\frac{75}{49} \Longrightarrow x_{50}=\frac{75}{98}</math>, and the answer is <math>75+98=\boxed{173}</math>.
  2 KB (319 words) - 22:26, 29 December 2022
• 2000 AIME I Problems/Problem 9
  <cmath>\begin{eqnarray*} -\log x \log y + \log x + \log y - 1 &=& 3 - \log 2000\\ Small note from different author: <math>-(3 - \log 2000) = \log 2000 - 3 = \log 2000 - \log 1000 = \log 2.</math>
  4 KB (623 words) - 15:56, 8 May 2021
• 2000 AIME I Problems/Problem 8
  ...math>n,</math> and <math>p</math> are positive integers and <math>p</math> is not divisible by the cube of any prime number. Find <math>m + n + p</math>. ...rac{3}{4}\right)^{3}\right)^{1/3}}{1}</math> of the height when the vertex is at the top.
  4 KB (677 words) - 16:33, 30 December 2023
• 2000 AIME I Problems/Problem 7
  note: this is the type of problem that makes you think symmetry, but actually can be solv == Solution 3 ==
  5 KB (781 words) - 15:02, 20 April 2024
• 2000 AIME I Problems/Problem 6
  ...ath> and that the [[arithmetic mean]] of <math>x</math> and <math>y</math> is exactly <math>2</math> more than the [[geometric mean]] of <math>x</math> a ...er of pairs of <math>(\sqrt{x},\sqrt{y})</math>, and so <math>(x,y)</math> is then <math>\boxed{997}</math>.
  6 KB (966 words) - 21:48, 29 January 2024
• 2000 AIME I Problems/Problem 5
  .../math> and <math>n</math> are [[relatively prime]] positive integers. What is <math>m + n</math>? If we work with the problem for a little bit, we quickly see that there is no direct combinatorics way to calculate <math>m/n</math>. The [[Principle
  7 KB (1,011 words) - 20:09, 4 January 2024
• 2000 AIME I Problems/Problem 4
  Expressing all terms 3 to 9 in terms of <math>a_1</math> and <math>a_2</math> and substituting the ...=69</math>. These numbers are relatively prime, as desired. The perimeter is <math>2(61)+2(69)=\boxed{260}</math>.
  3 KB (485 words) - 00:31, 19 January 2024
• 2000 AIME I Problems/Problem 2
  ...th>D</math> across the y-axis. The area of [[pentagon]] <math>ABCDE</math> is <math>451</math>. Find <math>u + v</math>. ...)</math> or <math>(11,10)</math>. Since <math>v < u</math> the latter case is the answer, and <math>u+v = \boxed{021}</math>.
  3 KB (434 words) - 22:43, 16 May 2021
• 2000 AIME I Problems/Problem 1
  ...ositive integer <math>n</math> such that no matter how <math>10^{n}</math> is expressed as the product of any two positive integers, at least one of thes <math>n = 3:</math> <cmath>2^3 = 8 , 5 ^3 = 125</cmath>
  1 KB (163 words) - 17:44, 16 December 2020
• 2001 AIME I Problems/Problem 15
  ...are considered to be consecutive, are written on faces that share an edge is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively pr ...resent adjacent octahedral faces. Each assignment of the numbers <math>1,2,3,4,5,6,7</math>, and <math>8</math> to the faces of the octahedron correspon
  11 KB (1,837 words) - 18:53, 22 January 2024
• 2001 AIME I Problems/Problem 14
  ...and <math>1</math> represent a house that does receive mail. This problem is now asking for the number of <math>19</math>-digit strings of <math>0</math n&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19\\\hline
  13 KB (2,298 words) - 19:46, 9 July 2020
• 2001 AIME I Problems/Problem 13
  ...h>d < 120.</math> The length of the chord of a <math>3d</math>-degree arc is <math>- m + \sqrt {n}</math> centimeters, where <math>m</math> and <math>n< ...5}}{2},</math> which equals <math>-9 + \sqrt{165}.</math> Thus, the answer is <math>9 + 165 = \boxed{174}</math>.
  3 KB (561 words) - 19:25, 27 November 2022
• 2001 AIME I Problems/Problem 12
  ...0,0,2),</math> and <math>D = (0,0,0).</math> The [[radius]] of the sphere is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively pr ...F = (0,2/3,0), G = (0,0,2/3), L = (0,2/3,2/3), M = (2/3,0,2/3), N = (2/3,2/3,0);
  6 KB (1,050 words) - 18:44, 27 September 2023
• 2014 USAJMO Problems
  ===Problem 3=== [[2014 USAJMO Problems/Problem 3|Solution]]
  3 KB (600 words) - 16:42, 5 August 2023
• 2001 AIME I Problems/Problem 11
  ...be the number associated with <math>P_i</math> after the renumbering. It is found that <math>x_1 = y_2,</math> <math>x_2 = y_1,</math> <math>x_3 = y_4, <cmath>\begin{align}x_1&=y_2 & \Longrightarrow & & c_1 &= 5 c_2-3\\
  3 KB (493 words) - 13:51, 22 July 2020
• 2001 AIME I Problems/Problem 10
  ...[[midpoint]] of the segment they determine also belongs to <math>S</math> is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively pr ...h>, <math>(0,2)</math>, <math>(2,0)</math>, <math>(1,3)</math>, and <math>(3,1)</math>, <math>8</math> possibilities.
  8 KB (1,187 words) - 02:40, 28 November 2020
• 2001 AIME I Problems/Problem 9
  ...<math>q</math>, and <math>r</math> are positive and satisfy <math>p+q+r=2/3</math> and <math>p^2+q^2+r^2=2/5</math>. The ratio of the area of triangle We let <math>[\ldots]</math> denote area; then the desired value is
  4 KB (673 words) - 20:15, 21 February 2024
• 2001 AIME I Problems/Problem 8
  ...e because its base-<math>7</math> representation is <math>102</math>. What is the largest 7-10 double? <cmath>2(a_na_{n-1}\cdots a_0)_7 = (a_na_{n-1}\cdots a_0)_{10}</cmath> (This is because the digits in <math>N</math> ' s base 7 representation make a numbe
  3 KB (502 words) - 11:28, 9 December 2023
• 2001 AIME I Problems/Problem 7
  ...h>\overline{AC}</math>, respectively, such that <math>\overline{DE}</math> is [[parallel]] to <math>\overline{BC}</math> and contains the center of the [ .../math> and <math>\triangle ABC</math> is <math>\frac{43}{63}</math>, which is the scale factor between the two similar triangles, and thus <math>DE = \fr
  9 KB (1,540 words) - 08:31, 1 December 2022
• 2001 AIME I Problems/Problem 6
  ...rolled four times. The [[probability]] that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form <ma The red path corresponds to the sequence of rolls <math>2, 3, 5, 5</math>. This establishes a [[bijection]] between valid dice roll seq
  11 KB (1,729 words) - 20:50, 28 November 2023
• 2001 AIME I Problems/Problem 5
  ...tude is contained in the y-axis, and the square of the length of each side is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relat path e = xscale(2)*unitcircle; real x = -8/13*3^.5;
  6 KB (1,043 words) - 10:09, 15 January 2024
• 2001 AIME I Problems/Problem 4
  ...ath>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>. ...and <math>\angle TCA=75^{\circ}</math>, meaning <math>\triangle TAC</math> is an isosceles triangle and <math>AC=24</math>.
  3 KB (534 words) - 03:22, 23 January 2023
• 2001 AIME I Problems/Problem 3
  ...nx^n + a_{n-1}x^{n-1} + \cdots + a_0 = 0</math>, then the sum of the roots is <math>\frac{-a_{n-1}}{a_n}</math>. ...but <math>x^{2001}+-x^{2001}=0</math>, so the term with the largest degree is <math>x^{2000}</math>. So we need the coefficient of that term, as well as
  2 KB (335 words) - 18:38, 9 February 2023
• 2001 AIME I Problems/Problem 2
  ...h>\mathcal{S}</math>, and the mean of <math>\mathcal{S}\cup\{2001\}</math> is <math>27</math> more than the mean of <math>\mathcal{S}</math>. Find the me ...1</math> as it is from <math>2001</math>. Thus, the mean of <math>S</math> is
  1 KB (225 words) - 07:57, 4 November 2022
• 2001 AIME I Problems/Problem 1
  ...ore the case <math>b = 0</math> as we have been doing so far, then the sum is <math>495 + 120 + 15 = \boxed{630}</math>. == Solution 3 ==
  4 KB (687 words) - 18:37, 27 November 2022
• 2002 AIME I Problems/Problem 15
  ...h>p, q,</math> and <math>r</math> are positive integers and <math>r</math> is not divisible by the square of any prime, find <math>p + q + r.</math> ...h>. Since <math>ABFG</math> is an isosceles trapezoid and <math>CDE</math> is an isosceles triangle, we have symmetry about the <math>xz</math>-plane.
  7 KB (1,181 words) - 20:32, 8 January 2024
• 2002 AIME I Problems/Problem 14
  ...th> and that 2002 is the largest element of <math>\mathcal{S},</math> what is the greatest number of elements that <math>\mathcal{S}</math> can have? ...mod <math>n</math>. Since they are positive integers, the largest element is at least <math>n^2+1</math>, the <math>(n+1)</math>th positive integer cong
  2 KB (267 words) - 19:18, 21 June 2021
• 2002 AIME I Problems/Problem 13
  ...<math>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>. ...= black; pen f = fontsize(8); pair A=(0,0), B=(24,0), E=(A+B)/2, C=IP(CR(A,3*70^.5),CR(E,27)), D=(B+C)/2, F=IP(circumcircle(A,B,C),E--C+2*(E-C));
  6 KB (974 words) - 13:01, 29 September 2023
• 2002 AIME I Problems/Problem 10
  ...{AB}</math> and <math>\overline{AC}</math>, respectively, so that <math>AE=3</math> and <math>AF=10</math>. Given that <math>EB=9</math> and <math>FC=27 ...ey share a common side and angle, so the area of triangle <math>AGF</math> is <math>10/13</math> the area of triangle <math>AEF</math>.
  4 KB (643 words) - 22:44, 8 August 2023
• 2002 AIME I Problems/Problem 9
  Note that it is impossible for any of <math>h,t,u</math> to be <math>1</math>, since then e ...t be <math>3</math> as well. This configuration works, so <math>333</math> is paintable.
  4 KB (749 words) - 19:44, 25 April 2024
• 2002 AIME I Problems/Problem 8
  (1) <math>a_1,a_2,a_3\cdots</math> is a nondecreasing sequence of positive integers (3) <math>a_9=k</math>
  1 KB (205 words) - 19:54, 4 July 2013
• 2002 AIME I Problems/Problem 7
  ...[[Binomial Expansion]] is valid for exponents that are not integers. That is, for all real numbers <math>x,y</math> and <math>r</math> with <math>|x|>|y ...=x^r+rx^{r-1}y+\dfrac{r(r-1)}{2}x^{r-2}y^2+\dfrac{r(r-1)(r-2)}{3!}x^{r-3}y^3 \cdots</cmath>
  2 KB (316 words) - 19:54, 4 July 2013
• 2002 AIME I Problems/Problem 5
  ...12 combinations of two distinct vertices that form a square side only form 3 squares, and all 12 combinations of two vertices that form a square diagona ...onals, meaning we counted them 6 times. Therefore, our answer is <math>198-3(6-1)=198-15=\boxed{183}.</math>
  1 KB (220 words) - 20:50, 12 November 2022
• 2002 AIME I Problems/Problem 4
  Since <math>m</math> is an integer, <math>t+1 = 29</math>, so <math>t=28</math>. It quickly follows ...=29(28)</math> and <math>n=28</math>, which implies <math>m>n</math>. This is impossible since <math>n-m>0</math>.
  2 KB (320 words) - 07:55, 4 November 2022
• 2002 AIME I Problems/Problem 3
  ...will both be two-digit numbers and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let <math>d</math> be D <center><math>(1,2), (1,3), (2,3), (1,4), (2,4), (3,4), \dots , (8,9)</math></center>
  2 KB (246 words) - 17:02, 21 May 2023
• 2002 AIME I Problems/Problem 2
  ...orean Theorem]], we find that the shorter dimension is <math>2r\left(\sqrt{3}+1\right)</math>. ...\cdot \left[\frac{\sqrt{3}-1}{\sqrt{3}-1}\right] = \frac{1}{2}\left(7\sqrt{3} - 7\right) = \frac{1}{2}\left(\sqrt{p}-q\right)</math>. Thus we have <math
  2 KB (287 words) - 19:54, 4 July 2013
• 2002 AIME I Problems/Problem 1
  ...it arrangement that reads the same left-to-right as it does right-to-left) is <math>\dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are rela ...is <math>\frac{10 \times 10}{10^3} = \frac 1{10}</math>. Similarly, there is a <math>\frac 1{26}</math> probability of picking the three-letter palindro
  3 KB (369 words) - 23:36, 6 January 2024
• 2003 AIME I Problems/Problem 15
  pair A=(0,0),C=(7.8,0),B=IP(CR(A,3.6),CR(C,5.07)), M=(A+C)/2, Da = bisectorpoint(A,B,C), D=IP(B--B+(Da-B)*10,A As <math>BD</math> is an angle bisector of both triangles <math>BAC</math> and <math>BF'F</math>,
  8 KB (1,382 words) - 14:23, 29 December 2022
• 2003 AIME I Problems/Problem 14
  ...in that order. Find the smallest value of <math> n </math> for which this is possible. ...th>10^k m - nX</math> is an integer and <math>\frac{10^k m - nX}{n}</math> is a fraction between <math>0</math> and <math>1</math>, we can rewrite this a
  3 KB (477 words) - 14:23, 4 January 2024
• 2003 AIME I Problems/Problem 13
  ...ath>'s than <math>0</math>'s. Find the [[remainder]] when <math> N </math> is divided by <math>1000</math>. ...are in the form <math>{2i \choose i}</math>, so the sum of these elements is <math>\sum_{i=0}^{5} {2i \choose i} = 1 + 2 +6 + 20 + 70 + 252 = 351</math>
  4 KB (651 words) - 19:42, 7 October 2023
• 2003 AIME I Problems/Problem 10
  ...</math> and <math> \angle ACB = 106^\circ. </math> Point <math> M </math> is in the interior of the triangle so that <math> \angle MAC = 7^\circ </math> ...congruent (by ASA), <math>CM = CN</math>. Hence <math>\triangle CMN</math> is an [[equilateral triangle]], so <math>\angle CNM = 60^\circ</math>.
  7 KB (1,058 words) - 01:41, 6 December 2022
• 2003 AIME I Problems/Problem 9
  An [[integer]] between <math>1000</math> and <math>9999</math>, inclusive, is called ''balanced'' if the sum of its two leftmost [[digit]]s equals the su ...balanced numbers. If the common sum of the first two and last two digits is <math>n</math>, such that <math>10 \leq n \leq 18</math>, there are <math>1
  4 KB (696 words) - 11:55, 10 September 2023
• 2003 AIME I Problems/Problem 7
  ...h <math> AB = 9 </math> and <math> BC = 21. </math> Point <math> D </math> is not on <math> \overline{AC} </math> so that <math> AD = CD, </math> and <ma ...<math>y^2 - 36 = x^2 - 225 \Longrightarrow x^2 - y^2 = 189</math>. The LHS is [[difference of squares]], so <math>(x + y)(x - y) = 189</math>. As both <m
  3 KB (490 words) - 18:13, 13 February 2021
• 2003 AIME I Problems/Problem 6
  ...also vertices of a <math>1</math> by <math>1</math> by <math>1</math> cube is <math> m + \sqrt{n} + \sqrt{p}, </math> where <math> m, n, </math> and <mat ...x | vertices]] of a [[cube (geometry) | cube]], there are <math>{8 \choose 3} = 56</math> total [[triangle]]s to consider. They fall into three categor
  3 KB (477 words) - 18:35, 27 December 2021
• 2003 AIME I Problems/Problem 5
  ...th>4</math> by <math>5</math> units. Given that the [[volume]] of this set is <math>\frac{m + n\pi}{p}, </math> where <math> m, n, </math> and <math> p < import three; currentprojection = perspective(5,4,3); defaultpen(linetype("8 8")+linewidth(0.6));
  2 KB (288 words) - 19:58, 4 July 2013
• 2003 AIME I Problems/Problem 4
  ...tity, <math>\sin^2 x + \cos^2 x = 1</math> and <math>2\sin x \cos x</math> is simply <math>\frac{1}{5}</math>, via substitution. Thus, substituting these ==Solution 3==
  3 KB (516 words) - 21:59, 22 October 2020
• 2003 AIME I Problems/Problem 3
  Let the [[set]] <math> \mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}. </math> Susan makes a list as follows: for each two-element subse ...the set from greatest to least to reduce error: <math>\{34, 21, 13, 8, 5, 3, 2, 1\}.</math>
  2 KB (317 words) - 00:09, 9 January 2024
• 2003 AIME I Problems/Problem 2
  ...th>1</math> is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. T ...>, while the total area is given by <math>100^{2} \pi</math>, so the ratio is
  4 KB (523 words) - 15:49, 8 March 2021
• 2024 AMC 8 Problems/Problem 8
  ...day. How many different dollar amounts could Taye have on Thursday, <math>3</math> days later? <math>\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 5\qquad\textbf{(D) } 6\qquad\textbf
  2 KB (384 words) - 22:57, 17 February 2024
• 2003 AIME II Problems/Problem 15
  where <math>m, n,</math> and <math>p</math> are integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p.</math> The five-element sum is just <math>\sin 30^\circ + \sin 60^\circ + \sin 90^\circ + \sin 120^\circ +
  4 KB (675 words) - 17:23, 30 July 2022
• 2003 AIME II Problems/Problem 14
  .../math> where <math>m</math> and <math>n</math> are positive integers and n is not divisible by the square of any prime. Find <math>m + n.</math> ...rac{8}{\sqrt{3}}, 4\right)</math> and that <math>B = \left(\frac{10}{\sqrt{3}}, 2\right)</math>.
  9 KB (1,461 words) - 15:09, 18 August 2023
• 2003 AIME II Problems/Problem 13
  ...he probability that the bug moves to its starting vertex on its tenth move is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively pr ...t the bug is at its starting vertex after <math>n</math> moves. If the bug is on its starting vertex after <math>n</math> moves, then it must be not on i
  15 KB (2,406 words) - 23:56, 23 November 2023
• 2003 AIME II Problems/Problem 12
  ...ath>s=v_1+\cdots+v_{27}</math> be the total number of votes cast. Our goal is to determine the smallest possible <math>s</math>. ...is means "For all <math>i</math>, <math>\frac{100v_i}s + 1 \leq v_i</math> is true")
  4 KB (759 words) - 13:00, 11 December 2022
• 2003 AIME II Problems/Problem 11
  ...<math>m</math> and <math>p</math> are relatively prime, and <math>n</math> is not divisible by the square of any prime, find <math>m + n + p.</math> Hence, the answer is <math>527+11+40=\boxed{578}.</math>
  5 KB (772 words) - 19:47, 1 August 2023
• 2003 AIME II Problems/Problem 9
  ...the roots of <math>Q(x) = 0,</math> find <math>P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).</math> ...g division to divide <math>P(x)</math> by <math>Q(x)</math>, the remainder is <math>x^2-x+1</math>.
  2 KB (376 words) - 11:47, 28 April 2024
• 2003 AIME II Problems/Problem 8
  ...ath> such that <math>f(1)=1440</math>, <math>f(2)=1716</math>, and <math>f(3)=1848</math>. Plugging in the values for x gives us a system of three equat ...gives <math>a=-72, b=492,</math> and <math>c=1020</math>. Thus, the answer is <math>-72(8)^2+492\cdot8+1020= \boxed{348}.</math>
  5 KB (793 words) - 15:18, 14 July 2023
• 2003 AIME II Problems/Problem 6
  ...tively, after a <math>180^\circ</math> rotation about <math>G.</math> What is the area of the union of the two regions enclosed by the triangles <math>AB Since a <math>13-14-15</math> triangle is a <math>5-12-13</math> triangle and a <math>9-12-15</math> triangle "glued"
  5 KB (787 words) - 17:38, 30 July 2022
• 2003 AIME II Problems/Problem 4
  .... The ratio of the volume of the smaller tetrahedron to that of the larger is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively pr ...rac{1}{3}, \frac{1}{3})</math>,<math>(0,\frac{1}{3}, \frac{1}{3}, \frac{1}{3})</math>.
  3 KB (563 words) - 17:36, 30 July 2022
• 2003 AIME II Problems/Problem 3
  .../math> is never immediately followed by <math>C</math>, and <math>C</math> is never immediately followed by <math>A</math>. How many seven-letter good wo ...s is restricted. Therefore, the number of seven-letter good words is <math>3*2^6=192</math>
  2 KB (336 words) - 17:29, 30 July 2022
• 2003 AIME II Problems/Problem 2
  ...wo of whose digits are the same. What is the remainder when <math>N</math> is divided by 1000? ...<math>0,1,2</math> is also divisible by 8. The only arrangement that works is <math>120</math>.
  1,013 bytes (162 words) - 09:00, 11 July 2023
• 2003 AIME II Problems/Problem 1
  ...eger]]s is <math>6</math> times their [[sum]], and one of the [[integer]]s is the sum of the other two. Find the sum of all possible values of <math>N</m ...<math>13, 8, 7</math> so the sum of all possible values of <math>N</math> is <math>12 \cdot (13 + 8 + 7) = 12(28) = \boxed{336}</math>.
  1 KB (174 words) - 08:56, 11 July 2023
• 2002 AIME II Problems/Problem 15
  ..., <math>b</math>, and <math>c</math> are positive integers, <math>b</math> is not divisible by the square of any prime, and <math>a</math> and <math>c</m ...and <math>b</math> respectively. We know that the point <math>(9,6)</math> is a point on both circles, so we have that
  7 KB (1,182 words) - 09:56, 7 February 2022
• 2002 AIME II Problems/Problem 14
  ...th center <math>O</math> on <math>\overline{AP}</math> is drawn so that it is [[Tangent (geometry)|tangent]] to <math>\overline{AM}</math> and <math>\ove ...Using the fact that the ratio of corresponding sides in similar triangles is equal to the ratio of their perimeters, we have
  4 KB (658 words) - 19:15, 19 December 2021
• 2002 AIME II Problems/Problem 13
  ...area of triangle <math>PQR</math> to the area of triangle <math>ABC</math> is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively pr C=(1.9375,3.4994);
  6 KB (935 words) - 13:23, 3 September 2021
• 2002 AIME II Problems/Problem 12
  ...h>a_n\le.4</math> for all <math>n</math> such that <math>1\le n\le9</math> is given to be <math>p^aq^br/\left(s^c\right)</math> where <math>p</math>, <ma ...h> to <math>(6,4)</math> that always stay below the line <math>y=\frac{2x}{3}</math>. We can find the number of such paths using a Pascal's Triangle typ
  7 KB (1,127 words) - 13:34, 19 June 2022
• 2002 AIME II Problems/Problem 11
  ...math>n</math>, and <math>p</math> are positive integers and <math>m</math> is not divisible by the square of any prime. Find <math>100m+10n+p</math>. ...>. Then, the common ratio is <math>\frac{1}{8x}</math>, and the first term is <math>8x^2</math>.
  4 KB (696 words) - 16:27, 22 March 2022
• 2002 AIME II Problems/Problem 9
  ...have no common elements.) Find the remainder obtained when <math>n</math> is divided by <math>1000</math>. ...</math>, <math>i \in B</math>, or <math>i \in C</math>. So there are <math>3^{10}</math> ways to organize the elements of <math>S</math> into disjoint <
  3 KB (404 words) - 23:07, 4 May 2024
• 2002 AIME II Problems/Problem 8
  ...<math>n(n+1)<2002</math>, which is easy to solve by trial, as the solution is obviously <math>\simeq \sqrt{2002}</math>.) ...2002}{n}\right\rfloor=k</math> has no integer solutions for <math>n</math> is <math>\boxed{049}</math>.
  6 KB (908 words) - 14:22, 14 July 2023
• 2002 AIME II Problems/Problem 7
  It is known that, for all positive integers <math>k</math>, <center><math>1^2+2^2+3^2+\ldots+k^{2}=\frac{k(k+1)(2k+1)}6</math>.</center>
  3 KB (403 words) - 12:10, 9 September 2023
• 2002 AIME II Problems/Problem 6
  Find the integer that is closest to <math>1000\sum_{n=3}^{10000}\frac1{n^2-4}</math>. ...1 = (n-2)A + (n+2)B</math> or <math>1 = n(A+B)+ 2(B-A)</math>. Since there is no n term on the left hand side, <math> A+B=0</math> and by inspection <mat
  2 KB (330 words) - 05:56, 23 August 2022
• 2002 AIME II Problems/Problem 5
  ...> and <math>m</math> are non-negative integers, for which <math>a^6</math> is not a divisor of <math>6^a</math>. ...l pairs of non-negative integers (n,m) for which <math>(2^n3^m)^{6}</math> is not a divisor of <math>6^{2^n3^m}</math>
  3 KB (515 words) - 14:46, 14 February 2021
• 2002 AIME II Problems/Problem 4
  ...ath>m</math> is a positive integer. Find the remainder when <math>m</math> is divided by <math>1000</math>. ...Thus, the area of the garden enclosed by the path when <math>n=202</math> is
  2 KB (268 words) - 07:28, 13 September 2020
• 2002 AIME II Problems/Problem 3
  ...r]]s that form an increasing [[geometric sequence]] and <math>b - a</math> is the [[Perfect square|square]] of an integer. Find <math>a + b + c.</math> ...the [[geometric mean]] of the [[product]] <math>abc</math>. <math>b=\sqrt[3]{abc}=6^2=36</math>.
  2 KB (263 words) - 22:50, 5 April 2024
• 2002 AIME II Problems/Problem 2
  ...7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R=(11,3,9)</math>. What is the [[surface area]] of the cube? <math>PR=\sqrt{(11-7)^2+(3-12)^2+(9-10)^2}=\sqrt{98}</math>
  709 bytes (103 words) - 03:32, 6 December 2019
• 2002 AIME II Problems/Problem 1
  &(2)& y\text{ is the number formed by reversing the digits of }x\text{; and}\\ &(3)& z=|x-y|.
  971 bytes (150 words) - 21:56, 11 February 2020
• 2001 AIME II Problems/Problem 15
  ...math>n</math>, and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>. import three; currentprojection = orthographic(camera=(1/4,2,3/4)); defaultpen(linewidth(0.7)); pen l = linewidth(0.5) + linetype("10 2");
  4 KB (518 words) - 15:01, 31 December 2021
• 2001 AIME II Problems/Problem 14
  ...{2}i</math> and <math>\text{cis\,} {8}\theta = -\frac{1}{2}- \frac{\sqrt{3}}{2}i</math> ...}}{2}i</math> and <math>\text{cis\,} {8}\theta = -\frac{1}{2}+\frac{\sqrt{3}}{2}i</math>
  2 KB (380 words) - 15:03, 22 July 2018
• 2001 AIME II Problems/Problem 13
  ...Hence <math>\angle ADB = \angle DEC</math>, and <math>\triangle BDE</math> is [[isosceles triangle|isosceles]]. Then <math>BD = BE = 10</math>. The answer is <math>m+n = \boxed{069}</math>.
  4 KB (743 words) - 03:32, 23 January 2023
• 2001 AIME II Problems/Problem 12
  ...has the midpoint triangle as a face. The [[volume]] of <math>P_{3}</math> is <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are rela ...ath>\left(\frac 12\right)^3 = \frac 18</math>. The total volume added here is then <math>\Delta P_1 = 4 \cdot \frac 18 = \frac 12</math>.
  2 KB (380 words) - 00:28, 5 June 2020
• 2001 AIME II Problems/Problem 11
  ...ty]] that Club Truncator will finish the season with more wins than losses is <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are rela ...and losses. Thus, by the [[complement principle]], the desired probability is half the probability that Club Truncator does not have the same number of w
  3 KB (415 words) - 23:25, 20 February 2023
• 2001 AIME II Problems/Problem 10
  ...0^{j-i} - 1</math>. From the factorization <math>10^6 - 1 = (10^3 + 1)(10^{3} - 1)</math>, we see that <math>j-i = 6</math> works; also, <math>a-b | a^n ...^{0},\dots\implies 94 - 6 = 88</math>, and so forth. Therefore, the answer is <math>94 + 88 + 82 + \dots + 4\implies 16\left(\dfrac{98}{2}\right) = \boxe
  4 KB (549 words) - 23:16, 19 January 2024
• 2001 AIME II Problems/Problem 9
  ...[[probability]] of obtaining a grid that does not have a 2-by-2 red square is <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are [[re ...easy: 4 ways to choose which the side the squares will be on, and <math>2^3</math> ways to color the rest of the squares, so 32 ways to do that. For th
  8 KB (1,207 words) - 20:04, 5 September 2023
• 2001 AIME II Problems/Problem 8
  ...<math>x</math>, and that <math>f(x) = 1-|x-2|</math> for <math>1\le x \le 3</math>. Find the smallest <math>x</math> for which <math>f(x) = f(2001)</ma ...h>f(2001) = 3^kf\left(\frac{2001}{3^k}\right),\ 1 \le \frac{2001}{3^k} \le 3 \Longrightarrow k = 6</math>. Indeed,
  3 KB (545 words) - 23:41, 14 June 2023
• 2001 AIME II Problems/Problem 7
  ...h> and <math>C_{3}</math> can be written as <math>\sqrt {10n}</math>. What is <math>n</math>? ...> and <math>b</math> are the legs of the right triangle and <math>c</math> is the hypotenuse. (This formula should be used ''only for right triangles''.)
  7 KB (1,112 words) - 02:15, 26 December 2022
• 2001 AIME II Problems/Problem 3
  x_{3}&=420,\\ x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4}\ \text{when}\ n\geq5, \end{align*}
  2 KB (300 words) - 01:28, 12 November 2022
• 2001 AIME II Problems/Problem 2
  ...5</math> percent of the school population, and the number who study French is between <math>30</math> percent and <math>40</math> percent. Let <math>m</m Therefore, the answer is <math>M - m = 499 - 201 = \boxed{298}</math>.
  2 KB (252 words) - 00:54, 10 January 2024
• 2000 AIME II Problems/Problem 15
  ...ac{1 - 0}{\sin 1} = \frac{1}{\sin 1^{\circ}}</math>. Therefore, the answer is <math>\boxed{001}</math>. The average term is around the 60's which gives <math>\frac{4}{3}</math>.
  3 KB (469 words) - 21:14, 7 July 2022
• 2000 AIME II Problems/Problem 14
  ..., and <math>0<f_m</math>. Given that <math>(f_1,f_2,f_3,\ldots,f_j)</math> is the factorial base expansion of <math>16!-32!+48!-64!+\cdots+1968!-1984!+20 ...k+1)\cdot k!- k!)} = 1+\sum_{k=1}^{n-1} {((k+1)!- k!)} = 1 + ((2! - 1!) + (3! - 2!) + \cdots + (n! - (n-1)!)) = n!</math>.
  7 KB (1,131 words) - 14:49, 6 April 2023
• 2000 AIME II Problems/Problem 13
  ...2000x^6+100x^5+10x^3+x-2=0</math> has exactly two real roots, one of which is <math>\frac{m+\sqrt{n}}r</math>, where <math>m</math>, <math>n</math> and < 2000x^6+100x^5+10x^3+x-2&=0\\
  6 KB (1,060 words) - 17:36, 26 April 2024
• 2000 AIME II Problems/Problem 11
  ...the absolute values of all possible slopes for <math>\overline{AB}</math> is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively pr ...ction of <math>D</math> across that perpendicular. Then <math>ABCD'</math> is a [[parallelogram]], and <math>\overrightarrow{AB} = \overrightarrow{D'C}</
  4 KB (750 words) - 22:55, 5 February 2024
• 2000 AIME II Problems/Problem 10
  A [[circle]] is [[inscribe]]d in [[quadrilateral]] <math>ABCD</math>, [[tangent]] to <math> == Solution 3 (Smart algebra to make 2 less annoying) ==
  2 KB (399 words) - 17:37, 2 January 2024
• 2000 AIME II Problems/Problem 9
  ...ch that <math>z+\frac 1z=2\cos 3^\circ</math>, find the least integer that is greater than <math>z^{2000}+\frac 1{z^{2000}}</math>. ...ac{2\cos 3 \pm \sqrt{4\cos^2 3 - 4}}{2} = \cos 3 \pm i\sin 3 = \text{cis}\,3^{\circ}</math>.
  4 KB (675 words) - 13:42, 4 April 2024
• 2000 AIME II Problems/Problem 8
  In [[trapezoid]] <math>ABCD</math>, leg <math>\overline{BC}</math> is [[perpendicular]] to bases <math>\overline{AB}</math> and <math>\overline{C ...<math>\overline{CD}</math>. Then <math>AE = x</math>, and <math>ADE</math> is a [[right triangle]]. By the [[Pythagorean Theorem]],
  4 KB (584 words) - 19:35, 7 December 2019
• 2000 AIME II Problems/Problem 7
  ...N{1!18!}</math></center> find the [[floor function|greatest integer]] that is less than <math>\frac N{100}</math>. <cmath>\frac {19!}{2!17!}+\frac {19!}{3!16!}+\frac {19!}{4!15!}+\frac {19!}{5!14!}+\frac {19!}{6!13!}+\frac {19!}{7
  2 KB (281 words) - 12:09, 5 April 2024
• 2000 AIME II Problems/Problem 6
  .../math> be the length of the segment joining the legs of the trapezoid that is [[parallel]] to the bases and that divides the trapezoid into two regions o ...length of the midline of the trapezoid is the average of its bases, which is <math>\frac{b+b+100}{2} = b+50</math>. The two regions which the midline di
  3 KB (433 words) - 19:42, 20 December 2021
• 2000 AIME II Problems/Problem 4
  What is the smallest positive integer with six positive odd integer divisors and tw ...math>18 = 2 \cdot 3 \cdot 3</math> factors, then it can have at most <math>3</math> distinct primes in its factorization.
  2 KB (397 words) - 15:55, 11 May 2022
• 2000 AIME II Problems/Problem 2
  A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola <math> {{AIME box|year=2000|n=II|num-b=1|num-a=3}}
  804 bytes (126 words) - 20:30, 4 July 2013
• 2000 AIME II Problems/Problem 1
  <center><math>\frac 2{\log_4{2000^6}} + \frac 3{\log_5{2000^6}}</math></center> <math>\frac 2{\log_4{2000^6}} + \frac 3{\log_5{2000^6}}</math>
  2 KB (292 words) - 13:33, 4 April 2024
• 2005 AMC 10B Problems/Problem 3
  ...the remaining paint is used. What fraction of the original amount of paint is available to use on the third day? ...frac{1}{10} \qquad \textbf{(B) } \frac{1}{9} \qquad \textbf{(C) } \frac{1}{3} \qquad \textbf{(D) } \frac{4}{9} \qquad \textbf{(E) } \frac{5}{9} </math>
  1 KB (163 words) - 14:00, 14 December 2021
• 2005 AMC 10B Problems/Problem 7
  ...ed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smallest circle to the area of the largest squ ...rac{\pi}{16} \qquad \textbf{(B) } \frac{\pi}{8} \qquad \textbf{(C) } \frac{3\pi}{16} \qquad \textbf{(D) } \frac{\pi}{4} \qquad \textbf{(E) } \frac{\pi}{
  2 KB (381 words) - 14:28, 14 December 2021
• 2005 AMC 10B Problems/Problem 12
  .... What is the probability that the product of the numbers on the top faces is prime? ...e which die will have the prime number, so the probability is <math>\dfrac{3}{6}\times\left(\dfrac{1}{6}\right)^{11}\times\dbinom{12}{1} = \dfrac{1}{2}\
  3 KB (385 words) - 14:03, 16 June 2022
• 2005 AMC 10B Problems/Problem 13
  ...etween <math>1</math> and <math>2005</math> are integer multiples of <math>3</math> or <math>4</math> but not <math>12</math>? ...12</math> are <math>\frac{2005}{12} = 167\text{ }R1.</math> So, the answer is <math>668+501-167-167 = \boxed{\textbf{(C) } 835}</math>
  1 KB (212 words) - 14:44, 15 December 2021
• 2005 AMC 10B Problems/Problem 14
  ..., and <math> C</math> is the midpoint of <math> \overline{BD}</math>. What is the area of <math> \triangle CDM</math>? ...}}{2}\qquad \textbf{(B) }\ \frac {3}{4}\qquad \textbf{(C) }\ \frac {\sqrt {3}}{2}\qquad \textbf{(D) }\ 1\qquad \textbf{(E) }\ \sqrt {2}</math>
  5 KB (882 words) - 22:12, 30 April 2024
• 2005 AMC 10B Problems/Problem 17
  ...<math>5^b = 6</math>, <math>6^c = 7</math>, and <math>7^d = 8</math>. What is <math>a \cdot b\cdot c \cdot d</math>? ...quad \textbf{(C) } 2 \qquad \textbf{(D) } \frac{5}{2} \qquad \textbf{(E) } 3 </math>
  2 KB (324 words) - 15:30, 16 December 2021
• 2005 AMC 10B Problems/Problem 18
  ..., and <math>g</math> are distinct digits and in increasing order, and none is either <math>0</math> or <math>1</math>. How many different telephone numbe ...r of ways to choose <math>7</math> numbers from <math>8</math>. The answer is then <math>\dbinom{8}{7}=\dfrac{8!}{7!\,(8-7)!}=\boxed{\textbf{(D) } 8}</ma
  1 KB (195 words) - 15:33, 16 December 2021
• 2005 AMC 10B Problems/Problem 20
  ...) of all 5-digit numbers that can be formed by using each of the digits 1, 3, 5, 7, and 8 exactly once? ...math> times. Therefore, the sum of all such numbers is <math> 24 \times (1+3+5+7+8) \times (11111) = 24 \times 24 \times 11111 = 6399936. </math> Since
  2 KB (356 words) - 19:15, 17 September 2023
• 2005 AMC 10B Problems/Problem 21
  ...<math>a</math> and the other two bear a number <math>b \neq a</math>. What is the value of <math>q/p</math>? ...ath> ways to pick the slips, so <math>q = \frac{45 \cdot 6 \cdot 4^2 \cdot 3^2}{40 \cdot 39 \cdot 38 \cdot 37}</math>.
  3 KB (398 words) - 19:17, 17 September 2023
• 2005 AMC 10B Problems/Problem 23
  ...The area of <math>ABEF</math> is twice the area of <math>FECD</math>. What is <math>AB/DC</math>? <math>\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 8 </math
  3 KB (489 words) - 19:22, 17 September 2023
• 2005 AMC 10B Problems/Problem 24
  ...x</math> and <math>y</math> be two-digit integers such that <math>y</math> is obtained by reversing the digits What is <math>x + y + m</math>?
  5 KB (845 words) - 19:23, 17 September 2023
• 2005 AMC 10B Problems/Problem 25
  ...perty that no two elements of <math>B</math> sum to <math>125</math>. What is the maximum possible number of elements in <math>B</math>? ...nd at most one number from each pair can be included in the set. The total is <math>24 + 38 = \boxed{\textbf{(C)}\ 62}</math>.
  3 KB (517 words) - 19:15, 15 October 2023
• 2006 USAMO Problems/Problem 2
  ...s of <math>k </math>, the minimum value of <math>N </math> for which there is a set of <math>2k+1 </math> distinct positive integers that has sum greater & = 2k^3 + 3k^2 + 3k
  2 KB (398 words) - 09:48, 5 August 2014
• 2006 USAMO Problems/Problem 4
  ...ve a sum and product of <math>n</math>. For <math>p_1+p_2=n</math>, which is only possible in one case, <math>n=4</math>, we consider <math>p_1=p_2=2</m ...we need to check for <math>n=1,2,3,5,7</math>. One is included because it is neither prime nor composite.
  3 KB (486 words) - 22:43, 5 August 2014
• 2006 USAMO Problems/Problem 6
  Thus, <math>XAE\sim XBF</math> by AA similarity, and <math>X</math> is the center of spiral similarity for <math>A,E,B,</math> and <math>F</math>. Thus, <math>YED\sim YFC</math> by AA similarity, and <math>Y</math> is the center of spiral similarity for <math>E,D,F,</math> and <math>C</math>.
  5 KB (986 words) - 22:46, 18 May 2015
• 2006 AMC 12A Problems/Problem 2
  Define <math>x\otimes y=x^3-y</math>. What is <math>h\otimes (h\otimes h)</math>? ...bf{(B)}\ 0\qquad\textbf{(C)}\ h\qquad\textbf{(D)}\ 2h\qquad\textbf{(E)}\ h^3</math>
  694 bytes (110 words) - 14:03, 27 July 2023
• 2006 AMC 12A Problems/Problem 3
  ...lems|2006 AMC 12A #3]] and [[2006 AMC 10A Problems/Problem 3|2006 AMC 10A #3]]}} ...ice's age is <math>3:5</math>. Alice is <math>30</math> years old. How old is Mary?
  1 KB (155 words) - 17:30, 16 December 2021
• 2006 AMC 12A Problems/Problem 6
  ...e two hexagons can be repositioned without overlap to form a square. What is <math>y</math>? label("$y$",(3,4),S);
  3 KB (528 words) - 18:14, 16 December 2021
• 2006 AMC 12A Problems/Problem 8
  <math> \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5</math> ...e must be either <math>1, 3, </math> or <math>5</math>, and <math>1</math> is clearly not possible. The other two possibilities both work:
  3 KB (450 words) - 02:00, 13 January 2024
• 2006 AMC 12A Problems/Problem 9
  ...more than an eraser, and both items cost a [[whole number]] of cents. What is the total cost, in cents, of one pencil and one eraser? ...1 \pmod 3</math> so <math>p</math> leaves a remainder of 1 on division by 3.
  3 KB (429 words) - 18:14, 26 September 2020
• Diagonal
  A '''diagonal''' of a [[polygon]] is any segment joining two [[vertex|vertices]] other than an [[edge]]. ...als of a polygon with <math>n</math> vertices is given by <math>\frac{n(n-3)}{2}</math>.
  2 KB (374 words) - 00:37, 25 January 2015
• Equation
  An '''equation''' is a [[relation]] which states that two [[expression]]s are equal, identical, A unique aspect to equations is the ability to modify an original equation by performing operations (such a
  5 KB (932 words) - 12:57, 26 July 2023
• Harvard-MIT Mathematics Tournament
  ...998. The contest takes place alternatively at Harvard or MIT each year. It is composed of two tournaments, the ''February Tournament'' and the ''November The February Tournament is the more difficult of the two tournaments, with its problems ranging from m
  4 KB (539 words) - 16:58, 19 February 2023
• Radius
  ...ons hold for the [[sphere]] and [[hypersphere]]. The plural form of radius is radii. The radius of a circle is often denoted using R or r.
  929 bytes (156 words) - 22:49, 5 January 2023
• Intersection
  ...math>\{1, 2, 3\}</math> and <math>\{1, 3, 5\}</math> is the set <math>\{1, 3\}</math>. ....e. <math>\bigcap_{i = 1}^n A_i = A_1 \cap A_2 \cap \ldots \cap A_n</math> is the intersection of the <math>n</math> sets <math>A_1, A_2, \ldots, A_n</ma
  1 KB (221 words) - 22:41, 11 April 2019
• Word problem
  A '''word problem''' is a problem posed in plain language, as opposed to a problem with only "math" Because word problems are based on a contextualized scenario, the context is important in interpreting the results. For instance, when determining the
  1 KB (200 words) - 15:05, 10 April 2020
• Multinomial Theorem
  where <math>\binom{n}{j_1; j_2; \ldots ; j_k}</math> is the [[multinomial coefficient]] <math>\binom{n}{j_1; j_2; \ldots ; j_k}=\df Note that this is a direct generalization of the [[Binomial Theorem]], when <math>k = 2</math
  3 KB (476 words) - 19:37, 4 January 2023
• Quadratic reciprocity
  <cmath> \genfrac{(}{)}{}{}{a}{p} =\begin{cases} 1 & \text{if } a \text{ is a quadratic residue modulo } p, \\ We say that <math>a</math> is a '''quadratic residue''' modulo <math>p</math> if there exists an integer
  7 KB (1,182 words) - 16:46, 28 April 2016
• 2006 AMC 10B Problems
  What is <math> (-1)^{1} + (-1)^{2} + ... + (-1)^{2006} </math> ? ...define <math> x \mathop{\spadesuit} y = (x+y)(x-y) </math>. What is <math> 3 \mathop{\spadesuit} (4 \mathop{\spadesuit} 5) </math>?
  14 KB (2,059 words) - 01:17, 30 January 2024
• 2005 AMC 10B Problems
  ...e property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>? == Problem 3 ==
  12 KB (1,874 words) - 21:20, 23 December 2020
• Noncommutative
  More formally, if <math>\star</math> is some [[binary operation]] on a [[set]], and <math>x</math> and <math>y</mat ...commutative]]. For example, <math>4\cdot3=3\cdot4=12</math>, and <math>2+3=3+2=5</math>.
  2 KB (257 words) - 15:30, 26 December 2017
• Platonic solid
  A '''Platonic solid''' is a [[polyhedron]], or 3 dimensional figure, in which all [[face]]s are [[congruent (geometry)|congr It is easy to verify that all five Platonic solids satisfy [[Euler's polyhedral f
  8 KB (1,168 words) - 22:48, 19 February 2022
• Law of Tangents
  ...'''Law of Tangents''' is a rather obscure [[trigonometric identity]] that is sometimes used in place of its better-known counterparts, the [[law of sine ...n a triangle, any ratio of linear combinations applied to lengths of sides is equal to the ratio of the same linear combinations applied to the sines of
  2 KB (306 words) - 16:11, 21 February 2023
• 2000 AMC 12 Problems/Problem 3
  ...0 AMC 12 Problems|2000 AMC 12 #3]] and [[2000 AMC 10 Problems|2000 AMC 10 #3]]}} ...h>. If she ate <math>20\%</math> of the jellybeans, then <math>80\%</math> is remaining. Hence, after day 1, there are:
  1 KB (159 words) - 01:34, 7 April 2023
• 2000 AMC 12 Problems/Problem 8
  Figures <math>0</math>, <math>1</math>, <math>2</math>, and <math>3</math> consist of <math>1</math>, <math>5</math>, <math>13</math>, and <mat draw((9,0)--(10,0)--(10,3)--(9,3)--cycle);
  7 KB (988 words) - 15:14, 10 April 2024
• Angle bisector theorem
  ...[[triangle]] <math>\triangle ABC</math> and [[angle bisector]] AD, where D is on side BC, then <math> \frac cm = \frac bn </math>. It follows that <math First, because <math>\overline{AD}</math> is an angle bisector, we know that <math>m\angle BAD = m\angle CAD</math> and
  3 KB (438 words) - 14:20, 4 March 2023
• 2000 AMC 12 Problems/Problem 4
  ...rwards is the sum of its two predecessors. Which one of the ten [[digit]]s is the last to appear in the units position of a number in the Fibonacci seque <math>1,1,2,3,5,8,3,1,4,5,9,4,3,7,0,7,7,4,1,5,6,....</math>
  1 KB (144 words) - 09:48, 8 November 2021
• Sigma notation
  ...ctly. Besides being easier to write than the explicit sum, sigma notation is also useful in that it shows the general form of each addend. ...er limit of summation is written ''under'' the sigma and the ''upper'' one is written above the sigma.
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• University of South Carolina High School Math Contest/1993 Exam/Problem 3
  If 3 circles of radius 1 are mutually tangent as shown, what is the area of the gap they enclose? <center>[[Image:Usc93.3.PNG]]</center>
  1 KB (173 words) - 17:09, 4 October 2016
• University of South Carolina High School Math Contest/1993 Exam/Problem 4
  If <math>(1 + i)^{100}</math> is expanded and written in the form <math>a + bi</math> where <math>a</math> a * [[University of South Carolina High School Math Contest/1993 Exam/Problem 3|Previous Problem]]
  872 bytes (114 words) - 13:36, 24 November 2016
• University of South Carolina High School Math Contest/1993 Exam/Problem 11
  ...e placed randomly into 4 boxes also labeled 1 to 4, one card per box. What is the probability that no card gets placed into a box having the same label a <center><math> \mathrm{(A) \ } 1/3 \qquad \mathrm{(B) \ }3/8 \qquad \mathrm{(C) \ }5/12 \qquad \mathrm{(D) \ } 1/2 \qquad \mathrm{(E)
  2 KB (334 words) - 16:27, 25 October 2023
• University of South Carolina High School Math Contest/1993 Exam/Problem 29
  If the sides of a triangle have lengths 2, 3, and 4, what is the radius of the circle circumscribing the triangle? ...>R = \frac{2\cdot3\cdot4}{4 (\frac{3}{4}\sqrt{15})} = \frac{6\cdot\frac{4}{3}}{\sqrt{15}} = \frac8{\sqrt{15}} \Longrightarrow \mathrm{(B)}</math>.
  2 KB (219 words) - 09:57, 31 August 2012
• University of South Carolina High School Math Contest/1993 Exam/Problem 20
  ...th> denote the product of all the elements in <math>A_i</math>. Then what is the value of <math>\pi(A_1)+\pi(A_2)+\cdots+\pi(A_{63})</math>? ...(1+5)(1+6)-1</math> (The <math>-1</math> since we have one less set). This is <math>7!-1=5039</math>.
  2 KB (263 words) - 18:13, 19 October 2021
• Canadian Mathematical Olympiad
  ...nadian Mathematical Olympiad Qualifying Repechage]] (CMOQR). The Olympiad is part of the selection process for the Canadian [[IMO]] team. ...5 - 8|breakdown=<u>Problem 1</u>: 5.5<br><u>Problem 2</u>: 6<br><u>Problem 3</u>: 6.5<br><u>Problem 4</u>: 7-7.5<br><u>Problem 5</u>: 7.5-8}}
  1 KB (177 words) - 00:02, 29 August 2018
• 1977 Canadian MO Problems/Problem 1
  ...rfect square when <math>b</math> is an integer. Hence, when <math>b</math> is a positive integer, <math>a</math> cannot be. ...Thus, <math>b^{2} + b + 1</math> is not a perfect square, and thus there is no <math>b</math> that satisfies <math>4f(a) = f(b)</math>.
  1 KB (245 words) - 10:04, 28 September 2023
• 1977 Canadian MO Problems/Problem 3
  ...Find the smallest positive integer <math>b</math> for which <math>N</math> is the fourth power of an integer. ...integer <math>a.</math> Because <math>7\mid a^4</math> and <math>7</math> is prime, <math>a^4 \ge 7^4.</math> Since we want to minimize <math>b,</math>
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• University of South Carolina High School Math Contest/1993 Exam/Problems
  ...s doubled and the length is increased by 3, then the area is tripled. What is the length of the rectangle? <cmath> \mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D) \ } 6 \qquad \mathrm{(E) \ } 9 </cmath>
  14 KB (2,102 words) - 22:03, 26 October 2018
• Mock AIME 1 2006-2007 Problems/Problem 2
  ...ath>\star (n)=12</math> and <math>0\le n< 10^{7}</math>. If <math>m</math> is the number of elements in <math>\mathcal{S}</math>, compute <math>\star(m)< ...h>m = 18564 - 7 - 42 - 42 - 105 = 18368</math> so <math>\star(m) = 1 + 8 + 3 + 6 + 8 = 026</math>.
  1 KB (188 words) - 15:53, 3 April 2012
• Mock AIME 1 2006-2007 Problems/Problem 3
  ...math>n</math> is an integer, find the remainder when <math>n^{2007}</math> is divided by <math>1000</math>. ...so <math>3^{2000}\equiv 1 \pmod{1000}</math> and so <math>3^{2007} \equiv 3^7 \equiv 2187 \equiv 187 \pmod{1000}</math>
  963 bytes (135 words) - 15:53, 3 April 2012
• Mock AIME 1 2006-2007 Problems/Problem 4
  ...respectively. If the <math>x</math>-coordinate of the triangle's centroid is <math>1</math>, find the area of <math>\triangle ABC</math>. ...^2)</math>. Then we have by the centroid condition that <math>a + b + c = 3</math>. From the first [[slope]] condition we have <math>10 = \frac{b^2 -
  1 KB (244 words) - 15:21, 5 November 2012
• Mock AIME 1 2006-2007 Problems/Problem 6
  ...as a [[rational number | rational]] [[slope]]. If <math>\mathcal{L}</math> is written in the form <math>ax+by=c</math> for [[positive integer]]s <math>a, ...: 5</math> and since the [[greatest common divisor]] of the three numbers is 1, <math>a = 1, b = 5, c = 5</math> and <math>a + b + c = 011</math>.
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• Mock AIME 1 2006-2007 Problems/Problem 7
  ...</math> have <math>AC=6</math> and <math>BC=3</math>. Point <math>E</math> is such that <math>CE=1</math> and <math>AE=5</math>. Construct point <math>F< ...\cdot 6 = CF\cdot 3</math>. <math>CF</math> is then 2, and <math>BF</math> is 1. We can now use Menelaus on line <math>DF</math> with respect to triangle
  3 KB (518 words) - 16:54, 25 November 2015
• Mock AIME 1 2006-2007 Problems/Problem 9
  ....</math>. If the sum of all possible [[distinct]] values of <math>S</math> is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are [[rela ...11}+a_{12}+...</math>. If the sum of all distinct values of <math>S</math> is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relati
  5 KB (744 words) - 19:46, 20 October 2020
• Mock AIME 1 2006-2007 Problems/Problem 12
  ...3}</math>, <math>10^6 \equiv 1 \pmod{13}</math> and the order of 10 mod 13 is 6. Thus, we get one value of <math>m</math> each time <math>n = 6j + 1</ma
  2 KB (249 words) - 18:14, 3 April 2012
• Mock AIME 1 2006-2007 Problems/Problem 13
  ...<math>n</math>. If <math>d_{1}=1</math>, <math>d_{2}=2</math>, <math>d_{3}=3</math>, <math>d_{4}=-7</math>, <math>d_{5}=13</math>, and <math>d_{6}=-16</ ...these variables, which does uniquely determine these variables - but there is no obvious way of computing them. We will show a different solution.
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• Mock AIME 1 2006-2007 Problems/Problem 15
  ...nd <math>n</math> are relatively prime positive integers. Compute the last 3 digits of <math>m+n</math> ...||||||</math> is the actual number of arrangements in which <math>A</math> is next to <math>1</math>. There are <math>\frac{12!}{10!}</math> <math>-</mat
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• 1977 Canadian MO Problems/Problem 2
  ...h> on the circumference of the circle such that the angle <math>OPA</math> is a maximum. ...ent that <math>O</math> is the midpoint of <math>AC</math>, <math>X</math> is the midpoint of <math>AB</math>, and hence <math>OX=\dfrac{BC}{2}</math>.
  2 KB (365 words) - 23:28, 21 September 2014
• Mock AIME 1 2006-2007/Problems
  ...ath>\star (n)=12</math> and <math>0\le n< 10^{7}</math>. If <math>m</math> is the number of elements in <math>\mathcal{S}</math>, compute <math>\star(m)< ==Problem 3==
  8 KB (1,355 words) - 14:54, 21 August 2020
• Mock AIME 2 2006-2007 Problems/Problem 2
  ...any [[element]]s <math>n</math> in <math>S</math> is <math>f(n) = \frac{2n^3+n^2-n-2}{n^2-1}</math> an integer? <math>f(n) = \frac{2n^3+n^2-n-2}{n^2-1} = \frac{(n - 1)(2n^2 + 3n + 2)}{(n - 1)(n + 1)} = \frac{2n^
  531 bytes (90 words) - 10:49, 4 April 2012
• Mock AIME 2 2006-2007 Problems/Problem 3
  .../math> is a [[perfect square]]. Find the [[remainder]] when <math>S</math> is divided by <math>1000.</math> ...>, which can be split into two [[factor]]s in 3 ways, <math>2043 \cdot 1 = 3 \cdot 681 = 227 \cdot 9</math>. This gives us three pairs of [[equation]]s
  1 KB (198 words) - 10:50, 4 April 2012
• Mock AIME 2 2006-2007 Problems/Problem 4
  ...ger]] such that <math>(a + bi)^n = (a - bi)^n</math>, where <math>n</math> is as small as possible and <math>i = \sqrt{-1}</math>. Compute <math>\frac{b ...r <math>b = 0</math> or <math>3a^2 = b^2</math>. Thus <math>n = b^2/a^2 = 3</math>.
  1 KB (240 words) - 10:50, 4 April 2012
• Mock AIME 2 2006-2007 Problems/Problem 5
  Given that <math> iz^2=1+\frac 2z + \frac{3}{z^2}+\frac{4}{z ^3}+\frac{5}{z^4}+\cdots</math> and <math>z=n\pm \sqrt{-i},</math> find <math> <center><math>iz^3 = z + 2 + \frac{3}{z} + \frac{4}{z^2} + \cdots</math>,</center>
  912 bytes (145 words) - 10:51, 4 April 2012
• Mock AIME 2 2006-2007 Problems/Problem 8
  .../math> and <math>x_{n+3} = x_{n+2}(x_{n+1}+x_n)</math> for <math>n = 1, 2, 3, 4</math>. Find the last three [[digit]]s of <math>x_7</math>. This solution is rather long and unpleasant, so a nicer solution may exist:
  3 KB (470 words) - 00:33, 10 August 2019
• Mock AIME 2 2006-2007 Problems/Problem 7
  ...If the length of this path is <math>m\sqrt{n},</math> where <math>n</math> is squarefree, find <math>m+n.</math> ...ius <math>51</math>, we must have that <math>\angle AOB=\frac{360^{\circ}}{3}=120^{\circ}</math>. We know that <math>AO=OB=51</math>, so we can use the
  1 KB (231 words) - 18:10, 10 July 2014
• Mock AIME 2 2006-2007 Problems/Problem 9
  ...ly such that <math>\frac{BX}{CX}=\frac23</math>, <math>\frac{AY}{CY}=\sqrt 3,</math> and <math>CY=CX-BX</math>. If <math>\tan \angle APB= -\frac{a+b\sqr pair A=(0,6),B=(5,0),C=origin,X=(3,0),Y=A/(sqrt(3)+1);
  2 KB (358 words) - 23:22, 3 May 2014
• Mock AIME 2 2006-2007 Problems/Problem 12
  ...math>BC=6</math>, <math>BO=1,</math> and the [[area]] of <math>ABCD</math> is <math>\frac{a\sqrt{b}}{c},</math> where <math>a,b,c</math> are [[relatively <math>m\angle DAC=m\angle DBC \Rightarrow ABCD</math> is a cylic quadrilateral.
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• Mock AIME 2 2006-2007 Problems/Problem 15
  ...ubes are colored red. An arrangement of the cubes is "intriguing" if there is exactly <math>1</math> red unit cube in every <math>1\times1\times4</math> ...3 squares to paint red as we can't select a red square from a square that is on the same row as the first column.
  4 KB (739 words) - 17:04, 24 November 2023
• Mock AIME 2 2006-2007 Problems
  A positive integer is called a ''dragon'' if it can be written as the sum of four positive intege ...ow many elements <math>n</math> in <math>S</math> is <math>f(n) = \frac{2n^3+n^2-n-2}{n^2-1}</math> an integer?
  5 KB (848 words) - 23:49, 25 February 2017
• Expected value
  ...erage outcome if the event were to be repeated many times. Note that this is ''not'' the same as the "most likely outcome." ...h> where the sum is over all outcomes <math>z</math> and <math>P(z)</math> is the probability of that particular outcome. If the event <math>Z</math> ha
  4 KB (719 words) - 19:41, 25 November 2020
• Base numbers/Conversion
  The next natural question is: how do we convert a number from another base into base 10? For example, w <center><math>4201_5 = (4\cdot 5^3 + 2\cdot 5^2 + 0\cdot 5^1 + 1\cdot 5^0)_{10}</math></center>
  7 KB (1,177 words) - 15:56, 18 April 2020
• 2005 AMC 10A Problems/Problem 9
  ...marked <math>O</math>. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads <math>XOXOX</math>? ...} \frac{1}{6}\qquad \textbf{(D) } \frac{1}{4}\qquad \textbf{(E) } \frac{1}{3} </math>
  764 bytes (112 words) - 12:01, 13 December 2021
• Arithmetico-geometric series
  An arithmetico-geometric series is the sum of consecutive terms in an arithmetico-geometric sequence defined a ...Or, <math>\frac{a_ng_{n+1}-x_1-drS_g}{r-1}</math>, where <math>S_g</math> is the sum of the first <math>n</math> terms of <math>g_n</math>.
  2 KB (477 words) - 19:39, 17 August 2020
• IMO Problems and Solutions
  * [[1959 IMO Problems/Problem 3 | Problem 3]] proposed by Hungary * [[1960 IMO Problems/Problem 3 | Problem 3]] proposed by Gheorghe D. Simionescu, Romania
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• 1959 IMO Problems
  Prove that <math>\frac{21n+4}{14n+3}</math> is irreducible for every natural number <math>n</math>. For what real values of <math>x</math> is
  3 KB (480 words) - 11:57, 17 September 2012
• 1959 IMO Problems/Problem 1
  Prove that the fraction <math>\frac{21n+4}{14n+3}</math> is irreducible for every natural number <math>n</math>. <cmath>(21n+4, 14n+3) = (7n+1, 14n+3) = (7n+1, 1) = 1</cmath>
  5 KB (767 words) - 10:59, 23 July 2023
• 1977 Canadian MO Problems
  ...h> on the circumference of the circle such that the angle <math>OPA</math> is a maximum. == Problem 3 ==
  3 KB (560 words) - 19:23, 10 March 2015
• 1971 Canadian MO Problems/Problem 1
  <math>DEB</math> is a chord of a circle such that <math>DE=3</math> and <math>EB=5 .</math> Let <math>O</math> be the center of the circ
  680 bytes (114 words) - 21:38, 9 July 2019
• 1959 IMO Problems/Problem 4
  ...ven hypotenuse <math>c</math> such that the median drawn to the hypotenuse is the [[geometric mean]] of the two legs of the triangle. ...o a segment from any point on the circle to the midpoint of the hypotenuse is a radius.)
  6 KB (939 words) - 17:31, 15 July 2023
• 1971 Canadian MO Problems/Problem 4
  It is easy to see that this value works for the second polynomial as well. ...)(x-t) </math> and <math> x^2+x+a = (x-s)(x-u)</math> where <math>s</math> is the common root. From Vieta's Formulas, we have: <math>-(s+t) = a,...
  2 KB (346 words) - 23:20, 18 January 2023
• 1970 IMO Problems/Problem 4
  ...gers <math>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 ...4, 5, 6 \}</math>, but that does not work because only one of the numbers is a multiple of 5. So there are no such sets.
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• 2006 Romanian NMO Problems
  ...math> and <math>NP</math> are perpendicular if and only if <math>PN</math> is the interior angle bisector of <math>\angle MPC</math>. ...uch that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or column).
  11 KB (1,779 words) - 14:57, 7 May 2012
• 2006 Romanian NMO Problems/Grade 9/Problem 2
  ...ath>\angle BDE = \angle ADP = \angle CDF</math>. Prove that <math>P</math> is the midpoint of <math>EF</math> and <math>DP \perp EF</math>. .../math>. Since P and P' are on the same ray (<math>DP</math>), P = P' and P is the midpoint of <math>EF</math>.
  3 KB (488 words) - 14:05, 15 December 2022
• 2006 Romanian NMO Problems/Grade 9/Problem 3
  ...[inscribe]]d in a [[circle]] of [[radius]] <math>r</math>, for which there is a [[point]] <math>P</math> on <math>CD</math> such that <math>CB=BP=PA=AB</ ...elow; <math>P=(0,0)</math>, <math>A=(-1,\sqrt{3})</math>, <math>B=(1,\sqrt{3})</math>, <math>C=(2,0)</math>, and <math>D=(-2,0)</math>.
  6 KB (1,080 words) - 19:28, 21 September 2014
• 1969 Canadian MO Problems/Problem 2
  ...numbers <math>\sqrt{c+1}-\sqrt{c}</math>, <math>\sqrt{c}-\sqrt{c-1}</math> is greater for any <math>c\ge 1</math>. Thus <math>\boxed{\sqrt{c}-\sqrt{c-1}}</math> is greater.
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• 1969 Canadian MO Problems/Problem 4
  ...<math>P</math> is chosen, <math>\frac{PD+PE+PF}{AB+BC+CA}=\frac{1}{2\sqrt{3}}</math>. ...<math>P</math>) Because the sum of the sides is <math>3s</math>, the ratio is always <math>\cfrac{s\frac{\sqrt3}{2}}{3s}=\frac{1}{2\sqrt3}.</math>
  1 KB (215 words) - 00:23, 9 October 2020
• 2005 Canadian MO Problems
  ...r row to a higher row) or revisits a triangle. An example of one such path is illustrated below for <math>n=5</math>. Determine the value of <math>f(2005 ==Problem 3==
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• 1969 Canadian MO Problems/Problem 6
  Find the sum of <math>1\cdot 1!+2\cdot 2!+3\cdot 3!+\cdots+(n-1)(n-1)!+n\cdot n!</math>, where <math> n!=n(n-1)(n-2)\cdots2\cd If <math> n</math> is [[odd integer | odd]], <math> (n+1)!-(n-1)!+(n-1)!-(n-3)!\cdots -2!+2!-0!.</math>
  2 KB (275 words) - 20:33, 27 November 2023
• 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. ...o <math>B</math> than any other point in <math>S</math> and <math>c</math> is closer to <math>C</math> than any other point in <math>S</math>.
  2 KB (460 words) - 13:35, 9 June 2011
• 2005 Canadian MO Problems/Problem 5
  ...ath> is divisible by <math>a+b+c</math>. For example, <math>(1,2,2)</math> is 5-powerful. <math>S_{k+3}-(a+b+c)S_{k+2}+(ab+bc+ca)S_{k+1}-(abc)S_k=0</math>.
  984 bytes (177 words) - 18:21, 28 November 2023
• Addition
  ...s called [[sum|a sum]]. For example, the sum of 3 and 2 is 5 because <math>3+2=5</math>. ...a+2), f(a+3), \cdots, f(b)</math>, where <math>f</math> is a [[function]], is denoted <math>\sum_{i=a}^bf(i)</math>. (See also [[Sigma notation]])
  2 KB (309 words) - 20:34, 4 July 2019
• Midpoint
  In [[Euclidean geometry]], the '''midpoint''' of a [[line segment]] is the [[point]] on the segment equidistant from both endpoints. ...<math>M</math> on <math>\overline{AB}</math> such that <math>AM=BM</math> is the midpoint of the segment.
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• P versus NP
  ...e solved by a deterministic algorithm in polynomial time. <math>NP</math> is the class of decision problems that can be solved by a ''non-deterministic' ...rministic - that, given the computer's present state and any inputs, there is only one possible action that the computer might take - and sequential - it
  6 KB (1,104 words) - 15:11, 25 October 2017
• 2005 AMC 10A Problems/Problem 2
  What is the value of <math> ((1 \star 2) \star 3)</math>? ...qquad \textbf{(D) } \frac{1}{2}\qquad \textbf{(E) } \textrm{This\, value\, is\, not\, defined.} </math>
  855 bytes (123 words) - 19:05, 25 December 2022
• 2005 AMC 10A Problems/Problem 4
  ...[[diagonal]] of length <math>x</math> is twice as long as it is wide. What is the area of the rectangle? ...tbf{(C) } \frac{1}{2}x^2\qquad \textbf{(D) } x^2\qquad \textbf{(E) } \frac{3}{2}x^2 </math>
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• 2005 AMC 10A Problems/Problem 5
  ...ore normally sells windows at <math>\$100</math> each. This week the store is offering one free window for each purchase of four. Dave needs seven window The store's offer means that every <math>5</math>th window is free.
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• 2005 AMC 10A Problems/Problem 10
  ...2 + ax + 8x + 9 = 0 </math> has only one solution for <math>x</math>. What is the sum of those values of <math>a</math>? A [[quadratic equation]] has exactly one [[root]] if and only if it is a [[perfect square]]. So set
  3 KB (443 words) - 18:05, 29 March 2023
• 2005 AMC 10A Problems/Problem 11
  ...ly one-fourth 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 \te
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• 2005 AMC 10A Problems/Problem 12
  ...cular sectors about the sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length <math>2</math>? ...2}{3}\pi+\frac{\sqrt{3}}{3}\qquad \textbf{(E) } \frac{2}{3}\pi+\frac{\sqrt{3}}{2} </math>
  2 KB (246 words) - 12:43, 13 December 2021
• 2005 AMC 10A Problems/Problem 13
  ...er of [[positive integer]]s over the interval <math> (4,130) </math> which is <math>129-5+1 = \boxed{\textbf{(E) }125}</math> ...</math> which contains the same number of elements as <math>\left\{1,\ 2,\ 3,\ \cdots ,\ 124,\ 125\right\}</math> which clearly contains <math>125</math
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• 2005 AMC 10A Problems/Problem 14
  How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits? If the middle digit is the average of the first and last digits, twice the middle digit must be eq
  4 KB (694 words) - 19:25, 13 December 2021
• 2005 AMC 10A Problems/Problem 15
  How many positive cubes divide <math> 3! \cdot 5! \cdot 7! </math> ? <math> \textbf{(A) } 2\qquad \textbf{(B) } 3\qquad \textbf{(C) } 4\qquad \textbf{(D) } 5\qquad \textbf{(E) } 6 </math>
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• 2005 AMC 10A Problems/Problem 17
  ...form an arithmetic sequence, although not necessarily in that order. What is the middle term of the arithmetic sequence? ...>DC</math> and <math>DE</math>). The sum of every term is equal to <math>2(3+5+6+7+9)=60</math>
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• 2005 AMC 10A Problems/Problem 18
  ...dependent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game? ...\frac{1}{3}\qquad \textbf{(D) } \frac{1}{2}\qquad \textbf{(E) } \frac{2}{3} </math>
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• 2005 AMC 10A Problems/Problem 21
  <math> \textbf{(A) } 3\qquad \textbf{(B) } 5\qquad \textbf{(C) } 7\qquad \textbf{(D) } 9\qquad \te ...evenly [[divide]]s <math>6n</math>, then <math>\frac{6n}{1+2+...+n}</math> is an [[integer]].
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• 2005 AMC 10A Problems/Problem 25
  ...respectively, with <math> AD = 19 </math> and <math> AE = 14 </math>. What is the [[ratio]] of the area of triangle <math>ADE</math> to the area of the [ ...c{266}{1521}\qquad \textbf{(B) } \frac{19}{75}\qquad \textbf{(C) }\frac{1}{3}\qquad \textbf{(D) } \frac{19}{56}\qquad \textbf{(E) } 1 </math>
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• Tetrahedron
  The '''tetrahedron''' (plural ''tetrahedra'') or ''triangular pyramid'' is the simplest [[polyhedron]]. Tetrahedra have four [[vertex|vertices]], fou triple[] P = {(0,0,(2/3)^.5),(3^(-0.5),0,0),(-1/2/3^.5,1/2,0),(-1/2/3^.5,-1/2,0)};
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• 2005 AMC 10A Problems/Problem 24
  ...me factor of <math>n</math>. For how many positive integers <math>n</math> is it true that both <math> P(n) = \sqrt{n} </math> and <math> P(n+48) = \sqrt <math> \textbf{(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
• Pi notation
  Pi notation is a method used to represent the product of terms. It is similar to [[Sigma notation]] but uses a capital letter Pi. ...^n a(k)</math> for the product <math>a(m)\cdot a(m+1)\cdot a(m+2)\cdot a(m+3)\cdots a(n-1)\cdot a(n)</math>.
  424 bytes (85 words) - 17:23, 8 February 2024
• Volume
  The '''volume''' of an object is a [[measure]] of the [[amount]] of [[space]] that it occupies. Note that vo ...[[prism]] of [[height]] <math>h</math> and base of [[area]] <math>b</math> is <math>b\cdot h</math>.
  3 KB (523 words) - 20:24, 17 August 2023
• Zero (constant)
  '''Zero''', or 0, is the name traditionally given to the additive [[identity]] in number systems ...ath>1/0</math> equals ∞! But that is not the case. 1 over <math>x</math> is the reciprocal or the multiplicative inverse of
  2 KB (371 words) - 12:55, 21 June 2023
• Multiset
  ...tained. In other words, a multiset is a set where duplication of elements is allowed. For example, <math>\{1, 1, 2, 3\} \neq \{1, 2, 3\}</math> as multisets.
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• Committee forming
  '''Committee forming''' is one technique for solving certain [[combinatorics]] problems. How many committees of 3 people can be formed from a group of 12 people?
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• System of equations
  ...ns''' is a set of [[equation]]s which share the same [[variable]]s. Below is an example of a system of equations. A system of [[linear]] equations is where all of the variables are to the power 1. There are three elementary
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• Graph of a function
  A '''graph''' is a visual representation of a [[function]]. If <math>f(x) = y</math> then t A single point is the simplest thing to graph. The graph of <math>(2,5)</math> would be a do
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• Obtuse triangle
  ...ch one [[angle]] is an [[obtuse angle]]. Any triangle which is not obtuse is either a [[right triangle]] or an [[acute triangle]]. D = (3, 0);
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• Divisibility rules/Rule for 11 proof
  ...ber <math>N</math> is divisible by 11 if the alternating sum of the digits is divisible by 11. ...nding of [[Introduction to modular arithmetic | basic modular arithmetic]] is necessary for this proof.''
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• Integral
  The '''integral''' is one of the two base concepts of [[calculus]], along with the [[derivative]] In introductory, high-school level texts, the integral is often presented in two parts, the '''indefinite integral''' and '''definite
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• Divisibility rules/Rule for 3 and 9 proof
  .... For instance, the sum of the digits of 1899 is divisible by 27, but 1899 is not itself divisible by 27. ...nding of [[Introduction to modular arithmetic | basic modular arithmetic]] is necessary for this proof.''
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• Power set
  ...is the set <math>\mathcal{P}(S)</math> of all [[subset]]s of that set. It is also sometimes denoted by <math>2^S</math>. ...h>S</math>, the cardinality <math>|\mathcal P (S)|</math> of the power set is strictly larger than the cardinality <math>|S|</math> of the set itself.
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• Orthocenter
  ...triangle]] is the point of intersection of its [[altitude|altitudes]]. It is [[mathematical convention | conventionally]] denoted <math>H</math>. dot((3,3));
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• MathPath
  ...o gain admission. MathPath is centered around mathematical exploration. It is generally located on a different college campus each year. The fee is <nowiki>$4500</nowiki>, but there are merit scholarships and financial aid.
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• AwesomeMath
  ...ve three-week online program for gifted students from around the globe. It is designed for bright middle and high school students who wish to sharpen the Step 3. Have Your Math Teacher/Mentor Submit a Recommendation Letter
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• 1970 Canadian MO Problems
  ...any of these numbers is added to the product of the other two, the result is 2. MP("b",(2.3/2-.05,7.5/8+.25),N);
  4 KB (604 words) - 04:32, 8 October 2014
• Alternating sum
  An '''alternating sum''' is a [[series]] of [[real number]]s in which the terms alternate sign. For example, the alternating [[harmonic series]] is <math>1 - \frac12 + \frac13 - \frac 14 + \ldots = \sum_{i = 1}^\infty \frac
  2 KB (301 words) - 22:13, 19 February 2022
• Free group
  ...up [[identity]] and is equal to the empty string. The group [[operation]] is concatenation. ...1, 2\}</math> is <math>X_1X_2^{-1}X_1^{-1}X_2^3</math> (where by <math>X_2^3</math> we mean <math>X_2X_2X_2</math>).
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• Perfect number
  A [[positive integer]] <math>n</math> is called a '''perfect number''' if it is the sum of its [[proper divisor]]s. * <math>6 = 3 + 2 + 1</math>
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• Stirling number
  The '''Stirling number of the first kind''' <math>c(n, k)</math> is the number of [[permutation]]s of an <math>n</math>-[[element]] [[set]] wit ...permutations <math>\{(1)(234), (1)(243), (134)(2),(143)(2),(124)(3),(142)(3),(123)(4),(132)(4), (12)(34), (13)(24), (14)(23)\}</math>.
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• United States Physics Olympics
  ...the semi-final exam, as it is the last of the national physics exams (and is thought of as analogous to [[USAMO]]). Any high school student who is either a US citizen or permanent resident is eligible to take this exam.
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• Derangement
  ...\{3,2, 1\}</math> is not a derangement of <math>\{1,2,3\}</math> because 2 is a fixed point. ...er or ''rencontres number'', or the ''subfactorial'' of <math>n</math> and is sometimes denoted <math>!n</math> or <math>D_n</math>. (Note that using th
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• 1970 IMO Problems
  === Problem 3 === [[1970 IMO Problems/Problem 3 | Solution]]
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• 1970 IMO Problems/Problem 2
  which is equivalent to {{IMO box|year=1970|num-b=1|num-a=3}}
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• Odd integer
  ...iple of <math>2</math>). The odd integers are <math>\ldots, -5, -3, -1, 1, 3, 5, \ldots.</math> Every odd integer can be written in the form <math>2k + ...d the result of a division where both the dividend and the divisor are odd is odd. But the sum and difference of any two odd integers are [[even integer
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• Proper fraction
  A '''proper fraction''' is a fraction such that the absolute value is less than 1. <math>\frac{3}{4}</math> Because <math>3\neq4</math> and <math>3<4.</math>
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• 2007 iTest Problems
  ...h that <math>q</math> is <math>2</math> greater than <math>p</math>. What is the arithmetic mean of the two primes in the smallest twin prime pair? ===Problem 3===
  33 KB (5,177 words) - 21:05, 4 February 2023
• Qq828sportsmy
  ...should visit QQ828 sportsbook Malaysia site and create your account! QQ828 is known for its promos and bonuses. Do you want to see some of the biggest pr ...omo. The maximum bonus that you can possibly get after availing this promo is MYR 300.
  2 KB (301 words) - 02:07, 9 December 2019
• 2005 Alabama ARML TST
  ...urnament]], one of two tests given to choose the [[Alabama ARML]] team. It is similar to an easy [[AIME]], Alabaman scores average about 5 right, while t * [[2005 Alabama ARML TST Problems/Problem 3]]
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• 1981 IMO Problems/Problem 2
  Clearly, the sum of the desired least elements is <math> \sum_{k=1}^{n} k {n-k \choose r-1} = \sum_{k=1}^{n} {k \choose 1}{n- ...the sets is <math>{n+1 \choose r+1}</math>, the mean of the least elements is <math>\frac{{n+1 \choose r+1}}{{n \choose r}} = \frac{n+1}{r+1}</math>, Q.E
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• 1987 IMO Problems/Problem 6
  ...at <math> 0 \leq k \leq \sqrt{n/3} </math>, then <math>k^2 + k + n </math> is prime for all integers <math>k </math> such that <math>0 \leq k \leq n - 2 ...lso have <math>(2b+1)^2>m</math>, then we can conclude that <math>m</math> is a prime. Since there must be a factor of <math>m</math> less than <math>\sq
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• 1981 IMO Problems/Problem 4
  ...consecutive [[positive integer]]s such that the largest number in the set is a [[divisor]] of the [[least common multiple]] of the remaining <math>n-1</ (b) For which values of <math>n>2</math> is there exactly one set having the stated property?
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• 1981 IMO Problems/Problem 5
  ...> is clearly the circumcenter of <math>O_A O_B O_C </math>, <math>O</math> is collinear with the incenter and circumcenter of <math>ABC</math>, as desire Suppose 3 congruent circles with centres P,Q,R lie inside ABC and are such that the c
  2 KB (373 words) - 23:09, 29 January 2021
• 1981 IMO Problems/Problem 6
  (3) <math>f(x+1,y+1)=f(x,f(x+1,y)), </math> ...ath> and <math>f(2, y+1) = f(2,y) + 2</math>, yielding <math>f(2,y) = 2y + 3</math>.
  2 KB (306 words) - 18:15, 12 April 2024
• Cube (geometry)
  ...al]] to the regular [[octahedron]] and has [[octahedral symmetry]]. A cube is a [[Platonic solid]]. All edges of cubes are equal to each other. ...iped]], an equilateral cuboid, and a right rhombohedron a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron
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• Element
  An '''element''', also called a '''member''', is an object contained within a [[set]] or class. ...,\,2,\,3,\,4\}</math> means set <math>A</math> contains the elements 1, 2, 3 and 4.
  704 bytes (102 words) - 15:59, 3 April 2012
• Asymptote: Basics
  ...interprets new commands. However, putting the commands on separate lines is often useful to separate your commands to organize your code and improve re This is a program
  5 KB (800 words) - 00:39, 11 March 2023
• Apothem
  ...[center]] of the [[polygon]] [[perpendicular]] to one of its [[edge]]s. It is also the [[radius]] of the [[incircle | inscribed circle]] of the polygon. ...<math>n</math>, and side length, <math>s</math>, the length of the apothem is <math>\frac{s}{2\tan\left(\frac{\pi}{n}\right)}</math>.
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• Dimension
  A 0-dimensional object is a [[point]], which has no [[length]], height, or [[depth]]. A 1-dimensional object is a [[line]], or [[line segment]], which has length, but no other characteris
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• 2005 AMC 10A Problems
  What is the value of <math> ((1 \star 2) \star 3)</math>? ...d \mathrm{(D) \ } \frac{1}{2}\qquad \mathrm{(E) \ } \textrm{This\, value\, is\, not\, defined.} </math>
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• 2005 AMC 10A Problems/Problem 20
  ...th>, arranged so that no two consecutive sides have the same length. What is the area of the octagon? Therefore, the area of the octagon is equal to the area of <math>5+4\left(\frac12\right)=7</math> squares.
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• Euler line
  ...nts. In particular, <math>\overline{OGNH}</math> and <math>OG:GN:NH = 2:1:3</math> Euler line is the central line <math>L_{647}</math>.
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• 2004 AMC 12A Problems/Problem 1
  ...lems|2004 AMC 12A #1]] and [[2004 AMC 10A Problems/Problem 3|2004 AMC 10A #3]]}} Alicia earns 20 dollars per hour, of which <math>1.45\%</math> is deducted to pay local taxes. How many cents per hour of Alicia's wages are
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• 2004 AMC 10A Problems/Problem 4
  What is the value of <math>x</math> if <math>|x-1|=|x-2|</math>? ...the distance between <math>x</math> and <math>1</math>; <math>|x-2|</math> is the distance between <math>x</math> and <math>2</math>.
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• 2004 AMC 12A Problems/Problem 4
  ...4</math> of Bertha's daughters, so the number of women having no daughters is <math>30 - 4 = \boxed{26}</math>. === Solution 3 ===
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• 2004 AMC 10A Problems/Problem 7
  ...the first level rests in a pocket formed by four oranges below. The stack is completed by a single row of oranges. How many oranges are in the stack? ...dded on top of the <math>2^{\text{nd}}</math>, it will be a layer of <math>3\times6=18</math> oranges, etc.
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• 2004 AMC 12A Problems/Problem 7
  A game is played with tokens according to the following rule. In each round, the pla ...fter <math>1</math> more round, player <math>A</math> will give away <math>3</math> tokens, leaving them empty-handed, and thus the game will end. We th
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• 2004 AMC 10A Problems/Problem 10
  ...ability that the number of heads obtained from flipping the two fair coins is the same? ...will be obtained; <math>0</math>, <math>1</math>, <math>2</math>, or <math>3</math> heads.
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• 1961 IMO Problems/Problem 1
  ...ath>. Mutiplying both sides by <math>(x-y) </math>, we have <math>x^3 - y^3 = 0 </math>. Since we want <math>x,y </math> to be real, this implies <mat ...c{a^2 - b^2}{2a}\right)^2 </math>. The [[discriminant]] for this equation is
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• 2004 AMC 10A Problems/Problem 15
  Given that <math>-4\leq x\leq-2</math> and <math>2\leq y\leq4</math>, what is the largest possible value of <math>\frac{x+y}{x}</math>? ...ed, which occurs when <math>|x|</math> is the largest and <math>|y|</math> is the smallest.
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• 2003 AMC 10A Problems
  What is the difference between the sum of the first <math>2003</math> even counting ...es and another pair of socks and a shirt for away games. If the total cost is &#36;2366, how many members are in the League?
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• Cyprus Mathematical Olympiad
  ...udents in every grade, organized by the Cyprus Mathematical Society. There is one set of problems for every grade in primary school (Dhitmotiko): 4th, 5t ...ontestants of each set of problems, and they are rewarded in the ratio 1:2:3 for gold, silver and bronze metal respectively.
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• 2003 AMC 10A Problems/Problem 5
  ...nd <math>e</math> denote the solutions of <math>2x^{2}+3x-5=0</math>. What is the value of <math>(d-1)(e-1)</math>? ...\mathrm{(A) \ } -\frac{5}{2}\qquad \mathrm{(B) \ } 0\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 5\qquad \mathrm{(E) \ } 6 </math>
  2 KB (240 words) - 15:37, 19 August 2023
• 2003 AMC 10A Problems/Problem 9
  <math>\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}</math>. <math> \mathrm{(A) \ } \sqrt{x}\qquad \mathrm{(B) \ } \sqrt[3]{x^{2}}\qquad \mathrm{(C) \ } \sqrt[27]{x^{2}}\qquad \mathrm{(D) \ } \sqrt[
  1 KB (230 words) - 18:47, 28 July 2023
• Pick's Theorem
  ...and the number of lattice points on the sides of the polygon. The formula is: where <math>I</math> is the number of lattice points in the interior and <math>B</math> being the n
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• Balkan Mathematical Olympiad
  The '''Balkan Mathematical Olympiad''' (BMO) is an annual contest for students from one of the [[#Member countries|member c ...6 - 8|breakdown=<u>Problem 1</u>: 6<br><u>Problem 2</u>: 6.5<br><u>Problem 3</u>: 7.5<br><u>Problem 4</u>: 8}}
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• 2003 AMC 10A Problems/Problem 12
  ...ath>, <math>(4,0)</math>, <math>(4,1)</math>, and <math>(0,1)</math>. What is the probability that <math>x<y</math>? ...\frac{3}{8}\qquad \mathrm{(D) \ } \frac{1}{2}\qquad \mathrm{(E) \ } \frac{3}{4} </math>
  1 KB (190 words) - 21:58, 19 July 2023
• Junior Balkan Mathematical Olympiad
  The '''Junior Balkan Mathematical Olympiad''' ('''JBMO''') is an annual contest for students under the age of 15.5 from one of the [[#Mem ...|type=Proof|difficulty=4 - 6|breakdown=<u>Problem 1</u>: 4<br><u>Problem 2/3</u>: 5<br><u>Problem 4</u>: 6}}
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• 2003 AMC 10A Problems/Problem 13
  ...times the sum of the other two. The second is seven times the third. What is the product of all three? Therefore, the product of all three numbers is <math>xyz=16\cdot\frac{7}{2}\cdot\frac{1}{2}=28 \Rightarrow \boxed{\mathrm{
  3 KB (383 words) - 15:45, 19 August 2023
• 2003 AMC 10A Problems/Problem 14
  ...+e</math>, where <math>d</math> and <math>e</math> are single digits. What is the sum of the digits of <math>n</math>? ...So <math>e = 3</math>. Therefore, <math>d \cdot e \cdot (10d+e) = 7 \cdot 3 \cdot 73 = 1533</math>.
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• 2003 AMC 10A Problems/Problem 15
  ......,100\}</math> is divisible by <math>2</math> and not divisible by <math>3</math>? Since every <math>2^{\text{nd}}</math> integer is divisible by <math>2</math>, there are <math>\lfloor\frac{100}{2}\rfloor=50
  2 KB (261 words) - 14:34, 17 August 2023
• 2003 AMC 10A Problems/Problem 16
  What is the units digit of <math>13^{2003}</math>? <math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 7\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 9 </m
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• 2003 AMC 10A Problems/Problem 17
  ...the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle? <!-- don't remove the following tag, ...} \sqrt{3}\qquad \mathrm{(D) \ } \frac{6}{\pi}\qquad \mathrm{(E) \ } \sqrt{3}\pi </math>
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• 2003 AMC 10A Problems/Problem 18
  What is the sum of the reciprocals of the roots of the equation The problem is asking for <math>\frac{1}{a}+\frac{1}{b}= \frac{a+b}{ab}</math>
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• 1997 AJHSME Problems/Problem 24
  ...on. The ratio of the area of the upper region to that of the lower region is A = (0,0); B = (2,2); C = (4,0); D = (7,-3); EE = (10,0);
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• 2003 AMC 10A Problems/Problem 20
  ...digit number <math>n</math> is selected at random. Which of the following is closest to the probability that the base-9 representation and the base-11 r <math> \mathrm{(A) \ } 0.3\qquad \mathrm{(B) \ } 0.4\qquad \mathrm{(C) \ } 0.5\qquad \mathrm{(D) \ } 0
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• Chinese Remainder Theorem
  The '''Chinese Remainder Theorem''' is a [[number theory | number theoretic]] result. Formally stated, the Chinese Remainder Theorem is as follows:
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• 1985 IMO Problems
  ...h>M = \{ 1,2, \ldots , n-1 \} </math> is colored either blue or white. It is given that === Problem 3 ===
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• 1985 IMO Problems/Problem 1
  ...math>, so <math>DCOT </math> is a cyclic quadrilateral and <math>T </math> is in fact the <math>T</math> of the previous solution. The conclusion follow === Solution 3 ===
  4 KB (684 words) - 07:28, 3 October 2021
• 2003 AMC 10A Problems/Problem 21
  Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, The assortments are: <math>\{(0,6,0); (0,5,1); (0,4,2); (0,3,3); (0,2,4); (0,1,5); (0,0,6)\} \rightarrow 7</math> assortments.
  7 KB (994 words) - 17:51, 11 April 2024
• 2003 AMC 10A Problems/Problem 22
  ...ath> intersects line <math>AH</math> at <math>G</math>, and <math>F</math> is on line <math>AD</math> with <math>GF \perp AF</math>. Find the length of < pair D=(0,0), Ep=(4,0), A=(9,0), B=(9,8), H=(3,8), C=(0,8), G=(-6,20), F=(-6,0);
  8 KB (1,308 words) - 18:25, 16 July 2023
• 1985 IMO Problems/Problem 2
  ...h>M = \{ 1,2, \ldots , n-1 \} </math> is colored either blue or white. It is given that ...that <math>i, -i </math>, and <math>i+k </math> have the same color, which is to say that all residues of the form <math> i + mk \; (m \in \mathbb{N}_0)<
  1 KB (267 words) - 00:11, 12 July 2020
• 2003 AMC 10A Problems/Problem 23
  ...of small equilateral triangles. For example, in the figure, we have <math>3</math> rows of small congruent equilateral triangles, with <math>5</math> s pair Ap=(0,0), Bp=(1,0), Cp=(2,0), Dp=(3,0), Gp=dir(60);
  5 KB (686 words) - 18:01, 28 January 2021
• 2003 AMC 10A Problems/Problem 25
  ...ath> is divided by <math>100</math>. For how many values of <math>n</math> is <math>q+r</math> divisible by <math>11</math>? <3
  6 KB (943 words) - 20:57, 29 May 2023
• 2004 AMC 10A Problems
  ...th>c</math>, with <math>b\neq c</math>, the operation <math>\otimes</math> is defined by: What is <math>\otimes ( \otimes (1,2,3), \otimes (2,3,1), \otimes (3,1,2))</math>?
  15 KB (2,092 words) - 20:32, 15 April 2024
• 2004 AMC 10A Problems/Problem 2
  ...th>c</math>, with <math>b\neq c</math>, the operation <math>\otimes</math> is defined by: What is <math>\otimes(\otimes(1,2,3),\otimes(2,3,1),\otimes(3,1,2))</math>?
  859 bytes (121 words) - 14:13, 21 April 2021
• 2001 AMC 8 Problems
  Casey's shop class is making a golf trophy. He has to paint <math>300</math> dimples on a golf ba ...inking of two whole numbers. Their product is 24 and their sum is 11. What is the larger number?
  13 KB (1,994 words) - 13:04, 18 February 2024
• 1985 AIME Problems
  ...volume of the cone produced is <math>1920\pi \;\textrm{cm}^3</math>. What is the length (in cm) of the hypotenuse of the triangle? ==Problem 3==
  7 KB (1,071 words) - 19:24, 23 February 2024
• 1985 AIME Problems/Problem 1
  Another way to do this is to realize that most of our numbers will be canceled out in the multiplicat ...{2}{x_1}}}}}}\cdot\dfrac{8}{\dfrac{7}{\dfrac{6}{\dfrac{5}{\dfrac{4}{\dfrac{3}{\dfrac{2}{x_1}}}}}}}</cmath>
  2 KB (279 words) - 15:07, 4 September 2020
• 1985 AIME Problems/Problem 2
  ...volume of the cone produced is <math>1920\pi \;\textrm{ cm}^3</math>. What is the length (in cm) of the [[hypotenuse]] of the triangle? ...h>\frac{1}{3} \pi \left(\frac{12}{5}b\right)b^2 = 800\pi</math> so <math>b^3 = 1000</math> and <math>b = 10</math> so <math>a = 24</math>. Then by the
  3 KB (463 words) - 15:10, 4 September 2020
• Pythagorean triple
  A '''Pythagorean triple''' is a triple of [[positive integer]]s, <math>(a, b, c)</math> such that <math>a <math>(3, 4, 5)</math><nowiki>*</nowiki>
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• 2003 AMC 12A Problems/Problem 1
  What is the difference between the sum of the first <math>2003</math> even counting The first <math>2003</math> odd counting numbers are <math>1,3,5,...,4005</math>.
  3 KB (436 words) - 20:31, 28 December 2021
• 2003 AMC 12A Problems/Problem 2
  ...es and another pair of socks and a shirt for away games. If the total cost is &#36;2366, how many members are in the League? ...ocks and <math>2</math> T-shirts, the total cost for <math>1</math> member is <math>2(4+9)=26</math> dollars.
  1 KB (185 words) - 10:16, 4 July 2013
• 2003 AMC 12A Problems/Problem 4
  ...</math> minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip? <math> \mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 3.125\qquad \mathrm{(C) \ } 3.5\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 4.5 </math>
  2 KB (246 words) - 04:55, 7 November 2020
• 2003 AMC 12A Problems/Problem 5
  ...ers <math>AMC10</math> and <math>AMC12</math> is <math>123422</math>. What is <math>A+M+C</math>? We know that <math>AMC12</math> is <math>2</math> more than <math>AMC10</math>. We set up <math>AMC10=x</math>
  2 KB (381 words) - 21:56, 19 July 2023
• 2003 AMC 12A Problems/Problem 7
  <math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 5 </math> ...a length greater than the semiperimeter, which is <math>\frac{1}{2}\cdot7=3.5</math>.
  1 KB (189 words) - 21:45, 19 July 2023
• 2003 AMC 12A Problems/Problem 8
  ...s the probability that a randomly drawn positive factor of <math>60</math> is less than <math>7</math>? ...rac{1}{6}\qquad \mathrm{(C) \ } \frac{1}{4}\qquad \mathrm{(D) \ } \frac{1}{3}\qquad \mathrm{(E) \ } \frac{1}{2} </math>
  2 KB (326 words) - 15:40, 19 August 2023
• 2003 AMC 12A Problems/Problem 12
  ...ed card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? <math>B_5</math> is the only blue card that <math>R_5</math> evenly divides, so <math>R_5</math
  2 KB (368 words) - 18:04, 28 December 2020
• 2003 AMC 12A Problems/Problem 13
  ...(geometry) | squares]] joined [[edge]]-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the <math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 5\qquad \mathrm{(E) \ } 6 </m
  3 KB (407 words) - 15:43, 19 August 2023
• 2003 AMC 12A Problems/Problem 15
  ...d [[area]] inside the smaller semicircle and outside the larger semicircle is called a ''lune''. Determine the area of this lune. filldraw(Arc((0,sqrt(3)),1,0,180)--cycle,mediumgray);
  3 KB (380 words) - 21:53, 19 March 2022
• 2003 AMC 12A Problems/Problem 18
  ...ath> is divided by <math>100</math>. For how many values of <math>n</math> is <math>q+r</math> divisible by <math>11</math>? ...math>5</math>-digit number is divided by <math>100</math>, the first <math>3</math> digits become the quotient, <math>q</math>, and the last <math>2</ma
  3 KB (426 words) - 18:20, 18 July 2022
• 2006 Cyprus Seniors Provincial/2nd grade/Problems
  ...ce of <math>A</math> form <math>\Gamma\Delta</math> and <math>x_{2}</math> is the distance of <math>\Gamma</math> form <math>\Alpha\Gamma</math> prove th == Problem 3 ==
  2 KB (266 words) - 22:07, 10 April 2013
• Specimen Cyprus Seniors Provincial/2nd grade/Problems
  a) the distance of <math>\Gamma</math> from <math>(\epsilon)</math> is equal to the sum of the distances <math>\text{B}</math> , <math>\Delta</mat ...}\cos x_{2} \sin x_{3}</math> and <math>\delta=\cos x_{1}\cos x_{2}\cos x_{3}</math> prove that <math>\alpha^2+\beta^2+\gamma^2+\delta^2=1</math>
  2 KB (266 words) - 20:19, 18 January 2021
• 2005 Alabama ARML TST Problems/Problem 1
  ...}8</math>. The numbers on the faces of the other die are <math>1, 2, 2, 3, 3,\text{ and }4</math>. Find the [[probability]] of rolling a sum of <math>9< ...sum of the two dice rolls: <center><math>(x+x^3+x^4+x^5+x^6+x^8)(x+2x^2+2x^3+x^4)=</math></center>
  1 KB (210 words) - 01:30, 3 January 2023
• 2005 Alabama ARML TST Problems/Problem 5
  .... The four squares are to be painted such that 2 are blue, 1 is red, and 1 is green. In how many ways can this be done? We choose two squares to be blue and one to be red; then the green's position is forced. There are <math>{4 \choose 2}{2 \choose 1}=12</math> ways to do thi
  1 KB (169 words) - 01:36, 3 January 2023
• Perfect power
  ...4^3 = 2^6</math>, so <math>64</math> is a perfect <math>2</math>nd, <math>3</math>rd and <math>6</math>th power. ...a perfect <math>1</math>st power" is a meaningless property: every integer is a <math>1</math>st power of itself.
  870 bytes (148 words) - 16:52, 18 August 2013
• 2002 AMC 12A Problems/Problem 1
  <math>(2x+3)(x-4)+(2x+3)(x-6)=0 </math> ...g the quadratic part of [[Vieta's Formulas]], we find the sum of the roots is <math>\frac{14}4 = \boxed{\textbf{(A) }7/2}</math>.
  979 bytes (148 words) - 13:06, 8 November 2021
• 1987 IMO Problems/Problem 2
  ...C]</math>. Thus, we are to prove that <math>[BNC]=[KNC]+[BMN]</math>. It is clear that since <math>\angle BAN=\angle NAC</math>, the segments <math>BN< ...e fact that <math>\angle BNC=180-\angle A</math> and that <math>BNC</math> is iscoceles, we find that <math>\angle NBC=\angle NCB=\frac{1}{2}A</math>. S
  4 KB (736 words) - 15:39, 21 September 2014
• 1987 IMO Problems/Problem 4
  Prove that there is no function <math>f </math> from the set of non-negative integers into its ...<math>f(f(n)) = n + k</math> for all <math>n</math>, where <math>k</math> is a fixed positive integer, then <math>k</math> must be even. If <math>k = 2h
  5 KB (923 words) - 19:51, 21 January 2024
• 1987 IMO Problems
  ...math>S </math> onto itself. An element <math>i </math> in <math>S </math> is called a fixed point of the permutation <math>f </math> if <math>f(i) = i < === Problem 3 ===
  3 KB (459 words) - 14:24, 17 September 2023
• 2003 AMC 12A Problems/Problem 25
  ...bx} </math>. For how many [[real number | real]] values of <math>a</math> is there at least one [[positive number | positive]] value of <math> b </math> ...0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ \mathrm{infinitely \ many} } </math>
  9 KB (1,606 words) - 11:34, 10 July 2020
• 2005 Cyprus Seniors TST/Day 1/Problem 2
  ...a-\alpha)+\beta^2 (\gamma+\alpha-\beta)+\gamma^2 (\alpha+\beta-\gamma)\leq 3 \alpha \beta \gamma</math> <math>2x(y+z)^2+2y(x+z)^2+2z(x+y)^2 \leq 3(x+y)(y+z)(z+x) </math>
  851 bytes (134 words) - 22:50, 12 October 2022
• 2002 IMO Shortlist Problems/N3
  ...math> p_{1}, p_{2}, \ldots , p_{n} </math> be distinct primes greater than 3. Show that <math> 2^{p_{1}p_{2} \cdots p_{n}} + 1 </math> has at least <ma ...'. For every divisor <math>a </math> of <math>m </math>, <math>ka </math> is clearly a divisor of <math>km </math>, but not <math>m </math>.
  10 KB (1,739 words) - 06:38, 12 November 2019
• ELMO
  The '''ELMO''' is an annual math olympiad that happens at [[MOP]]. Its initials have stood f ...the [[IMO]] (that is, 6 problems over two days with 4.5 hours per day); it is a tradition for all new students not in Black (and occasionally a couple re
  5 KB (773 words) - 19:16, 17 June 2022
• Mock AIME 4 2006-2007 Problems/Problem 10
  ...e 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}</math></p></center> is divided by 1000. ...+ \zeta + 1 = 0</math>. If <math>i</math> is two more than a multiple of 3, <math>\omega^i + \zeta^i + 1 = \omega^2 + \zeta^2 + 1= \zeta + \omega + 1
  4 KB (595 words) - 12:14, 25 November 2023
• Mock AIME 4 2006-2007 Problems/Problem 12
  ...<math>a</math> and <math>b</math> are positive integers and <math>a</math> is prime. Find the sum of the digits of <math>a + b</math>. ...a few small values for <math>x</math>, we see that <math>f(1)=0, f(2)=0, f(3)=0, f(4)=1, f(5)=1, f(6)=2, f(7)=2,
  992 bytes (156 words) - 20:34, 27 September 2019
• Mock AIME 4 2006-2007 Problems/Problem 15
  ...h>m</math> and <math>n</math> are [[positive integer]]s and <math>n</math> is not [[divisibility | divisible]] by the [[square]] of any [[prime]]. Find t ...ius is 43sqrt3.Then the coordinate of point D are 24, -18sqrt3. The answer is then 6 + sqrt43, which yields 12.
  3 KB (532 words) - 20:29, 31 August 2020
• Sphere
  ...e sphere. The common distance of the points of the sphere from the center is called the ''[[radius]]''. Spheres are the natural 3-dimensional analog of [[circle]]s.
  1 KB (247 words) - 20:36, 11 December 2020
• Asymptote (Vector Graphics Language)
  ...patible with the standard mathematics typesetting language, [[LaTeX]]. It is also a complete programming language, and has cleaner syntax than its prede Here is an example of an image that can be produced using Asymptote:
  2 KB (245 words) - 11:42, 14 January 2024
• 2020 AMC 10A Problems
  <cmath>x- \frac{3}{4} = \frac{5}{12} - \frac{1}{3}?</cmath> ...\ \frac{7}{36}\qquad\textbf{(C)}\ \frac{7}{12}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{5}{6}</math>
  13 KB (1,968 words) - 18:32, 29 February 2024
• Asymptote: Advanced Configuration
  ...choose <tt>gsNNNw32.exe</tt>; for 64-bit, <tt>gsNNNw64.exe</tt>, where NNN is version N.NN of Ghostscript). # When the download is complete, browse to <tt>D:\downloads\Ghostscript</tt> in your files and dou
  12 KB (1,931 words) - 13:53, 26 January 2020
• Asymptote: Reference
  A '''point''' is associated with a cartesian coordinate pair in Asymptote. There are two us returns the point (cos(theta),sin(theta)) where theta is in degrees, and
  7 KB (1,205 words) - 21:38, 26 March 2024
• Asymptote: Logical Operators and Loops
  fill(smiley,Circle((-.3,.4),.1),black); fill(smiley,Circle((.3,.4),.1),black);
  2 KB (324 words) - 16:50, 2 October 2016
• Asymptote: Useful commands and their Output
  ...ontests. Clicking on them will reveal the code used to make them, so this is an excellent resource for students looking to make more advanced diagrams. ...ate plane. In other words, assume that before each of the examples below is the command
  6 KB (871 words) - 21:14, 12 June 2023
• Asymptote: Help
  ...XnicCenter, but when I hit the hotkey to compile my code, all that happens is that a window flashes briefly then disappears. Nothing else seems to happe '''What might be going on:''' Your file is saved as a .tex file, not a .asy file. (TeXnicCenter will even make it loo
  6 KB (1,055 words) - 23:30, 6 October 2019
• Power's of 2 in pascal's triangle
  Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers above it. It Looks something like this: 1 3 3 1
  2 KB (341 words) - 16:57, 16 June 2019
• Carmichael function
  The Carmichael function <math>\lambda</math> is defined at <math>n</math> to be the smallest [[positive integer]] <math>\la This function is also known as the ''reduced totient function'' or the ''least universal exp
  2 KB (258 words) - 11:56, 1 August 2022
• Asymptote (geometry)
  An '''asymptote''' is a [[line]] or [[curve]] that a certain [[function]] approaches. ...g values of <math>x</math> that make the function undefined. Generally, it is found by setting the denominator of a rational function to zero.
  4 KB (664 words) - 11:44, 8 May 2020
• Pigeonhole Principle/Solutions
  Also, note that it is possible to pull out 4 socks without obtaining a pair. ...number of friends one person in the group can have is n-1, and the minimum is 0. If all of the members have at least one friend, then each individual can
  10 KB (1,617 words) - 01:34, 26 October 2021
• 2006 Canadian MO Problems/Problem 2
  ...a [[rectangle]] <math>DEFG</math> in this triangle so that <math>D</math> is on <math>AB</math>, <math>E</math> on <math>AC</math>, and <math>F</math> a The locus is the [[line segment]] which joins the [[midpoint]] of side <math>BC</math> t
  2 KB (416 words) - 20:00, 21 September 2014
• 2000 AMC 12 Problems/Problem 12
  ...e [[nonnegative integer]]s such that [mathjax]A + M + C=12[/mathjax]. What is the maximum value of [mathjax]A \cdot M \cdot C + A \cdot M + M \cdot C + A It is not hard to see that
  4 KB (623 words) - 15:45, 18 February 2024
• 2000 AMC 12 Problems/Problem 7
  ...integer]]s <math>b</math> have the property that <math>\log_{b} 729</math> is a positive integer? ...0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 4 } </math>
  616 bytes (86 words) - 23:49, 29 December 2023
• 2002 AMC 12P Problems/Problem 3
  ...positive integers and the volume of the box is <math>2002</math> in<math>^3</math>. Find the minimum possible sum of the three dimensions. ...th>2002</math> are <math>11, 13,</math> and <math>14</math>, so our answer is <math>11+13+14=\boxed{\textbf{(B) } 38}</math>
  1 KB (197 words) - 22:07, 30 December 2023
• 2007 AIME II Problems/Problem 9
  ...oint <math>P,</math> and the inscribed circle of triangle <math>DEF</math> is tangent to <math>EF</math> at [[point]] <math>Q.</math> Find <math>PQ.</mat ...st amongst the numbers given. <math>BE = DF = \sqrt{63^2 + 84^2} = 21\sqrt{3^2 + 4^2} = 105</math>. Also, the length of <math>EF = \sqrt{63^2 + (448 - 2
  5 KB (818 words) - 11:05, 7 June 2022
• 2007 AIME II Problems/Problem 10
  ...e [[relatively prime]]. Find <math>m+n+r.</math> (The set <math>S-A</math> is the set of all elements of <math>S</math> which are not in <math>A.</math>) ...th>A</math> must have either 0, 6, or 1, 5 elements. The total probability is <math>\frac{2}{64} + \frac{2}{64} = \frac{4}{64}</math>.
  8 KB (1,367 words) - 11:48, 23 October 2022
• 2005 AMC 10A Problems/Problem 23
  ...at <math>DC \perp AB</math> and <math>DE</math> is a second diameter. What is the ratio of the area of <math>\triangle DCE</math> to the area of <math>\t dotfactor=3;
  6 KB (1,045 words) - 09:46, 4 April 2023
• 2002 USAMO Problems/Problem 2
  ...\cot \frac{A}{2} \right)^2 + \left( 2 \cot \frac{B}{2} \right)^2 + \left( 3 \cot \frac{C}{2} \right)^2 = \left( \frac{6s}{7r} \right)^2 ...r]] and [[inradius]], respectively. Prove that triangle <math>ABC </math> is similar to a triangle <math>T </math> whose side lengths are all positive i
  2 KB (298 words) - 22:32, 6 April 2016
• 2002 USAMO Problems/Problem 3
  ...th leading coefficient 1) of degree <math>n </math> with real coefficients is the average of two monic polynomials of degree <math>n </math> with <math>n ...han <math>n </math> such that <math>Q(x_i) = y_i - x_i^n </math>. Then it is necessary and sufficient that <math>P(x) = x^n + Q(x) </math>.
  4 KB (688 words) - 13:38, 4 July 2013
• 2002 USAMO Problems/Problem 5
  ...<math>n_1 = a</math>, <math>n_k = b </math>, and <math>n_in_{i+1} </math> is divisible by <math>n_i + n_{i+1} </math> for each <math>i </math> (<math> 1 ...ote that <math> \leftrightarrow </math> is an [[equivalence relation]]: it is [[reflexive]] (<math> a \leftrightarrow a</math>), [[symmetric]] (<math> a
  7 KB (1,194 words) - 15:39, 28 March 2015
• 2002 USAMO Problems/Problem 4
  We first prove that <math>f </math> is [[odd function | odd]]. ...th>, which implies <math>f(-y) = -f(y) </math>. Therefore <math>f </math> is odd. Henceforth, we shall assume that all variables are non-negative.
  3 KB (490 words) - 22:09, 18 September 2018
• Mock AIME 1 Pre 2005 Problems
  ...gers with three distinct digits. Compute the remainder when <math>S</math> is divided by <math>1000</math>. == Problem 3 ==
  6 KB (1,100 words) - 22:35, 9 January 2016
• Mock AIME 3 Pre 2005 Problems
  ...ecting their centers is <math>84</math>, then the area of the third circle is <math>k\pi</math> for some integer <math>k</math>. Determine <math>k</math> [[Mock AIME 3 Pre 2005/Problem 1|Solution]]
  7 KB (1,135 words) - 23:53, 24 March 2019
• Mock AIME 3 Pre 2005 Problems/Problem 2
  ...in increasing order. Determine the remainder obtained when <math>N</math> is divided by <math>1000</math>. (Repeated digits are allowed.) ...to these urns. Using the ball-and-urn argument, having <math>9</math> urns is equivalent to <math>8</math> dividers, and there are <math>{8 + 7 \choose 7
  950 bytes (137 words) - 10:16, 29 November 2019
• Mock AIME 3 Pre 2005 Problems/Problem 3
  A [[function]] <math>f(x)</math> is defined for all real numbers <math>x</math>. For all non-zero values <math> 0 &= x^2 - \frac{3 \times 2004 - 4}{10}x + \frac 52\end{align*}</cmath>
  1 KB (191 words) - 10:22, 4 April 2012
• Mock AIME 3 Pre 2005 Problems/Problem 5
  ...ter Zuminglish words. Determine the remainder obtained when <math>N</math> is divided by <math>1000</math>. ...lowed by a constant (<tt>VC</tt> - the only other combination, two vowels, is impossible due to the problem statement). Then, note that:
  5 KB (795 words) - 16:03, 17 October 2021
• Mock AIME 3 Pre 2005 Problems/Problem 6
  ...h>p, q,</math> and <math>r</math> are positive integers and <math>r</math> is not divisible by the square of any prime. Determine <math>p + q + r</math>. This is a [[telescoping series]]; note that when we expand the summation, all of th
  3 KB (501 words) - 14:48, 29 November 2019
• Mock AIME 3 Pre 2005 Problems/Problem 7
  ...ect at <math>P</math>. If <math>AB = 1, CD = 4,</math> and <math>BP : DP = 3 : 8,</math> then the area of the inscribed circle of <math>ABCD</math> can ...th>, so we can solve (algebraically or by inspection) to get that <math>BC=3</math> and <math>AD=2</math>.
  2 KB (330 words) - 10:23, 4 April 2012
• Mock AIME 3 Pre 2005 Problems/Problem 8
  Determine the remainder obtained when <math>N</math> is divided by <math>1000</math>. ...5</math> so the only restrictions are imposed by <math>10 \pm 1 \pm 2 \pm 3 \pm 4 \pm 5 \pm 6</math> being equal to either <math>9,11,25</math>. If we
  3 KB (520 words) - 12:55, 11 January 2019
• Mock AIME 3 Pre 2005 Problems/Problem 9
  ...overline{BD}</math> past <math>D</math> such that <math>\angle{BAE}</math> is right. If <math>BD = 15, DE = 2,</math> and <math>BC = 16</math>, then <mat ...e (2x/17)/AD=(x/2)/16. Hence, AD=64/17, and CD=16-AD=208/17, so the answer is 225.
  2 KB (278 words) - 16:32, 27 December 2019
• Mock AIME 3 Pre 2005 Problems/Problem 10
  <math>\{A_n\}_{n \ge 1}</math> is a sequence of positive integers such that ...th>n > 1</math>. Compute the remainder obtained when <math>a_{2004}</math> is divided by <math>1000</math> if <math>a_1 = 1</math>.
  2 KB (306 words) - 10:36, 4 April 2012
• Mock AIME 3 Pre 2005 Problems/Problem 12
  .../math> such that <math>1 \le n \le 1000</math> and <math>n^{12} - 1</math> is divisible by <math>73</math>. ...</math>, and the rest being the previous: <math>+2, +5, +1, +15, +3, +19, +3, +15, +1, +5, +2</math>. This sequence then repeats itself. We hence find t
  714 bytes (105 words) - 23:59, 24 April 2013
• Mock AIME 3 Pre 2005 Problems/Problem 13
  <math>\left(\frac{2}{3}\right)^{2005} \cdot \sum_{k=1}^{2005} \frac{k^2}{2^k} \cdot {2005 \choose Determine the remainder obtained when <math>S</math> is divided by <math>1000</math>.
  3 KB (502 words) - 14:53, 19 July 2020
• Mock AIME 3 Pre 2005 Problems/Problem 14
  ...such that <math>p</math> and <math>r</math> are coprime and <math>q</math> is not divisible by the square of any prime. Determine <math>p + q + r</math>. Since <math>r_1</math> is the circumcenter of <math>\Delta PDA</math>,
  3 KB (563 words) - 02:05, 25 November 2023
• Mode
  ...ost times. For example, in <math>\{1,1,3,5,6,7,7,7,7,8\}</math>, the mode is <math>7</math>.
  328 bytes (44 words) - 15:31, 12 November 2023
• 1960 IMO Problems/Problem 3
  ...math>ABC</math>, the hypotenuse <math>BC</math>, of length <math>a</math>, is divided into <math>n</math> equal parts (<math>n</math> an odd integer). Le C=(0,3);
  3 KB (501 words) - 00:14, 17 May 2015
• 1960 IMO Problems
  ...rty that <math>N</math> is divisible by 11, and <math>\dfrac{N}{11}</math> is equal to the sum of the squares of the digits of <math>N</math>. === Problem 3 ===
  3 KB (511 words) - 21:21, 20 August 2020
• Mock AIME 5 2005-2006 Problems/Problem 1
  ...0</math>). Evaluate the remainder when <math>f(1)+f(2)+\cdots+f(99)</math> is divided by <math>1000</math>. ...are divisible by <math>2</math>, the number of numbers divisible by <math>3</math>, etc.
  2 KB (209 words) - 12:43, 10 August 2019
• 2007 AIME I Problems
  ...rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the distance in == Problem 3 ==
  7 KB (1,218 words) - 15:28, 11 July 2022
• 2007 AIME I Problems/Problem 1
  ...math>2</math> factors of <math>2</math> and <math>1</math> factor of <math>3</math>. ...t each square is in the form <math>(12c)^2</math>, where <math>12 c</math> is a positive integer less than <math>\sqrt{10^6}</math>. There are <math>\lef
  1 KB (204 words) - 13:56, 7 February 2023
• 2007 AIME I Problems/Problem 2
  ...rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the [[distance]] ...st along. Thus it is reasonable to assume that there is some point when Al is halfway between Cy and Bob. At this time <math>s</math>, we have that
  2 KB (301 words) - 08:09, 4 November 2022
• 2007 AIME I Problems/Problem 3
  ...nary parts of <math>z^{2}</math> and <math>z^{3}</math> are the same, what is <math>b</math> equal to?<!-- don't remove the following tag, for PoTW on th ...h> and <math>b = -15, 0, 15</math>. Since <math>b > 0</math>, the solution is <math>\boxed{015}</math>.
  1,003 bytes (163 words) - 15:34, 18 February 2017
• 2007 AIME I Problems/Problem 5
  ...is <math>C = \frac{5}{9}(F-32).</math> An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fah ...]</math> is an integer from <math>0</math> to <math>968</math>. This value is computed as <math>\left[968*\frac{5}{9}\right]+1 = \boxed{539}</math>, addi
  7 KB (1,076 words) - 00:10, 29 November 2023
• 2007 AIME I Problems/Problem 8
  The [[polynomial]] <math>P(x)</math> is [[cubic polynomial | cubic]]. What is the largest value of <math>k</math> for which the polynomials <math>Q_1(x) We then know that <math>a</math> is a root of
  4 KB (728 words) - 00:11, 29 November 2023
• 2007 AIME I Problems/Problem 10
  ...er of shadings with this property. Find the remainder when <math>N</math> is divided by 1000. Now consider the 3x3 that is next to the 3 boxes we have filled in. We must put one ball in each row (since there mus
  13 KB (2,328 words) - 00:12, 29 November 2023
• 2007 AIME I Problems/Problem 6
  ...nd ending with 39. For example, <math>0,\ 3,\ 6,\ 13,\ 15,\ 26,\ 39</math> is a move sequence. How many move sequences are possible for the frog? ...th>0 \Rightarrow 13</math>, we can end at either <math>12</math> (mult. of 3) or <math>13</math> (mult. of 13).
  10 KB (1,519 words) - 00:11, 29 November 2023
• 2007 AIME I Problems/Problem 9
  ...extension of leg <math>CA</math>, the circle with center <math>O_2</math> is [[tangent]] to the hypotenuse and to the extension of [[leg]] <math>CB</mat ...both [[tangent]]s to the circle, we see that <math>\overline{O_1A}</math> is an [[angle bisector]]. Thus, <math>\triangle AFO_1 \cong \triangle ADO_1</m
  11 KB (1,851 words) - 12:31, 21 December 2021
• 2007 AIME I Problems/Problem 11
  ...\sum_{p=1}^{2007} b(p),</math> find the [[remainder]] when <math>S</math> is divided by 1000. ...th> numbers in this range, so the sum of <math>b(p)</math> over this range is <math>(2k)k=2k^2</math>. <math>44<\sqrt{2007}<45</math>, so all numbers <ma
  3 KB (562 words) - 20:02, 30 December 2023
• 2007 AIME I Problems/Problem 12
  ...original and the rotated triangle is in the form <math>p\sqrt{2} + q\sqrt{3} + r\sqrt{6} + s</math>, where <math>p,q,r,s</math> are integers. Find <ma ...,C,B,4); MA("30^\circ",A,Cp,Bp,4); MA("45^\circ",A,extension(A,C,Bp,Cp),Bp,3); MA("15^\circ",C,Y,Cp,8); MA("15^\circ",C,A,Cp,9);
  10 KB (1,458 words) - 20:50, 3 November 2023
• 2007 AIME I Problems/Problem 14
  ...[[non-negative]] integral indexes in the following way: <math>a_{0}=a_{1}=3</math>, <math>a_{n+1}a_{n-1}=a_{n}^{2}+2007</math>. ...<math>a_1/a_0 = 1</math>, this means that the sequence <math>(a_n)</math> is increasing. Since <math>a_3=672</math> already, we must have <math>a_{2006}
  13 KB (2,185 words) - 23:30, 5 December 2022
• 2007 AIME I Problems/Problem 13
  Note first that the intersection is a [[pentagon]]. ...{2} - \frac{d}{2} + \sqrt{2}c = d \Longrightarrow c = d\sqrt{2}</math>. It is then <math>x - y + 2\sqrt{2}z = 2</math>.
  7 KB (1,034 words) - 21:56, 22 September 2022
• 2007 AIME I Problems/Problem 15
  ..., where <math>p</math> and <math>q</math> are rational, and <math>r</math> is an [[integer]] not divisible by the [[square]] of a [[prime]]. Find <math> ...- 2y + 56)</math>. Some terms will cancel out, leaving <math>y = \frac{5}{3}x - 22</math>.
  4 KB (673 words) - 22:14, 6 August 2022
• LaTeX:Math
  The solution to $\sqrt{x}=5$ is $x=25$. ...ere<nowiki>\]</nowiki> or <nowiki>$$math stuff here$$</nowiki> (the former is usually preferred now) to put the expression in display math mode:
  8 KB (1,323 words) - 15:09, 3 December 2020
• 2004 USAMO Problems
  \frac{1}{3} | AB^3 - AD^3 | \le | BC^3 - CD^3 | \le 3 |AB^3 - AD^3 | ...<math> a_1, \ldots, a_n </math> are integers whose greatest common divisor is 1. Let <math>S </math> be a set of integers with the following properties:
  3 KB (474 words) - 09:20, 14 May 2021
• 2004 USAMO Problems/Problem 4
  ...numbers written in them, in each row, the square with the greatest number is colored black. Alice wins if she can then draw a line from the top of the ...4,3), (4,4), (5,4), (5,5), (6,5), (6,6) useless, where <math>(m,n) </math> is the square in the <math>m</math>th column and <math>n</math>th row. Thus B
  2 KB (430 words) - 13:40, 4 July 2013
• 2004 USAMO Problems/Problem 5
  (a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \ge (a+b+c)^3 We first note that for positive <math>x </math>, <math> x^5 + 1 \ge x^3 + x^2 </math>. We may prove this in the following ways:
  4 KB (626 words) - 12:58, 27 August 2023
• 2007 AIME II Problems/Problem 1
  A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequ ...0</math> places to place the <math>0</math>s. There are <math>\frac{6!}{(6-3)!} = 120</math> ways to place the remaining three characters. In total, tha
  1 KB (180 words) - 14:01, 3 December 2023
• 2007 AIME II Problems/Problem 3
  <b>The sum of the squares of the sides of a parallelogram is the sum of the squares of the diagonals.</b> ...he diagonal of the square, with length <math>17\sqrt{2}</math>; the answer is <math>EF^2 = (17\sqrt{2})^2 = 578</math>.
  6 KB (933 words) - 00:05, 8 July 2023
• 2007 AIME II Problems/Problem 4
  ...a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the t Therefore, we can write that (note that two hours is similar to having twice the number of workers, and so on):
  1 KB (207 words) - 11:12, 20 April 2024
• 2007 AIME II Problems/Problem 5
  The [[graph]] of the [[equation]] <math>9x+223y=2007</math> is drawn on graph paper with each [[square]] representing one [[unit square|un The number of non-diagonal squares is <math>2007 - 231 = 1776</math>. Divide this in 2 to get the number of squar
  4 KB (559 words) - 00:38, 3 January 2023
• 2007 AIME II Problems/Problem 6
  ...i}</math> is [[odd]], and <math>a_{i}>a_{i+1}</math> if <math>a_{i}</math> is [[even]]. How many four-digit parity-monotonic integers are there? ...ater than <math>1</math>, so there are four possible places to align <math>3</math> as the second digit.
  2 KB (338 words) - 15:30, 7 August 2022
• 2007 AIME II Problems/Problem 7
  ...t[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor</math> and <math>k</math> divides <math>n_{i}</math> for al ...math>, all of the [[even]] numbers work, giving 10 integers. For <math>x = 3</math>, we get 13, and so on. We can predict that at <math>x = 22</math> we
  3 KB (428 words) - 18:04, 4 December 2020
• 2007 AIME II Problems/Problem 2
  ...teger]]s, <math>a</math> is a [[factor]] of <math>b</math>, <math>a</math> is a factor of <math>c</math>, and <math>a+b+c=100</math>. ...uces to <math>a(1 + x + y) = 100</math>. Therefore, <math>1 + x + y</math> is equal to one of the 9 factors of <math>100 = 2^25^2</math>.
  1 KB (228 words) - 08:41, 4 November 2022
• 2007 AIME II Problems/Problem 12
  ...2},\ldots</math> consists entirely of [[integer|integral]] powers of <math>3.</math> Given that <math>\sum_{n=0}^{7}\log_{3}(x_{n}) = 308</math> and <math>56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57,</math>
  5 KB (829 words) - 12:22, 8 January 2024
• 2007 AIME II Problems/Problem 13
  ...in the bottom row is the number in the top square a [[multiple]] of <math>3</math>? ...the number of ways a number can "travel" to the top position going only up is equal to the number of times it will be counted in the final sum.)
  3 KB (600 words) - 11:10, 22 January 2023
• 2007 AIME II Problems/Problem 15
  ...{C}</math> to <math>CA</math> and <math>CB</math>, and <math>\omega</math> is [[externally tangent]] to <math>\omega_{A},</math> <math>\omega_{B},</math> pen p=fontsize(10)+linewidth(3);
  11 KB (2,099 words) - 17:51, 4 January 2024
• 2007 AIME II Problems/Problem 14
  ...(2)+f(3)=125,</math> and for all <math>x</math>, <math>f(x)f(2x^{2})=f(2x^{3}+x).</math> Find <math>f(5).</math> ...\cdot a(2x^2)^m = 2^ma^2x^{3m}</math>, and the leading term of <math>f(2x^3 + x) = 2^max^{3m}</math>. Hence <math>2^ma^2 = 2^ma</math>, and <math>a = 1
  7 KB (1,335 words) - 17:44, 25 January 2022
• 2007 AIME II Problems
  A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequ ...teger]]s, <math>a</math> is a [[factor]] of <math>b</math>, <math>a</math> is a factor of <math>c</math>, and <math>a+b+c=100</math>.
  9 KB (1,435 words) - 01:45, 6 December 2021
• 2005 USAMO Problems
  ...g'') Determine all composite positive integers <math>n</math> for which it is possible to arrange all divisors of <math>n</math> that are greater than 1 x^6+x^3+x^3y+y & = 147^{157} \\
  4 KB (609 words) - 09:24, 14 May 2021
• 2002 USA TST Problems
  ...ath>, when written in lowest terms, is divisible by <math> \displaystyle p^3 </math>. === Problem 3 ===
  3 KB (544 words) - 06:58, 3 August 2017
• 1961 IMO Problems/Problem 2
  ...<math>b</math>, and <math>c</math> be the lengths of a triangle whose area is ''S''. Prove that <math>a^2 + b^2 + c^2 \ge 4S\sqrt{3}</math>
  5 KB (860 words) - 13:12, 13 February 2024
• 1961 IMO Problems/Problem 4
  ...interior of [[triangle]] <math>P_1P_2P_3</math> a [[point]] <math>P</math> is given. Let <math>Q_1,Q_2,Q_3</math> be the [[intersection]]s of <math>PP_1 ...an <math>\frac{1}{3}</math>, and at least one not less than <math>\frac{1}{3}</math>. These correspond to ratios <math>\frac{PP_i}{PQ_i}</math> being le
  2 KB (399 words) - 00:34, 4 September 2023
• Newton's Inequality
  ...<math> x_i = x_{i+1} = \cdots = x_{i+k} </math>, then <math>x_{i} </math> is a root of <math>P </math> <math>k+1 </math> times, so it must be a root of ...symmetric sum <math>s'_k </math> for <math> x'_1, \ldots, x'_{m-1} </math> is <math> \frac{m-k}{m}s_k </math>, so the symmetric average <math>d'_k = \fra
  5 KB (830 words) - 04:05, 28 January 2023
• 2007 AMC 12A Problems
  ...with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise? ==Problem 3==
  11 KB (1,750 words) - 13:35, 15 April 2022
• 2007 AMC 12A Problems/Problem 20
  ...d off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra? ...mathrm{(D)}\ \frac{8\sqrt{2}-11}{3}\qquad \mathrm{(E)}\ \frac{6-4\sqrt{2}}{3}</math>
  2 KB (317 words) - 10:10, 16 September 2022
• Pyramid
  ...'' is a 3-dimensional [[geometric solid]]. It consists of a [[base]] that is a [[polygon]] and a [[point]] not on the plane of the polygon, called the [ ...bh</math>, where <math>b</math> is the area of the base and <math>h</math> is the [[height]].
  2 KB (329 words) - 21:05, 20 August 2021
• 1989 USAMO Problems/Problem 2
  ...vertices and <math>14</math> edges. The sum of the degrees of the vertices is <math>28</math>; by the [[Pigeonhole Principle]] at least <math>12</math> v === Solution 3===
  3 KB (438 words) - 01:19, 27 December 2023
• 1989 USAMO Problems/Problem 1
  U_n &= \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}. .../math>, we see that <math>(a,b,c,d) = (1989,1989,1990, 2\cdot 1989)</math> is a suitable solution. <math>\blacksquare</math>
  2 KB (267 words) - 19:10, 18 July 2016
• 1989 USAMO Problems
  <math>U_n = \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}</math>. ==Problem 3==
  2 KB (326 words) - 18:52, 18 July 2016
• 2003 USAMO Problems
  A convex polygon <math> \mathcal{P} </math> in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The === Problem 3 ===
  3 KB (487 words) - 09:21, 14 May 2021
• 2003 USAMO Problems/Problem 1
  ...ach of the residue classes mod 5, <math>k </math> exists and the induction is complete. ...hen, subtracting the larger from the smaller would yield a new number that is a multiple of <math> 5^n</math> and has only even digits. We could then ha
  4 KB (736 words) - 22:17, 3 March 2023
• 2003 USAMO Problems/Problem 2
  A convex polygon <math> \mathcal{P} </math> in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The ...olygon be <math>AC </math> and <math>BD </math>. Since <math>ABCD </math> is a convex quadrilateral with sides and diagonals of rational length, we cons
  2 KB (335 words) - 20:50, 29 April 2014
• 2003 USAMO Problems/Problem 3
  '''Lemma 3.''' If <math>c_k =j </math> is not stable, then <math>t(c_k) > c_k </math>. ...ve <math> t(c_k) \ge c_k </math>. Equality implies that <math>c_k </math> is stable.
  3 KB (636 words) - 13:39, 4 July 2013
• 2007 USAMO Problems
  ...ce, when <math>n=9</math> the obtained sequence is <math>9, 1, 2, 0, 3, 3, 3, \ldots</math>. Prove that for any <math>n</math> the sequence <math>a_1, ...<math>(m,n)</math>, where <math>m</math> and <math>n</math> are integers. Is it possible to cover all grid points by an infinite family of discs with no
  3 KB (539 words) - 13:42, 4 July 2013
• 2007 USAMO Problems/Problem 1
  ...ce, when <math>n=9</math> the obtained sequence is <math>9, 1, 2, 0, 3, 3, 3, \ldots</math>. Prove that for any <math>n</math> the sequence <math>a_1, .../math> to satisfy the conditions, provided that <math>k\leq n</math>. This is true since <math>k\leq\frac{n + 1}{2}\leq n</math>, so the sequence must ev
  6 KB (1,204 words) - 20:06, 23 August 2023
• 2007 USAMO Problems/Problem 2
  ...h>(m,n)</math>, where <math>m</math> and <math>n</math> are [[integer]]s. Is it possible to cover all grid points by an infinite family of [[circle|disc ...cle with radius greater than <math>\frac{1}{\sqrt{2}}</math> between those 3 circles.
  5 KB (754 words) - 03:41, 7 August 2014
• 2007 USAMO Problems/Problem 3
  Base Case: <math>k=1</math> is trivial. The question is asking for the case when <math>k=n</math> which we have proved.
  6 KB (1,071 words) - 08:40, 21 October 2020
• 2007 USAMO Problems/Problem 4
  (''Reid Barton'') An ''animal'' with <math>n</math> ''cells'' is a connected figure consisting of <math>n</math> equal-sized [[Square (geome A ''dinosaur'' is an animal with at least 2007 cells. It is said to be ''primitive'' if its cells cannot be partitioned into two or mor
  10 KB (1,878 words) - 14:56, 30 June 2021
• 2007 USAMO Problems/Problem 5
  ...the number <math>7^{7^n}+1</math> is the [[product]] of at least <math>2n+3</math> (not necessarily distinct) [[prime]]s. ...an does <math>x + 1</math>. To confirm that <math>(x^7 + 1)/(x + 1)</math> is composite, observe that
  2 KB (240 words) - 09:47, 7 August 2014
• 2007 USAMO Problems/Problem 6
  ...tangent externally]] to <math>\omega</math>. Circle <math>\Omega_A</math> is [[internally tangent|tangent internally]] to <math>\Omega</math> at <math>A <cmath>8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3,</cmath>
  7 KB (1,274 words) - 15:11, 31 August 2017
• Nesbitt's Inequality
  '''Nesbitt's [[Inequality]]''' is a theorem which, although rarely cited, has many instructive proofs. It st \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} \ge \frac{3}{2}
  7 KB (1,224 words) - 16:21, 24 October 2022
• 2007 BMO Problems/Problem 3
  is a rational number. '''''Note''': A permutation of the set <math> \{ 1, 2, \ldots, n \} </math> is a one-to-one function of this set to itself.''
  4 KB (682 words) - 10:53, 13 January 2016
• Pascal Triangle Related Problems
  3: 1 3 3 1 1. You need to find the 6th number (remember the first number in each row is considered the 0th number) of the 10th row in Pascal's triangle.
  4 KB (513 words) - 20:18, 3 January 2023
• 2007 Cyprus MO/Lyceum/Problem 12
  ...here <math>k \in \Re</math> is a constant. The value of <math>f(-1)</math> is ...-1\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } -2\qquad \mathrm{(E) \ } 3</math>
  982 bytes (165 words) - 00:19, 19 January 2024
• 2004 IMO Shortlist Problems/N2
  ...ath> from the set <math> \mathbf{N} </math> of positive integers to itself is defined by the equality ...</math> exactly <math>\phi(n/d )</math> times, where <math>\phi(m) </math> is defined as the number of natural numbers less than or equal to <math>m </ma
  6 KB (1,007 words) - 09:10, 29 August 2011
• 2005 IMO Shortlist Problems/C2
  ''This was also Problem 3, Day 2, of the 2006 Australia [[TST]].'' ...this is clear for <math>m=0</math>, let us WLOG assume that <math>m</math> is positive. Under this assumption, we will now prove the stronger bound
  3 KB (538 words) - 09:12, 29 August 2011
• 2007 AMC 12B Problems/Problem 1
  Isabella's house has <math>3</math> bedrooms. Each bedroom is <math>12</math> feet long, <math>10</math> feet wide, and <math>8</math> fe ...eilings). Therefore, we calculate the number of square feet of walls there is in one bedroom:
  1 KB (167 words) - 10:19, 7 March 2022
• 2007 AMC 12B Problems/Problem 6
  ...site directions at the same speed. They meet at point <math>D</math>. What is <math>BD</math>? <math>\mathrm {(A)}\ 1 \qquad \mathrm {(B)}\ 2 \qquad \mathrm {(C)}\ 3 \qquad \mathrm {(D)}\ 4 \qquad \mathrm {(E)}\ 5</math>
  792 bytes (121 words) - 04:21, 15 December 2020
• 2005 IMO Shortlist Problems/A3
  '''Lemma.''' <math> \max (p,q,r,s) \le 3 </math>. ...tion of <math>(3,2,2,2) </math>, a contradiction, since we assumed <math>p>3 </math>. {{Halmos}}
  3 KB (542 words) - 17:09, 19 December 2018
• 2005 IMO Shortlist Problems/N3
  ...</math> and <math>ab+bc+ca - de-ef-fd </math>. Prove that <math>S </math> is composite. <math> (x+a)(x+b)(x+c) \equiv x^3 + (a+b+c)x^2 + (ab+bc+ca)x + abc \equiv x^3 - (d+e+f)x^2 + (de+ef+fd)x - def </math> <math> \equiv (x-d)(x-e)(x-f) \pmo
  2 KB (271 words) - 09:13, 29 August 2011
• Order of operations
  The '''order of operations''' is a [[mathematical convention]] for [[arithmetic]] computation. The below li ...m PEMDAS. An AoPS mnemonic you can use to remember the order of operations is "Please Evaluate, My Dear AoPS Students".
  2 KB (271 words) - 13:19, 5 March 2022
• Octahedron
  An '''octahedron''' is a type of [[polyhedron]]. ...eometry]], an octahedron is any polyhedron with eight [[face]]s. The term is most frequently to refer to a polyhedron with eight [[triangular]] faces, w
  1 KB (155 words) - 12:19, 21 July 2023

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