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1988 AIME Problems/Problem 9
A little bit of checking tells us that the units digit must be 2. Now our cub ...value for the hundreds digit is <math>4</math>, and so <math>442</math> is a valid solution.

6 KB (893 words) - 08:15, 2 February 2023
1988 AIME Problems/Problem 8
...ty in terms of <math>x</math> and <math>z,</math> then solve for <math>f(x,z):</math> zf(x,z-x) &= (z-x)f(x,z) \\

4 KB (538 words) - 13:24, 12 October 2021
1989 AIME Problems/Problem 15
...th>RST</math>. We'll make use of the following fact: if <math>P</math> is a point in the interior of triangle <math>XYZ</math>, and line <math>XP</math pair X = (1,2), Y = (0,0), Z = (3,0);

13 KB (2,091 words) - 00:20, 26 October 2023
1989 AIME Problems/Problem 8
...tic function of <math>k:</math> <cmath>f(k)=ak^2+bk+c,</cmath> where <math>a,b,</math> and <math>c</math> are linear combinations of <math>x_1,x_2,x_3,x f(1)&=\phantom{42}a+b+c&&=1, \\

8 KB (1,146 words) - 04:15, 20 November 2023
1990 AIME Problems/Problem 14
...ments <math>\overline{CP}</math> and <math>\overline{DP}</math>, we obtain a [[triangular pyramid]], all four of whose faces are [[isosceles triangle]]s pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6);

7 KB (1,086 words) - 08:16, 29 July 2023
1990 AIME Problems/Problem 13
...tive integers n, there's exactly one n-digit power of 9 that does not have a left-most digit 9 ...hat would result in <math>\frac{m}{9}</math> also being an n-digits, hence a contradiction. Therefore, this single n-digit power of 9 must not begin wit

5 KB (762 words) - 01:18, 10 February 2023
1990 AIME Problems/Problem 10
...The set <math>C = \{zw : z \in A ~ \mbox{and} ~ w \in B\}</math> is also a set of complex roots of unity. How many distinct elements are in <math>C_{ ...ine <math>n = e^{2\pi i/144}</math>. We can write the numbers of set <math>A</math> as <math>\{n^8, n^{16}, \ldots n^{144}\}</math> and of set <math>B</

3 KB (564 words) - 04:47, 4 August 2023
1992 AIME Problems/Problem 14
In triangle <math>ABC^{}_{}</math>, <math>A'</math>, <math>B'</math>, and <math>C'</math> are on the sides <math>BC</ma Let <math>K_A=[BOC], K_B=[COA],</math> and <math>K_C=[AOB].</math> Due to triangles <math>BOC</math> and <math>ABC</math> having the same base, <cma

4 KB (667 words) - 01:26, 16 August 2023
1992 AIME Problems/Problem 10
...1</math>, inclusive. What is the integer that is nearest the area of <math>A</math>? ...q 1</math> we have the inequality <cmath>0\leq a,b \leq 40</cmath>which is a square of side length <math>40</math>.

2 KB (323 words) - 12:05, 16 July 2019
1992 AIME Problems/Problem 9
...with center <math>P^{}_{}</math> on <math>AB^{}_{}</math> is drawn tangent to <math>BC^{}_{}</math> and <math>AD^{}_{}</math>. Given that <math>AP^{}_{}= ...<math>AP=x</math> so that <math>PB=92-x.</math> Extend <math>AD, BC</math> to meet at <math>X,</math> and note that <math>XP</math> bisects <math>\angle

5 KB (874 words) - 10:27, 22 August 2021
1993 AIME Problems/Problem 6
...ying, <math>9a = 10b + 9 = 11c + 19</math>. The relationship between <math>a,\ b</math> suggests that <math>b</math> is divisible by <math>9</math>. Als ...information given, it can be determined that, for positive integers <math>a, \ b, \ c</math>:

3 KB (524 words) - 18:06, 9 December 2023
1994 AIME Problems/Problem 10
In triangle <math>ABC,\,</math> angle <math>C</math> is a right angle and the altitude from <math>C\,</math> meets <math>\overline{AB ...ath>, <math>29^2 = x^2 - y^2 = (x-y)(x+y)</math>. As <math>x,y \in \mathbb{Z}</math>, the pairs of factors of <math>29^2</math> are <math>(1,29^2)(29,29

3 KB (534 words) - 16:23, 26 August 2018
1995 AIME Problems/Problem 12
// n = normal to plane // projection of point A onto line BC

8 KB (1,172 words) - 21:57, 22 September 2022
1995 AIME Problems/Problem 11
...1995,</math> for how many ordered triples <math>(a, b, c)</math> does such a plane exist? ...ath>P'</math> be of length <math>x,y,z</math>, such that <math>x \le y \le z</math>. Then

2 KB (292 words) - 19:30, 4 July 2013
1996 AIME Problems/Problem 14
A <math>150\times 324\times 375</math> [[rectangle|rectangular]] [[solid]] is ...oordinates of the diagonally opposite corner of the rectangle, where <math>a, b, c \in \mathbb{Z_{+}}</math>.

5 KB (923 words) - 21:21, 22 September 2023
1996 AIME Problems/Problem 13
...th> [[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 t pair B=(0,0), C=(15^.5, 0), A=IP(CR(B,30^.5),CR(C,6^.5)), E=(B+C)/2, D=foot(B,A,E);

3 KB (521 words) - 01:18, 25 February 2016
1996 AIME Problems/Problem 11
...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 <math>\mathrm {P}=r(\cos{\the 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
1997 AIME Problems/Problem 14
...math>w</math> be distinct, randomly chosen [[root]]s of the equation <math>z^{1997}-1=0</math>. Let <math>\frac{m}{n}</math> be the [[probability]] tha :<math>z^{1997}=1=1(\cos 0 + i \sin 0)</math>

5 KB (874 words) - 22:30, 1 April 2022
1997 AIME Problems/Problem 12
...math>f</math> defined by <math>f(x)= \frac{ax+b}{cx+d}</math>, where <math>a</math>,<math>b</math>,<math>c</math> and <math>d</math> are nonzero real nu ...x + b}{cx + d} + b}{c\frac {ax + b}{cx + d} + d} = x</math>, which reduces to

11 KB (2,063 words) - 22:59, 21 October 2023
1997 AIME Problems/Problem 3
...he two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah sho ...0</math>. Using [[Simon's Favorite Factoring Trick|SFFT]], this factorizes to <math>(9x-1)\left(y-\dfrac{1000}{9}\right)=\dfrac{1000}{9}</math>, and <mat

2 KB (375 words) - 19:34, 4 August 2021

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