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2005 AIME I Problems/Problem 1
...ts of external tangency, we get a [[regular polygon|regular]] [[hexagon]]. If we connect the [[vertex|vertices]] of the hexagon to the [[center]] of the

1 KB (213 words) - 13:17, 22 July 2017
2005 AIME I Problems/Problem 2
Any term in the sequence <math>S_k</math> can be written as 1+kx. If this is to equal 2005, then the remainder when 2005 is divided by k is 1.

2 KB (303 words) - 01:31, 5 December 2022
2005 AIME I Problems/Problem 4
...quare formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there a ...between <math>(n + 3)^2 + 5</math> and <math>(n + 4)^2 + 5</math>. Thus, if the number of columns is <math>n</math>, the number of students is <math>n(

8 KB (1,248 words) - 11:43, 16 August 2022
2005 AIME I Problems/Problem 5
...be <math>1\text{'s}</math>; therefore, the number of possible permutations if all the coins are indistinguishable is <math>9</math> (there are <math>8</m ...he coins below it to be arranged. If it is facing down however, there will only be one way to arrange the coins. We can the recurrence: <cmath>a_n=a_{n-1}+

5 KB (830 words) - 01:51, 1 March 2023
2005 AIME I Problems/Problem 6
If you think of each part of the product as a quadratic, then <math>((x-1)^2+\ If we don't see the fourth power, we can always factor the LHS to try to creat

4 KB (686 words) - 01:55, 5 December 2022
2005 AIME I Problems/Problem 7
...[[right triangle]]s, so <math>AE = 5</math> and <math>BF = 4</math>. Also, if we draw the horizontal line extending from <math>C</math> to a point <math>

4 KB (567 words) - 20:20, 3 March 2020
2005 AIME I Problems/Problem 8
...^3 + 4y = 2y^2 + 1</math> or <math>y^3 - 8y^2 + 16y - 4 = 0</math>. Thus, if this equation has roots <math>r_1, r_2</math> and <math>r_3</math>, by [[Vi

1 KB (161 words) - 19:50, 2 January 2022
2005 AIME I Problems/Problem 9
...a pair for each edge, and 7 edges border one of the unpainted faces while only 5 border two painted faces. Thus, the probability that two orange faces sh ...t the placement of the two unpainted faces is in fact of vital importance: if they were on opposite faces, the answer would be 0 because any placement of

4 KB (600 words) - 21:44, 20 November 2023
2005 AIME I Problems/Problem 10
...}{2}, \frac{5}{2}\right)</math>, and the sum of coordinates will be larger if we take the positive value, so <math>A = \left(\frac{35}{2} - \frac{5}2, \f

5 KB (852 words) - 21:23, 4 October 2023
2005 AIME I Problems/Problem 11
...its size when it touches both other sides of the square. This can happen only when it is arranged so that the center of the semicircle lies on one diagon Now, if we draw a line through the center, <math>O</math>, of the semicircle and it

4 KB (707 words) - 11:11, 16 September 2021
2005 AIME I Problems/Problem 12
...se product is <math>n</math>.) Thus, <math>S(n)</math> is odd if and only if there are an odd number of perfect squares less than <math>n</math>. So <m So, for a given <math>n</math>, if we choose the positive integer <math>m</math> such that <math>m^2 \leq n <

4 KB (647 words) - 02:29, 4 May 2021
2005 AIME I Problems/Problem 13
# From any lattice point <math> (a,b), </math> the particle may only move to <math> (a+1,b), (a,b+1), </math> or <math>(a+1,b+1). </math> *'''Case ''' <math>d = 1</math>: It is easy to see only <math>2</math> cases.

5 KB (897 words) - 00:21, 29 July 2022
2005 AIME I Problems/Problem 15
...enly [[trisect]]s the [[median of a triangle | median]] <math> AD. </math> If the area of the triangle is <math> m \sqrt{n} </math> where <math> m </math <math>m = 4\sqrt{2}</math> or <math>m = 0</math> if <math>m = 0</math>, we get a degenerate triangle, so <math>m = 4\sqrt{2}</m

5 KB (906 words) - 23:15, 6 January 2024
2005 AIME II Problems/Problem 15
...nt, then the distance between their centers is <math>r_1 + r_2</math>, and if they are internally tangent, it is <math>|r_1 - r_2|</math>. So we have ...e ellipse, which will mean that for that choice of <math>a</math> there is only one solution to the most recent equation. But a quadratic has one solution

12 KB (2,000 words) - 13:17, 28 December 2020
2004 AIME I Problems/Problem 15
f(x)=\begin{cases}1 & \text{if }x = 1\\ \frac x{10} & \text{if }x\text{ is divisible by 10}\\ x+1 & \text{otherwise}\end{cases} ...of <math>n</math>. Using [[complementary counting]], we see that there are only <math>2^9 - 1</math> ways.

9 KB (1,491 words) - 01:23, 26 December 2022
2004 AIME I Problems/Problem 14
...}=4\sqrt{5},</math> however this is length of the rope projected into 2-D. If we let <math>\theta</math> be the angle between the horizontal and the rope

4 KB (729 words) - 01:00, 27 November 2022
2004 AIME I Problems/Problem 10
For this solution, if you didn't know the formula for the distance from a point to a line, you ca

5 KB (836 words) - 07:53, 15 October 2023
2004 AIME I Problems/Problem 9
*If <math>D'</math> lies on <math>\overline{EF}</math>, then <math>ED' = \frac{ *If <math>D'</math> lies on <math>\overline{FG}</math>, then <math>GD' = \frac{

4 KB (618 words) - 20:01, 4 July 2013
2004 AIME I Problems/Problem 2
...s <math>1 + \frac{2m-2}{2} = m</math>. <math>2 + \frac{m-1}{2} > m</math> if <math>m < 3</math>, which is clearly not possible, thus <math>2 + \frac{m-1 ...)+m=-3</math>. Now, we make a guess and assume that <math>b-a+m=99</math> (if we get a negative value for <math>m</math>, we can try <math>b-a+m=-99</mat

8 KB (1,437 words) - 21:53, 19 May 2023
2004 AIME I Problems/Problem 4
If we imagine an arbitrary line with length <math>2</math> connecting two side ...that as a ladder slides down a wall, its midpoint traces a quarter circle (if you don't believe me, try it with your pencil). There are <math>4</math> qu

3 KB (532 words) - 09:22, 11 July 2023
2004 AIME I Problems/Problem 6
...f <math> i </math> is [[odd integer | odd]] and <math> a_i>a_{i+1} </math> if <math> i </math> is [[even integer | even]]. How many snakelike integers be ...s create the snakelike number from digits <math>a , and, if we already picked the digits there are 5 ways to do so, as said in the firs

3 KB (562 words) - 18:12, 4 March 2022
2004 AIME I Problems/Problem 8
If we join the adjacent vertices of the regular <math>n</math>-star, we get a A regular <math>n</math>-star will be formed if we choose a vertex number <math>m</math>, where <math>0 \le m \le n-1</math

4 KB (620 words) - 21:26, 5 June 2021
2004 AIME I Problems
...<a_{i+1} </math> if <math> i </math> is odd and <math> a_i>a_{i+1} </math> if <math> i </math> is even. How many snakelike integers between 1000 and 9999 f(x)=\begin{cases}1 &\mbox{if }x = 1\\ \frac x{10} &\mbox{if }x\mbox{ is divisible by 10}\\ x+1 &\mbox{otherwise}\end{cases}

9 KB (1,434 words) - 13:34, 29 December 2021
2004 AIME II Problems/Problem 15
..., k - 1}</math> values - specifically, double, and maybe <math>+ 1</math> (if the member of the pair that you're looking for is the top one at the final ...adding one every time we pair an odd to an even (but ignoring the pairing if our current square is the smaller of the two):

6 KB (899 words) - 20:58, 12 May 2022
2004 AIME II Problems/Problem 14
...particular, this invalidates the values of <math>n</math> for which their only expressions in terms of <math>(b,c)</math> fall into the inequality <math>9 All other elements of <math>S</math> are possible because if any element <math>n</math> of <math>S</math> between <math>46</math> and <m

11 KB (1,857 words) - 21:55, 19 June 2023
2004 AIME II Problems/Problem 11
If the starting point <math>A</math> is on the positive <math>x</math>-axis at

2 KB (268 words) - 22:20, 23 March 2023
2004 AIME II Problems/Problem 10
...2^{40} </math> whose binary expansions have exactly two <math>1</math>'s. If a number is chosen at random from <math> S, </math> the [[probability]] tha ...ry with 40 digits (because <math>2^{40}</math> has 41) with leading zeroes if necessary. Therefore the number of sets where there are exactly two 1’s i

8 KB (1,283 words) - 19:19, 8 May 2024
2004 AIME II Problems/Problem 7
Firstly, observe that if we are given that <math>AE=8</math> and <math>BE=17</math>, the length of t

9 KB (1,501 words) - 05:34, 30 October 2023
2004 AIME II Problems/Problem 4
First, let's count numbers with only a single digit. We have nine of these for each length, and four lengths, s If the number has a single digit, namely the number <math>n \in [1,9],</math>

3 KB (508 words) - 01:16, 19 January 2024
2004 AIME II Problems/Problem 3
...hey form a rectangular solid which is one unit shorter in each dimension. If the original block has dimensions <math>l \times m \times n</math>, we must

2 KB (377 words) - 11:53, 10 March 2014
2004 AIME II Problems
...1 and <math> 2^{40} </math> whose binary expansions have exactly two 1's. If a number is chosen at random from <math> S, </math> the probability that it

9 KB (1,410 words) - 05:05, 20 February 2019
1983 AIME Problems
...bility that at least two of the three had been sitting next to each other. If <math>P</math> is written as a fraction in lowest terms, what is the sum of ...ne of the central angle of minor arc <math>AB</math> is a rational number. If this number is expressed as a fraction <math>\frac{m}{n}</math> in lowest t

7 KB (1,104 words) - 12:53, 6 July 2022
2019 AIME I Problems/Problem 2
...m <math>1</math> through <math>18</math> such that <math>B-J\geq 2</math>. If <math>B</math> chooses <math>19</math>, <math>J</math> has choices <math>1< ...<math>B</math> can choose anything from <math>3</math> to <math>20</math>. If <math>J</math> chooses <math>2</math>, then <math>B</math> can choose anyth

5 KB (830 words) - 22:15, 28 December 2023
1984 AIME Problems
Find the value of <math>a_2+a_4+a_6+a_8+\ldots+a_{98}</math> if <math>a_1</math>, <math>a_2</math>, <math>a_3\ldots</math> is an arithmetic ...hmetic mean) of the numbers in <math>S</math> is <math>56</math>. However, if <math>68</math> is removed, the average of the remaining numbers drops to <

6 KB (933 words) - 01:15, 19 June 2022
1986 AIME Problems
If <math>\tan x+\tan y=25</math> and <math>\cot x + \cot y=30</math>, what is Determine <math>3x_4+2x_5</math> if <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, <math>x_4</math>, and

5 KB (847 words) - 15:48, 21 August 2023
1987 AIME Problems
...rdered pair <math>(m,n)</math> of non-negative integers is called "simple" if the addition <math>m+n</math> in base <math>10</math> requires no carrying. ...d the number itself. A natural number greater than 1 will be called "nice" if it is equal to the product of its distinct proper divisors. What is the sum

6 KB (869 words) - 15:34, 22 August 2023
1988 AIME Problems
Find <math>(\log_2 x)^2</math> if <math>\log_2 (\log_8 x) = \log_8 (\log_2 x)</math>. ...ane is called a mean line for the points <math>w_1, w_2, \dots, w_n</math> if <math>L</math> contains points (complex numbers) <math>z_1, z_2, \dots, z_n

6 KB (902 words) - 08:57, 19 June 2021
1989 AIME Problems
...ath>d_{}^{}</math> is a single digit in base 10. Find <math>n_{}^{}</math> if If <math>a<b<c<d<e^{}_{}</math> are consecutive positive integers such that <m

7 KB (1,045 words) - 20:47, 14 December 2023
1990 AIME Problems
If the rules are followed, in how many different orders can the eight targets ...</math> and <math>\overline{BD}</math> intersect at <math>P^{}_{}</math>. If triangle <math>ABP^{}_{}</math> is cut out and removed, edges <math>\overli

6 KB (870 words) - 10:14, 19 June 2021
1991 AIME Problems
Find <math>x^2+y^2_{}</math> if <math>x_{}^{}</math> and <math>y_{}^{}</math> are positive integers such th .../math> does the quadratic equation <math>x^2 + ax^{}_{} + 6a=0</math> have only integer roots for <math>x^{}_{}</math>?

7 KB (1,106 words) - 22:05, 7 June 2021
1992 AIME Problems
A positive integer is called ascending if, in its decimal representation, there are at least two digits and each digi Define a positive integer <math>n^{}_{}</math> to be a factorial tail if there is some positive integer <math>m^{}_{}</math> such that the decimal r

8 KB (1,117 words) - 05:32, 11 November 2023
1993 AIME Problems
...n the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went <math>\frac{n^{2}}{2}</math> miles on the <math>n^{\mbox ...b_3</math>, with the sides of the brick parallel to the sides of the box. If <math>p</math> is written as a fraction in lowest terms, what is the sum of

8 KB (1,275 words) - 06:55, 2 September 2021
1994 AIME Problems
If <math>f(19)=94,\,</math> what is the remainder when <math>f(94)\,</math> is ...duct of the non-zero digits of <math>n\,</math>. (If <math>n\,</math> has only one digit, then <math>p(n)\,</math> is equal to that digit.) Let

7 KB (1,141 words) - 07:37, 7 September 2018
1996 AIME Problems
...d yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane boar

6 KB (931 words) - 17:49, 21 December 2018
1997 AIME Problems
...on represented. A set of three cards from the deck is called complementary if all of the following statements are true: ...ig|\big| |x|-2\big|-1\Big|+\Big|\big| |y|-2\big|-1\Big|=1.</math></center> If a model of <math>S</math> were built from wire of negligible thickness, the

7 KB (1,098 words) - 17:08, 25 June 2020
1998 AIME Problems
If <math>\{a_1,a_2,a_3,\ldots,a_n\}</math> is a [[set]] of [[real numbers]], i

7 KB (1,084 words) - 02:01, 28 November 2023
2001 AIME I Problems
Call a positive integer <math>N</math> a <math>\textit{7-10 double}</math> if the digits of the base-7 representation of <math>N</math> form a base-10 nu

7 KB (1,212 words) - 22:16, 17 December 2023
2003 AIME I Problems
...een <math>1000</math> and <math>9999,</math> inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digi

6 KB (965 words) - 16:36, 8 September 2019
2001 AIME II Problems
A set of positive numbers has the <math>triangle~property</math> if it has three distinct elements that are the lengths of the sides of a trian

8 KB (1,282 words) - 21:12, 19 February 2019
2002 AIME II Problems
If <math>n=202</math>, then the area of the garden enclosed by the path, not i

7 KB (1,177 words) - 15:42, 11 August 2023
1983 AIME Problems/Problem 1
<math>x^{24}=w</math>, <math>y^{40}=w</math>, and <math>(xyz)^{12}=w</math>. If we now convert everything to a power of <math>120</math>, it will be easy t ...x^{24}=w</math>, <math>y^{40}=w</math>, and <math>(xyz)^{12}=w</math>. The only expression containing <math>z</math> is <math>(xyz)^{12}=w</math>. It now b

4 KB (642 words) - 03:14, 17 August 2022
1983 AIME Problems/Problem 3
If we were to expand by squaring, we would get a [[quartic Equation|quartic]] ...</math>, which is obviously false). Hence we have <math>y=10</math> as the only solution for <math>y</math>. Substituting <math>x^2+18x+30</math> back in f

3 KB (532 words) - 05:18, 21 July 2022
1983 AIME Problems/Problem 4
...h>OB</math>. We know how long <math>OA</math> and <math>AB</math> are, so if we can find <math>\cos \angle OAB</math>, we'll be in good shape. If we let <math>M</math> be the midpoint of <math>AC</math>, that mean that <m

11 KB (1,741 words) - 22:40, 23 November 2023
1983 AIME Problems/Problem 7
...bility that at least two of the three had been sitting next to each other. If <math>P</math> is written as a fraction in lowest terms, what is the sum of ...her, then two of the pairs from the previous case are counted. However, we only want to count this as one case, so we need to subtract the number of instan

9 KB (1,392 words) - 20:37, 19 January 2024
1983 AIME Problems/Problem 8
...et the required prime be <math>p</math>; then <math>10 \le p < 100</math>. If <math>p > 50</math>, then the factor of <math>p</math> appears twice in the

2 KB (249 words) - 23:25, 11 May 2024
1983 AIME Problems/Problem 9
Thus, if <math>3x \sin x-2=0</math>, then the minimum is obviously <math>12</math>. <math>f'(y)</math> is zero only when <math>y = \frac{2}{3}</math> or <math>y = -\frac{2}{3}</math>. It can

4 KB (722 words) - 20:25, 14 January 2023
1983 AIME Problems/Problem 10
...are both <math>1</math>. Since the thousands digit must be <math>1</math>, only one of the other three digits can be <math>1</math>. This means the possibl ...equence of <math>4</math> digits instead of a <math>4</math>-digit number. Only looking at the sequences which have one digit repeated twice, we notice tha

5 KB (855 words) - 20:26, 14 January 2023
1983 AIME Problems/Problem 13
...contains all the elements of <math>T</math> except for <math>7</math>, and only those elements . Since each element of <math>T'</math> has one fewer elemen If we look at when 6 appears, which it also does 64 times, whether it comes as

5 KB (894 words) - 22:02, 5 April 2024
1983 AIME Problems/Problem 14
...line connecting the centers, which is not true in general. It is true here only because the perpendicular from <math>P</math> passes through through the po Firstly, notice that if we reflect <math>R</math> over <math>P</math>, we get <math>Q</math>. Since

13 KB (2,149 words) - 18:44, 5 February 2024
1983 AIME Problems/Problem 15
...ne of the central angle of minor arc <math>AB</math> is a rational number. If this number is expressed as a fraction <math>\frac{m}{n}</math> in lowest t Applied to this problem, this means that <math>D</math> is the only point that lies on both (1) the given circle, and (2) the line through <mat

19 KB (3,221 words) - 01:05, 7 February 2023
1984 AIME Problems/Problem 1
Find the value of <math>a_2+a_4+a_6+a_8+\ldots+a_{98}</math> if <math>a_1</math>, <math>a_2</math>, <math>a_3\ldots</math> is an [[arithmet If <math> a_1 </math> is the first term, then <math> a_1+a_2+a_3 + \cdots + a_

4 KB (576 words) - 21:03, 23 December 2023
1984 AIME Problems/Problem 2
Any multiple of 5 ends in 0 or 5; since <math>n</math> only contains the digits 0 and 8, the units [[digit]] of <math>n</math> must be The sum of the digits of any multiple of 3 must be [[divisible]] by 3. If <math>n</math> has <math>a</math> digits equal to 8, the sum of the digits

1 KB (187 words) - 20:05, 29 May 2021
1984 AIME Problems/Problem 4
...hmetic mean) of the numbers in <math>S</math> is <math>56</math>. However, if <math>68</math> is removed, the average of the remaining numbers drops to <

2 KB (319 words) - 03:38, 16 January 2023
1984 AIME Problems/Problem 5
Determine the value of <math>ab</math> if <math>\log_8a+\log_4b^2=5</math> and <math>\log_8b+\log_4a^2=7</math>. ...nd that <math>\log a^3 b = 21\log 2 \Longrightarrow a^3 b = 2^{21}</math>. If we multiply the two equations together, we get that <math>a^4b^4 = 2^{36}</

6 KB (863 words) - 16:10, 16 May 2024
1984 AIME Problems/Problem 6
.../math> (the centers of the other two circles), and call it <math>M</math>. If we draw the feet of the [[perpendicular]]s from <math>A,C</math> to the lin

6 KB (1,022 words) - 19:29, 22 January 2024
1984 AIME Problems/Problem 7
n-3&\mbox{if}\ n\ge 1000\\ f(f(n+5))&\mbox{if}\ n<1000\end{cases}</math>

4 KB (617 words) - 22:09, 15 May 2024
1984 AIME Problems/Problem 8
We shall introduce another factor to make the equation easier to solve. If <math>r</math> is a root of <math>z^6+z^3+1</math>, then <math>0=(r^3-1)(r^ ...</math>. But <math>\theta</math> can't be <math>120^{\circ}</math> because if <math>r=\cos 120^\circ +i\sin 120^\circ </math>, then <math>r^6+r^3+1=3</ma

3 KB (430 words) - 19:05, 7 February 2023
1984 AIME Problems/Problem 9
if (i%2==0) if (n%2==0 && n!=0)

6 KB (947 words) - 20:44, 26 November 2021
1984 AIME Problems/Problem 10
..., John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over <math>80</math>, John could ...ath>s=119\Rightarrow 4c-w=89</math> so <math>w\equiv 3\pmod{4}</math>. But if <math>w=3</math>, then <math>c=23</math>, which was the result given; other

7 KB (1,163 words) - 23:53, 28 March 2022
1984 AIME Problems/Problem 11
...the oak trees; we have only two types, birch trees and "non-birch" trees. (If you don't believe this reasoning, think about it. You could also differenti ...</math> denote birch tree and not-birch tree, respectively. Notice that we only need <math>4</math> <math>n</math>s to separate the <math>5</math> <math>b<

7 KB (1,115 words) - 00:52, 7 September 2023
1984 AIME Problems/Problem 12
...(2+x)=f(2-x)</math> and <math>f(7+x)=f(7-x)</math> for all <math>x</math>. If <math>x=0</math> is a root for <math>f(x)=0</math>, what is the least numbe If <math>f(2+x)=f(2-x)</math>, then substituting <math>t=2+x</math> gives <mat

3 KB (588 words) - 14:37, 22 July 2020
1984 AIME Problems/Problem 14
If <math>x \ge 18</math> and is <math>0 \bmod{6}</math>, <math>x</math> can be If <math>x\ge 44</math> and is <math>2 \bmod{6}</math>, <math>x</math> can be

8 KB (1,346 words) - 01:16, 9 January 2024
1984 AIME Problems/Problem 15
Determine <math>x^2+y^2+z^2+w^2</math> if ...learn the above solutions. This is only a last resort and only to be used if you have too much time left. The exact amount of time this bash takes depen

6 KB (1,051 words) - 04:52, 8 May 2024
1985 AIME Problems/Problem 14
...points, and each of the two players earned <math>\frac{1}{2}</math> point if the game was a tie. After the completion of the tournament, it was found th ...=0</math>. Thus, <math>n=16</math> or <math>n=25</math>. Note however that if <math>n=16</math>, then the strongest <math>16</math> people get a total of

5 KB (772 words) - 22:14, 18 June 2020
1985 AIME Problems/Problem 13
...divisor | divides]] <math>100+n^2</math>, it must divide <math>2n+1</math> if it is going to divide the entire [[expression]] <math>100+n^2+2n+1</math>. Because <math>n</math> and <math>2n+1</math> will be coprime, the only thing stopping the GCD from being <math>1</math> is <math>n-200.</math>

4 KB (671 words) - 20:04, 6 March 2024
1985 AIME Problems/Problem 12
1 & \mathrm{if} \ k=0 \\ \frac13(1-P(k-1)) & \mathrm{if} \ k\geq1

17 KB (2,837 words) - 13:34, 4 April 2024
1985 AIME Problems/Problem 11
...s, we must have <math>F_2 Y \leq F’_2 Y</math> with equality if and only if <math>Y</math> is on the <math>x</math>-axis. Now, we have ...path from <math> F_1</math> to <math>F_2</math>., so this is only possible if we have equality and thus <math>X = Y</math>). Finding the optimal location

5 KB (932 words) - 17:00, 1 September 2020
1985 AIME Problems/Problem 10
.../math> for any integer <math>n</math> (same reasoning as above). So now we only need to test every 10 numbers; and our answer will be 100 times the number ...according to this the maximum we can get is <math>1+2+3 = 6</math>, so we only need to try the first 6 numbers.

12 KB (1,859 words) - 18:16, 28 March 2022
1985 AIME Problems/Problem 9
.../math> [[radian]]s, respectively, where <math>\alpha + \beta < \pi</math>. If <math>\cos \alpha</math>, which is a [[positive]] [[rational number]], is e

5 KB (763 words) - 16:20, 28 September 2019
1985 AIME Problems/Problem 8
...ll be larger than 1, so the largest error will be larger than 1. However, if all of the <math>A_i</math> are 2 or 3, the largest error will be less than

2 KB (377 words) - 02:17, 16 February 2021
1985 AIME Problems/Problem 5
...h>n \ge 3</math>. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 term

2 KB (410 words) - 13:37, 1 May 2022
1985 AIME Problems/Problem 4
...points closest to the opposite vertices. Find the value of <math>n</math> if the the [[area]] of the small square is exactly <math>\frac1{1985}</math>.

3 KB (484 words) - 21:40, 2 March 2020
1985 AIME Problems/Problem 3
Find <math>c</math> if <math>a</math>, <math>b</math>, and <math>c</math> are [[positive integer]] ...[prime number]]. Thus either <math>b = 1</math> or <math>b = 107</math>. If <math>b = 107</math>, <math>3a^2 - 107^2 = 1</math> so <math>3a^2 = 107^2 +

1 KB (205 words) - 18:58, 10 March 2024
1986 AIME Problems/Problem 15
The only relevant part about the xy plane here is that the slopes of the medians det ...</math>). It follows that if <math>CG = 2x</math>, <math>GD = x</math> and if <math>GB = 2y</math>, <math>GE = y</math>. By the Law of Cosines on <math>\

11 KB (1,722 words) - 09:49, 13 September 2023
1986 AIME Problems/Problem 13
...e T's into smaller groups, we will always have <math>5</math> TT sequences if we divide the additional T's among the first <math>4</math> spaces. We can

4 KB (772 words) - 21:09, 7 May 2024
1986 AIME Problems/Problem 12
...those subsets were disjoint, we would directly arrive at a contradiction; if not, we could remove the common elements to get two disjoint subsets. ...th>S</math> could have sum at least <math>62=15+14+13+11+9</math> would be if <math>S=\{ 15,14,13,11,9\}</math>. But the subsets <math>\{9,15\}</math> an

2 KB (364 words) - 19:41, 1 September 2020
1986 AIME Problems/Problem 11
Now, notice that if <math>x = -1</math>, then <math>y = 0</math>, so <math>f^{''}(-1) = g^{''}( ...end{bmatrix} ,</math> where the term <math>\dbinom{n}{k}</math> is negated if <math>n+k</math> is odd.

6 KB (872 words) - 16:51, 9 June 2023
1986 AIME Problems/Problem 10
...bc)</math>. Play the role of the magician and determine <math>(abc)</math> if <math>N= 3194</math>. ...digits the three digit number <math>(abc)</math>, so of the four options, only <math>m = \boxed{358}</math> satisfies this inequality.

3 KB (565 words) - 16:51, 1 October 2023
1986 AIME Problems/Problem 9
...rawn through <math>P</math> [[parallel]] to the sides of the [[triangle]]. If these three segments are of an equal length <math>d</math>, find <math>d</m (Note: I chose <math>F'P</math> to be <math>x</math> only because that is what I had written when originally solving. The solution wo

11 KB (1,850 words) - 18:07, 11 October 2023
1986 AIME Problems/Problem 7
...'distinct'' powers of 3, in base 3 each number is a sequence of 1s and 0s (if there is a 2, then it is no longer the sum of distinct powers of 3). Theref

5 KB (866 words) - 00:00, 22 December 2022
1986 AIME Problems/Problem 6
...63</math>. Quickly testing, we find that <math>63</math> is too large, but if we plug in <math>62</math> we find that our answer is <math>\frac{62(63)}{2 ...math>62.5</math> and <math>61.5</math> with the quadratic formula, and the only integer between the two is <math>62</math>.

2 KB (336 words) - 14:13, 6 September 2020
1986 AIME Problems/Problem 5
If <math>n+10 \mid n^3+100</math>, <math>\gcd(n^3+100,n+10)=n+10</math>. Using ...row n+10 \mid 900</math>. This is because of the property that states that if <math>a \mid b</math> and <math>a \mid c</math>, then <math>a \mid b \pm c<

2 KB (338 words) - 19:56, 15 October 2023
1986 AIME Problems/Problem 4
Determine <math>3x_4+2x_5</math> if <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, <math>x_4</math>, and

1 KB (212 words) - 16:25, 17 November 2019
1986 AIME Problems/Problem 3
If <math>\tan x+\tan y=25</math> and <math>\cot x + \cot y=30</math>, what is ...{\tan x + \tan y}{1-\tan x \tan y} = \frac{25}{1-\tan x \tan y}</math>. We only need to find <math>\tan x \tan y</math>.

3 KB (545 words) - 23:44, 12 October 2023
1987 AIME Problems/Problem 15
...<math>ABC</math>, as shown in the figures below. Find <math>AC + CB</math> if area <math>(S_1) = 441</math> and area <math>(S_2) = 440</math>.

5 KB (838 words) - 18:05, 19 February 2022
1987 AIME Problems/Problem 14
...pothesize that this will happen with the fraction as a whole. Then we will only be left with the largest term in the numerator, which is <math>(58^2 + 2 \c

7 KB (965 words) - 10:42, 12 April 2024
1987 AIME Problems/Problem 13
...h>r_n</math>, with its current predecessor and exchanging them if and only if the last term is smaller. If any of <math>r_1, \ldots, r_{19}</math> is larger than <math>r_{20}</math>,

3 KB (514 words) - 21:27, 31 December 2023
1987 AIME Problems/Problem 12
...c{3}{10^6} n + \frac{1}{10^9}</math>. For a given value of <math>n</math>, if <math>(n + \frac{1}{1000})^3 - n^3 > 1</math>, there exists an integer betw Because if its equal to, then there is no integer in between the two values. - resourc

4 KB (673 words) - 19:48, 28 December 2023
1987 AIME Problems/Problem 11
...dd, and <math>n</math> is the middle term, the sum equals <math>kn</math>. If <math>k</math> is even, then we have the sum equal to <math>kn+k/2</math>, ...ote that the factors of <math>m(2n + m + 1)</math> are of opposite parity (if <math>m</math> is odd, then <math>(2n + m + 1)</math> is even and vice vers

3 KB (418 words) - 18:30, 20 January 2024
1987 AIME Problems/Problem 10
...His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking ...pting the isolation of the <math>150</math> on one side, and the fact that if we could cancel out the <math>b</math>'s, then the <math>e</math>'s in the

7 KB (1,187 words) - 16:21, 27 January 2024
1987 AIME Problems/Problem 8
...n}{8}</math>. Thus, <math>48n < 56k < 49n</math>. <math>k</math> is unique if it is within a maximum [[range]] of <math>112</math>, so <math>n = 112</mat ...6 within the open interval between <math>48n</math> and <math>49n</math>. If <math>n=112,</math> then <math>98>k>96</math> and <math>k=97</math> is the

2 KB (393 words) - 16:59, 16 December 2020
1987 AIME Problems/Problem 7
...> and <math>2^y5^3</math>, with <math>x < 3</math> and <math>y < 3</math> (if they were equal to 3, it would overlap with case 1). Thus, there are <math> If <math>c</math> does not have a factor of <math>5^3</math>, then at least on

3 KB (547 words) - 22:54, 4 April 2016
1987 AIME Problems/Problem 6
...llel]] to <math>AB</math>. Find the [[length]] of <math>AB</math> (in cm) if <math>BC = 19</math> cm and <math>PQ = 87</math> cm.

3 KB (530 words) - 07:46, 1 June 2018
1987 AIME Problems/Problem 5
Find <math>3x^2 y^2</math> if <math>x</math> and <math>y</math> are [[integer]]s such that <math>y^2 + 3x If we move the <math>x^2</math> term to the left side, it is factorable with [

1 KB (160 words) - 04:44, 21 January 2023
1987 AIME Problems/Problem 3
...the number itself. A natural number greater than 1 will be called ''nice'' if it is equal to the product of its distinct proper divisors. What is the sum #:If we let <math>n = pq</math>, where <math>p,q</math> are the prime factors, t

3 KB (511 words) - 09:29, 9 January 2023
1987 AIME Problems/Problem 1
...r]] <math>(m,n)</math> of [[non-negative]] [[integer]]s is called "simple" if the [[addition]] <math>m+n</math> in base <math>10</math> requires no carry ...on, consider this. For every [[positive integer]] <math>m</math>, there is only one [[whole number]] <math>n</math> that you can add to it to obtain the re

1 KB (191 words) - 14:42, 17 September 2016
1988 AIME Problems/Problem 15
...hen we are duplicating an ordering from <math>\textbf{Case 1}</math>. Thus if <math>k</math> letters are in the basket after returning from lunch, then t ...nsert letter 9, counting the possibility of not having to insert it (i.e., if it arrived before lunch) as one of the cases. This yields

7 KB (1,186 words) - 10:16, 4 June 2023
1988 AIME Problems/Problem 14
...of <math>C</math> are given by <math>x=0</math> and <math>y=0</math>. Now if we represent the line <math>y=2x</math> by the complex number <math>1+2i</m Therefore, if <math>(x', y')</math> is mapped to <math>(x, y)</math> under the reflection

4 KB (700 words) - 17:21, 3 May 2021
1988 AIME Problems/Problem 13
Find <math>a</math> if <math>a</math> and <math>b</math> are integers such that <math>x^2 - x - 1< If <math>ax^n+bx^{n-1}+1</math> has <math>x^2-x-1</math> as a factor, then <ma

10 KB (1,585 words) - 03:58, 1 May 2023
1988 AIME Problems/Problem 12
...f the segments indicated in the figure. Find the product <math>abc</math> if <math>a + b + c = 43</math> and <math>d = 3</math>. If we add the equations together, we get <math>\frac{a+b+c}{3}=\frac{A^2B+A^2C

4 KB (727 words) - 23:37, 7 March 2024
1988 AIME Problems/Problem 11
...alled a mean [[line]] for the [[point]]s <math>w_1, w_2, \dots, w_n</math> if <math>L</math> contains points (complex numbers) <math>z_1, z_2, \dots, z_n ...h> must pass through the mean (the center of mass) of these points, which, if we graph them on the complex plane, is <math>(\frac{3}{5}, \frac{504i}{5})<

2 KB (422 words) - 00:22, 6 September 2020
1988 AIME Problems/Problem 9
...100}</math> we get <math>20k + 8 \equiv 88 \pmod{100}</math>. This is true if the tens digit is either <math>4</math> or <math>9</math>. Casework: ...realize that <math>600,120</math> are both multiples of <math>8</math>, we only need that <math>600k^2+120k \equiv 5\pmod{125}</math>, so we write <math>60

6 KB (893 words) - 08:15, 2 February 2023
1988 AIME Problems/Problem 4
...ximately <math>2</math> every time we choose a negative and a positive, so if we have <math>20</math> numbers, namely <math>10</math> positive and <math> ...alues. It then becomes clear that <math>|x_1|+|x_2|+...+|x_n|\ge19</math>. If each <math>x_n</math> were equal to 1, then <math>n=19</math>. However, <ma

2 KB (394 words) - 10:21, 27 January 2024
1988 AIME Problems/Problem 3
Find <math>(\log_2 x)^2</math> if <math>\log_2 (\log_8 x) = \log_8 (\log_2 x)</math>.

3 KB (481 words) - 21:52, 18 November 2020
1989 AIME Problems/Problem 15
...e area of polygon <math>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 In Solution 5, instead of finding all of <math>x, y, z</math>, we only need <math>y, z</math>. This is because after we solve for <math>y, z</math

13 KB (2,091 words) - 00:20, 26 October 2023
1989 AIME Problems/Problem 13
...s into groups of 11. Now let's examine <math>\{1, 2, \ldots , 20\}</math>. If we pick <math>1, 3, 4, 6, 9</math> from the first <math>11</math> numbers, ...89 = 180 \times 11 + 9</math>, this is not maximized. It is only maximized if we include the last element of the final set of 11, which is 10 (this is <m

2 KB (274 words) - 04:07, 17 December 2023
1989 AIME Problems/Problem 10
...th>, <math>\beta</math>, <math>\gamma</math>, be the angles opposite them. If <math>a^2+b^2=1989c^2</math>, find

8 KB (1,401 words) - 21:41, 20 January 2024
1989 AIME Problems/Problem 9
If <math>n\geq174,</math> then ...n divided by <math>5.</math> It is obvious that <math>n>133,</math> so the only possibilities are <math>n = 144</math> or <math>n \geq 174.</math> It quick

6 KB (874 words) - 15:50, 20 January 2024
1989 AIME Problems/Problem 7
If the integer <math>k</math> is added to each of the numbers <math>36</math>,

2 KB (320 words) - 15:21, 14 May 2024
1989 AIME Problems/Problem 6
Now, if we drop an altitude from point<math>M</math>, we get :

6 KB (980 words) - 15:08, 14 May 2024
1989 AIME Problems/Problem 4
If <math>a<b<c<d<e</math> are [[consecutive]] [[positive]] [[integer]]s such t ...]s. <math>c</math> is minimized if it’s [[prime factorization]] contains only <math>3,5</math>, and since there is a cubed term in <math>5^2 \cdot y^3</m

3 KB (552 words) - 12:41, 3 March 2024
1989 AIME Problems/Problem 3
...d <math>d</math> is a single [[digit]] in [[base 10]]. Find <math>n</math> if ...latively prime]], <math>4d + 1</math> must be [[divisible]] by 37, and the only digit for which this is possible is <math>d = 9</math>. Thus <math>4d + 1

3 KB (499 words) - 22:17, 29 March 2024
1990 AIME Problems/Problem 15
Find <math>ax^5 + by^5</math> if the real numbers <math>a,b,x,</math> and <math>y</math> satisfy the equatio

4 KB (644 words) - 16:24, 28 May 2023
1990 AIME Problems/Problem 14
...</math> and <math>\overline{BD}</math> intersect at <math>P^{}_{}</math>. If triangle <math>ABP^{}_{}</math> is cut out and removed, edges <math>\overli ...the Pythagorean Theorem, <math>AP = CP = DP = \frac{\sqrt{939}}{2}</math>. If we let <math>P</math> be the center of a [[sphere]] with radius <math>\frac

7 KB (1,086 words) - 08:16, 29 July 2023
1990 AIME Problems/Problem 13
If there is exactly 1 n-digit power of 9, then such a number <math>m</math> ca ...{i+1}</math>). Which is a contradiction to our assumption that the section only has 1 element. So in this case, the number doesn't have 9 as the leftmost d

5 KB (762 words) - 01:18, 10 February 2023
1990 AIME Problems/Problem 10
...plex numbers for reasoning is sharp, but note that the Frobenius Number is only constrained to positive integer pairs. Please check on "Comment of S2" belo ...d integers x and y so that ax + by = c for some integer constants a, b, c: if gcd(a, b) = 1, then any arbitrary integer c could by formed with some combi

3 KB (564 words) - 04:47, 4 August 2023
1990 AIME Problems/Problem 9
There are six slots for the heads to be placed, but only <math>5</math> heads remaining. Thus, using [[stars-and-bars]] there are <m If the first coin flipped is a T, then the remaining <math>n-1</math> flips mu

3 KB (425 words) - 12:36, 12 May 2024
1990 AIME Problems/Problem 8
If the rules are followed, in how many different orders can the eight targets

3 KB (491 words) - 04:24, 4 November 2022
1990 AIME Problems/Problem 7
...e bisector theorem as in solution 1, we find that <math>QP' = 25/2</math>. If we draw the right triangle formed by <math>Q, P',</math> and the point dire ...is, substitute values into the <math>[\triangle ABC] = rs</math> equation. If lines are drawn from the incenter perpendicular to <math>PR</math> and <mat

8 KB (1,319 words) - 11:34, 22 November 2023
1990 AIME Problems/Problem 3
...h>. Although, the problem asks for <math>s</math>, not <math>r</math>. The only conceivable reasoning behind this is that <math>r</math> is greater than 10

3 KB (516 words) - 19:18, 16 April 2024
1990 AIME Problems/Problem 1
...n for the smallest square or cube greater than <math>500</math>. However, if we use <math>n=8^3=512</math>, the number of terms in <math>T</math> will e

2 KB (283 words) - 23:11, 25 June 2023
2006 AMC 10B Problems/Problem 5
Note: If you do this, always check to see if it fits, because this doesn't always work. For example, a 3x3 and a 3x4 doe

1 KB (242 words) - 18:35, 15 August 2023
2006 AMC 10B Problems/Problem 7
...ng a negative value into <math>x</math> can help. For ease of computation, if <math>x=-3</math>, <math>\sqrt{\frac{-3}{1-\frac{4}{3}}}=\sqrt{\frac{-3}{\f Note that if <math>x=-1</math> was chosen, C would have fit as well. Make sure to avoid

1 KB (179 words) - 10:33, 19 August 2022
1991 AIME Problems/Problem 15
...math>(m - n^2)(m + n^2) = 289\cdot 1 = 17\cdot 17 = 1\cdot 289.</math> The only possible value, then, for <math>m</math> is <math>145</math>, in which case ...thus the minimum value, with equality when all the tangents are equal. The only value for which <math>\sqrt {289 + n^{4}}</math> is an integer is <math>n =

4 KB (658 words) - 16:58, 10 November 2023
1991 AIME Problems/Problem 12
...is <math>12</math> away from point <math>E</math>. Repeating the process, if we break down triangle <math>DER</math> into two more similar triangles, we .... We observe that <math>[PQRS]=\frac{1}{2}\cdot30\cdot40=600</math>. Thus, if we drop an altitude from <math>P</math> to <math>\overline{SR}</math> to po

8 KB (1,270 words) - 23:36, 27 August 2023
2006 AMC 10B Problems/Problem 16
There are <math>365</math> days in a year, plus <math>1</math> extra day if there is a Leap Day, which occurs on years that are multiples of <math>4</m ...there is one day extra after the <math>52</math> weeks, we can deduce that if we were to travel forward <math>x</math> amount of non-leap years, then the

2 KB (336 words) - 10:51, 11 May 2024
1991 AIME Problems/Problem 10
If <math>S(a,1)=S(b,1)</math>, then you do the same for the second letters of Similarly, if <math>S(a,2)=S(b,2)</math>, then there is a <math>(4/9)^3</math> chance tha

5 KB (813 words) - 06:10, 25 February 2024
1991 AIME Problems/Problem 9
If we square the given <math>\sec x = \frac{22}{7} - \tan x</math>, we find th ...h>0 = 435y^2 - 616y - 435 = (15y - 29)(29y + 15)</math>. It turns out that only the [[positive]] root will work, so the value of <math>y = \frac{29}{15}</m

10 KB (1,590 words) - 14:04, 20 January 2023
1991 AIME Problems/Problem 7
...</math>, from which we realize that <math>f(x) = x</math>. This is because if we expand the entire expression, we will get a fraction of the form <math>\

2 KB (285 words) - 05:15, 13 June 2022
1991 AIME Problems/Problem 6
THIS SOLUTION IS INCORRECT, PLEASE CORRECT IT IF YOU HAVE TIME!

3 KB (447 words) - 17:02, 24 November 2023
1991 AIME Problems/Problem 5
...bination]] of numbers for <math>a</math>; however, since <math>a<b</math>, only half of them will be between <math>0 < \frac{a}{b} < 1</math>. Therefore, t

919 bytes (141 words) - 20:00, 4 July 2022
1991 AIME Problems/Problem 3
...ing the well-known fact <math>\log(a_{}^{}b)=\log a + \log b</math> (valid if <math>a_{}^{},b_{}^{}>0</math>), we have ...th>\log\big[\;\big]</math> terms (recall that <math>\log y_{}^{} <0</math> if <math>0<y_{}^{}<1</math>). Therefore, the integer <math>k_{}^{}</math> that

5 KB (865 words) - 12:13, 21 May 2020
1991 AIME Problems/Problem 2
...would have length <math>\frac{1}{168}\cdot{5}</math> by similar triangles. If you add the two lengths together, it is <math>\frac{167}{168}\cdot{5} + \fr

4 KB (595 words) - 12:51, 17 June 2021
1991 AIME Problems/Problem 1
Find <math>x^2+y^2_{}</math> if <math>x_{}^{}</math> and <math>y_{}^{}</math> are positive integers such th ...uation factors to <math>(x + y)xy = 880 = 2^4 \cdot 5 \cdot 11</math>. The only set with a factor of <math>11</math> is <math>(5,11)</math>, and checking s

4 KB (628 words) - 22:05, 7 June 2021
1992 AIME Problems/Problem 15
Define a positive integer <math>n^{}_{}</math> to be a factorial tail if there is some positive integer <math>m^{}_{}</math> such that the decimal r Note that if <math>m</math> is a multiple of <math>5</math>, <math>f(m) = f(m+1) = f(m+2

2 KB (358 words) - 01:54, 2 October 2020
1992 AIME Problems/Problem 13
...b^2</math> (this follows directly from the [[trivial inequality]], because if <math>{x^2 \ge 0}</math> then plugging in <math>a+n</math> for <math>x</mat ...ions -- the writer of Solution 2 gave us a pretty good example of checking if the AM-GM equality can be obtained. ~Will_Dai

4 KB (703 words) - 02:40, 29 December 2023
1992 AIME Problems/Problem 12
...square is eaten, the squares to the right of it must also be eaten. Thus, if <math>a_i</math> is the number of squares remaining on row <math>i</math>,

2 KB (443 words) - 22:41, 22 December 2021
1992 AIME Problems/Problem 9
...<math>CD</math> are parallel, <math>\triangle XCD ~ \triangle XAB</math>. If <math>AX</math> is further extended to a point <math>A'</math> and <math>XB

5 KB (874 words) - 10:27, 22 August 2021
1992 AIME Problems/Problem 6
...>c \in \{0, 1, 2, 3, 4\}</math>. <math>1abc + 1ab(c+1)</math> has no carry if <math>a, b \in \{0, 1, 2, 3, 4\}</math>. This gives <math>5^3=125</math> po ...abc</math> ends in <math>00</math>, which we will address later. Clearly, if <math>c \in \{0, 1, 2, 3, 4 ,5\}</math>, then adding <math>(1abc) + (1abc -

3 KB (455 words) - 02:03, 10 July 2021
1992 AIME Problems/Problem 5
If <math>abc</math> is not divisible by <math>3</math> or <math>37</math>, the

2 KB (277 words) - 20:45, 4 March 2024
1992 AIME Problems/Problem 2
A [[positive integer]] is called ascending if, in its [[decimal representation]], there are at least two digits and each ...d by its [[digit]]s: for any [[set]] of digits (not including 0, since the only position for 0 is at the leftmost end of the number, i.e. a leading 0), the

2 KB (336 words) - 05:18, 4 November 2022
1992 AIME Problems/Problem 1
Note that if <math>x</math> is a solution, then <math>(300-x)</math> is a solution. We k

1 KB (190 words) - 20:02, 23 February 2022
2006 AMC 10B Problems/Problem 20
...u know that, you can guarantee that the answer must be a multiple of 101. (If you're wondering why there can't be a fraction like <math>1/101</math>, the

4 KB (594 words) - 15:45, 30 July 2023
2006 AMC 10B Problems/Problem 25
...et of the ages of Mr. Jones' children (in other words <math>i \in S</math> if Mr. Jones has a child who is <math>i</math> years old). Then <math>|S| = 8< ...</math>. Since <math>11\cdot504 = 5544</math> and <math>5544</math> is the only <math>4</math> digit multiple of <math>504</math> that fits all the conditi

5 KB (878 words) - 14:39, 3 December 2023
1993 AIME Problems/Problem 15
...ath> and <math>BCH^{}_{}</math> are tangent to <math>\overline{CH}</math>. If <math>AB = 1995\,</math>, <math>AC = 1994\,</math>, and <math>BC = 1993\,</ ...be shown that in any triangle with side lengths <math>n-1, n, n+1</math>, if you draw an altitude from the vertex to the side of <math>n+1</math>, and d

3 KB (449 words) - 21:39, 21 September 2023
1993 AIME Problems/Problem 14
...bed in a larger rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its Note that such a rectangle is unstuck if its four vertices are in or on the edge of all four quadrants, and it is no

3 KB (601 words) - 09:25, 19 November 2023
1993 AIME Problems/Problem 13
...ount of time, in seconds, before Jenny and Kenny can see each other again. If <math>t\,</math> is written as a fraction in lowest terms, what is the sum If we bring <math>\overline{AB}</math> up to where the point J is, we have by

8 KB (1,231 words) - 20:06, 26 November 2023
1993 AIME Problems/Problem 12
...</math> are generated by rolling the die repeatedly and applying the rule: If the die shows label <math>L\,</math>, where <math>L \in \{A, B, C\}</math>, If we have points <math>(p,q)</math> and <math>(r,s)</math> and we want to fin

4 KB (611 words) - 13:59, 15 July 2023
1993 AIME Problems/Problem 11
...occurs and is a head is <math>\frac{1}{2^n}</math>. The first person wins if the coin lands heads on an odd numbered flip. So, the probability of the fi ...that Alfred wins the <math>n+1</math>th game is <math>\frac{1}{3}</math>. If Bonnie wins the <math>n</math>th game, then the probability that Alfred win

7 KB (1,058 words) - 20:57, 22 December 2020
1993 AIME Problems/Problem 10
...ns are concurrent. Now, the number of edges <math>E</math> can be obtained if we count the number of sides that each triangle and pentagon contributes: < ...-30 \implies V = E-30</math> based off the problem condition. Furthermore, if we draw out a few pentagons as well as triangles on each of side of the pen

4 KB (716 words) - 20:50, 17 April 2022
1993 AIME Problems/Problem 9
...m 1) A number <math>n</math> will only occupy the same point on the circle if <math>\frac12(n)(n + 1)\equiv \frac12(1993)(1994) \pmod{2000}</math>. ...on the same point if <math>\ a(a+1)/2\equiv \ b(b+1)/2\pmod{2000}</math>. If we assume the final answer be <math>n</math>, then we have <math>\frac12(n)

3 KB (488 words) - 02:06, 22 September 2023
1993 AIME Problems/Problem 8
...hile the rest of the “different” cases are counted twice, this one was only counted once so we should subtract one from the total count, divide by two, ...lement set, namely <math>\{\{1,2,\ldots,n-1\},\{1,2,\ldots,n-1\}\},</math> only contributes two unordered pairs to the <math>n</math>-element set. The tabl

9 KB (1,400 words) - 14:09, 12 January 2024
1993 AIME Problems/Problem 7
...b_3</math>, with the sides of the brick parallel to the sides of the box. If <math>p</math> is written as a fraction in lowest terms, what is the sum of *If <math>x_2</math> is a dimension of the box, then any of the other three rem

5 KB (772 words) - 09:04, 7 January 2022
1955 AHSME Problems/Problem 27
If <math>r</math> and <math>s</math> are the roots of <math>x^2-px+q=0</math>,

600 bytes (108 words) - 10:47, 15 February 2021
1993 AIME Problems/Problem 4
...th>. The last two solutions don't follow <math>a , so we only need to consider the first two solutions. ...a+b</math> must differ by 2 as well, but an odd number <math>93</math> can only result from two numbers of different parity. <math>c-d</math> will be even,

8 KB (1,343 words) - 16:27, 19 December 2023
1993 AIME Problems/Problem 2
...n the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went <math>n^{2}_{}/2</math> miles on the <math>n^{\mbox{th}}

2 KB (241 words) - 11:56, 13 March 2015
1994 AIME Problems/Problem 15
...us call <math>P_{}^{}</math> a fold point of <math>\triangle ABC\,</math> if these creases, which number three unless <math>P^{}_{}</math> is one of the ...shaded region below) is simply the sum of two [[segment]]s of the circles. If we construct the midpoints of <math>M_1, M_2 = \overline{AB}, \overline{BC}

4 KB (717 words) - 22:20, 3 June 2021
1994 AIME Problems/Problem 12
...ired to partition the field is clearly <math>52*(24+1) + 24(52+1)</math>. (If you are confused, just draw the square out). This is clearly greater than <

3 KB (473 words) - 17:06, 1 January 2024
1994 AIME Problems/Problem 9
...are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the dra ...c{1}{11}</math> chance for this to happen. We remain with the case [BCDEF] if this is the case.

3 KB (589 words) - 14:18, 21 July 2019
1994 AIME Problems/Problem 6
if((i%10) == 0){draw(pica,(-20/sqrt(3)-abs((0,i))/sqrt(3),i)--(20/sqrt(3)+abs( ...of <math>2\cdot 21^2=882</math> unit equilateral triangles. Of course, we only draw <math>21</math> horizontal lines, so we are overcounting the triangles

4 KB (721 words) - 16:14, 8 March 2021
1994 AIME Problems/Problem 5
...duct of the non-zero digits of <math>n\,</math>. (If <math>n\,</math> has only one digits, then <math>p(n)\,</math> is equal to that digit.) Let

2 KB (275 words) - 19:27, 4 July 2013
1994 AIME Problems/Problem 4
Note that if <math>2^x \le a<2^{x+1}</math> for some <math>x\in\mathbb{Z}</math>, then <

2 KB (264 words) - 13:33, 11 August 2018
1994 AIME Problems/Problem 3
If <math>f(19)=94,\,</math> what is the remainder when <math>f(94)\,</math> is

2 KB (252 words) - 11:12, 3 July 2023
1995 AIME Problems/Problem 15
...f the form <math>(1/2)^a</math> is <math>1/2+1/4+1/8+1/16=15/16</math>, so if there are n blocks of <tt>H</tt>'s before the final five <tt>H</tt>'s, the ...ve tails. Specifically, <math>h_{5} = 0</math> and <math>t_{2} = 1</math>. If we can solve for <math>h_{1}</math> and <math>t_{1}</math>, we are done; th

6 KB (979 words) - 13:20, 11 April 2022
1995 AIME Problems/Problem 12
// if you see this

8 KB (1,172 words) - 21:57, 22 September 2022
1995 AIME Problems/Problem 11
...ths in common. Since <math>x < a, y < b, z < c</math>, it follows that the only possibility is <math>y=a,z=b=1995</math>. Then, ...s of <math>3^25^27^219^2</math> is <math>(2+1)(2+1)(2+1)(2+1) = 81</math>. Only in <math>\left\lfloor \frac {81}2 \right\rfloor = 40</math> of these cases

2 KB (292 words) - 19:30, 4 July 2013
1995 AIME Problems/Problem 10
...ath>0\mod{5}</math>, then <math>b = 5</math> because <math>5</math> is the only prime divisible by <math>5</math>. We get <math> n = 215</math> as our larg If <math>b</math> is <math>1\mod{5}</math>, then <math>b + 2 \times 42</math>

3 KB (436 words) - 19:26, 2 September 2023
1995 AIME Problems/Problem 9
...rg}(11+xi)</math> and <math>\angle BDM=\text{Arg}(1+xi)</math>. So we need only find <math>x</math> such that <math>\text{Arg}((11+xi)^3)=\text{Arg}(1331-3

7 KB (1,181 words) - 13:47, 3 February 2023
1995 AIME Problems/Problem 8
 {{note|1}} Another way of stating this is to note that if <math>\frac{x}{y}</math> and <math>\frac{x+1}{y+1}</math> are integers, the ...for <math>y</math>, then <math>x = ky = ((y+1)a+1)y = y(y+1)a+y</math>. We only have to check values of <math>y</math> when <math>y(y+1)<100</math>. This y

4 KB (646 words) - 17:37, 1 January 2024
1995 AIME Problems/Problem 6
...ath>(62+1)\times (38+1)</math> [[factor]]s by its [[prime factorization]]. If we group all of these factors (excluding <math>n</math>) into pairs that mu

2 KB (407 words) - 08:14, 4 November 2022
1995 AIME Problems/Problem 3
If the object took <math>4</math> steps, then it must have gone two steps <tt> .../tt> can be permuted in <math>\frac{6!}{3!2!1!} = 60</math> ways. However, if the first four steps of the sequence are <tt>N,N,E,E</tt> in some permutati

3 KB (602 words) - 23:15, 16 June 2019
1995 AIME Problems/Problem 1
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 14
...pectively. The total cuts by the diagonal is therefore <math>a+b+c</math>, if we can ensure that no more than <math>1</math> positive integer is present

5 KB (923 words) - 21:21, 22 September 2023
1996 AIME Problems/Problem 12
...of times this value appears. Each occurs <math>5 \cdot 8!</math>, because if you fix <math>a_n</math> and <math>a_{n + 1}</math> there are still <math>8 ..._4 + a_6 + a_8 + a_{10})</math>. Due to the symmetry of this situation, we only need to compute the [[expected value]] of the result. <math>10</math> must

5 KB (879 words) - 11:23, 5 September 2021
1996 AIME Problems/Problem 11
...sixth roots of unity that are not the second or third roots of unity. The only roots in this category are <math>\mathrm{cis} (60, 72, 144)</math>, so our

6 KB (1,022 words) - 20:23, 17 April 2021
1996 AIME Problems/Problem 10
== Solution 3 (Only sine and cosine sum formulas) == ...ce <math>\sin{45^{\circ}} = \cos{45^{\circ}} = \tfrac{\sqrt{2}}{2}</math>, if <math>\alpha = \tfrac{\sqrt{2}}{2}</math> we have

4 KB (503 words) - 15:46, 3 August 2022
1996 AIME Problems/Problem 9
...d. Then he opens the lockers that are multiples of <math>4</math>, leaving only lockers <math>2 \pmod{8}</math> and <math>6 \pmod{8}</math>. Then he goes a ...h>2^{n-1}</math> lockers) corresponds to the locker <math>2^n+2-2y</math> (if he started with <math>2^n</math> lockers). It follows that <math>L_{n} = 2^

3 KB (525 words) - 23:51, 6 September 2023
1996 AIME Problems/Problem 7
...d yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a [[rotation]] in the plane ...tyle="font-size:85%">For those symmetric about the center, there is only one other.</td></tr></table></center>

4 KB (551 words) - 11:44, 26 June 2020
1996 AIME Problems/Problem 3
If we expand the exponent the expression becomes <math>\underbrace{(xy+x+y+1)\

3 KB (515 words) - 04:29, 27 November 2023
1997 AIME Problems/Problem 13
...\Big|\big||x|-2\big|-1\Big|+\Big|\big||y|-2\big|-1\Big|=1.</math></center> If a model of <math>S</math> were built from wire of negligible thickness, the ...= 1 \Longrightarrow f(x), f(y) \le 1 \Longrightarrow x, y \le 4</math>. We only have a <math>4\times 4</math> area, so guessing points and graphing won't b

7 KB (1,225 words) - 19:56, 4 August 2021
1997 AIME Problems/Problem 12
The only value that is not in the range of this function is <math>\frac {a}{c}</math considered equivalent if they are scalar multiples of each other. Similarly,

11 KB (2,063 words) - 22:59, 21 October 2023
1997 AIME Problems/Problem 10
...on represented. A set of three cards from the deck is called complementary if all of the following statements are true: ...thermore, should the two be different there is only one option- choose the only value that is remaining. In this way, every two card pick corresponds to ex

3 KB (585 words) - 19:37, 25 April 2022
1997 AIME Problems/Problem 8
...d and fourth columns, the two rows with shared 1s or -1s are fixed, so the only things that can be changed is the orientation of the mixed rows, in <math>2 ...he grid. If it goes like 1,1, -1, -1, there is 1 way to complete the grid. If it goes like 1, -1, -1, 1, then there are 2 ways to complete the grid. So o

4 KB (638 words) - 16:41, 22 January 2024
1997 AIME Problems/Problem 4
If (in the diagram above) we draw the line going through the centers of the ci

2 KB (354 words) - 22:33, 2 February 2021
1997 AIME Problems/Problem 3
...or on factors of 1000. If <math>9x - 1 = 100</math>, we get a non-integer. If <math>9x - 1 = 125</math>, we get <math>x=14</math> and <math>y=112</math>, ...e factors 2 or 5 and is <math>8\pmod9</math>. It is quick to find there is only one: 125. That gives 14 as <math>10a+b</math> and 112 as <math>100x+10y+z</

2 KB (375 words) - 19:34, 4 August 2021
1998 AIME Problems/Problem 15
Thus, if <math>n</math> is even, each integer different from <math>i</math> and <mat ...ervation argument, but you probably shouldn't) <math>n</math> is even, and if you try to apply the odd case to the even case, (namely that there is <math

9 KB (1,671 words) - 22:10, 15 March 2024
1998 AIME Problems/Problem 13
If <math>\{a_1,a_2,a_3,\ldots,a_n\}</math> is a [[set]] of [[real numbers]], i ...f size 1 has a 9, then its power sum must be <math>9i</math>, and there is only <math>1</math> of these such subsets. There are <math>{8\choose1}</math> wi

2 KB (384 words) - 19:02, 20 October 2023
1998 AIME Problems/Problem 10
...d the radii [[perpendicular]] to the flat surface, we get a [[trapezoid]]; if we draw the segment parallel to the surface that connects the center of the

3 KB (496 words) - 13:02, 5 August 2019
1998 AIME Problems/Problem 9
...math>M_1</math> comes (a total range of <math>2m</math> minutes). However, if <math>M_1</math> comes into the cafeteria in the first or last <math>m</mat ...ave to enter the cafeteria within <math>m</math> minutes of each other; so if we fix point <math>M_1</math> then <math>M_2</math> has a <math>\frac{2m}{6

4 KB (624 words) - 18:34, 18 February 2018
1998 AIME Problems/Problem 5
...and -1 if <math>n</math> is odd. <math>\frac {k(k-1)}2</math> will be even if <math>4|k</math> or <math>4|k-1</math>, and odd otherwise. If we group the terms in pairs, we see that we need a formula for <math>-\frac

1 KB (225 words) - 02:20, 16 September 2017
1998 AIME Problems/Problem 4
...ters in what order the people pick the tiles; the final answer is the same if we assume the opposite, that order doesn't matter.) In this case, there are only two ways to get an odd sum. Either have the sequence <math>OOO</math> or <m

5 KB (917 words) - 02:37, 12 December 2022
1998 AIME Problems/Problem 3
...the second one, we get <math>y = 20</math> or <math> y = -2x - 20</math>. If we graph these four equations, we see that we get a parallelogram with base

1 KB (198 words) - 20:13, 23 February 2018
1998 AIME Problems/Problem 2
...and the above conditions are satisfied. But <math>2y\le60</math>, so this only works for <math>x\le15</math>. Thus, there are ...ing the difference between adjacent terms (1,3,3,5,5,...). We suspect that if 60 was replaced with 2n, we will find 1+3+3+5+5+7+7 ...., where there will

6 KB (913 words) - 16:34, 6 August 2020
1999 AIME Problems/Problem 11
*Only if <math>x</math> is in radians, which it is not. However, the solution is sti

4 KB (614 words) - 04:38, 8 December 2023
1999 AIME Problems/Problem 10
...d form a triangle by 4. We can pick any segment for the first choice, then only segments that share an endpoint with the first one, then the one segment th If we have <math>6, 7,</math> or <math>8</math> endpoints, it is easy to see t

3 KB (524 words) - 17:25, 17 July 2023
1999 AIME Problems/Problem 5
...don't, the difference must be bumping the number up a ten, a hundred, etc. If we take <math>T(a999)</math> as an example,

1 KB (170 words) - 19:40, 4 July 2013
1999 AIME Problems/Problem 3
If <math>n^2-19n+99=x^2</math> for some positive integer <math>x</math>, then So if <math> n \geq 12</math> and <math> n^2 -19n + 99 </math> is a perfect squar

2 KB (296 words) - 01:18, 29 January 2021
1999 AIME Problems/Problem 2
...th>(28, 153-a)</math>. For two given points, the line will pass the origin if the coordinates are [[proportion]]al (such that <math>\frac{y_1}{x_1} = \fr Edit by ngourise: if u dont think this is the best solution then smh

3 KB (423 words) - 11:06, 27 April 2023
1999 AIME Problems/Problem 1
...be a number in the sequence that is divisible by <math>3</math>. However, if the common difference is <math>6</math>, we find that <math>5,11,17,23</mat If we let the arithmetic sequence to be <math>p, p+a, p+2a, p+3a</math>, and <

2 KB (332 words) - 13:22, 3 August 2020
2000 AIME I Problems/Problem 15
...o on (which is a very simple bash, as <math>2000</math> is a small number. If you don't want to do this, define sequence <math>a_n = 2a_{n-1} - 1</math>, ...th>n-1</math> card, and the new card "<math>1</math>" is added to the top. If <math>a_n = n-1</math> (meaning the card <math>n-1</math> is the bottom car

15 KB (2,673 words) - 19:16, 6 January 2024
2000 AIME I Problems/Problem 13
...>x</math> miles, <math>t=\frac{d}{r}=\frac{x}{50}</math> hours has passed. If the truck leaves the highway it can travel for at most <math>t=\frac{1}{10} ...the line <math>y=5-\frac{24}{7}x</math> bounds the region the truck can go if it moves in positive y direction. The intersection of these two lines is <m

3 KB (571 words) - 00:38, 13 March 2014
2000 AIME I Problems/Problem 11
...mber is in the denominator. Now we simply combine and reduce these groups. If the powers of 2 are on the denominator, then every power of five will be mu ...econd case can actually be discarded, but still can be found. Realize that if we include the powers of 2 and 5, then the second case is (sum of all divis

4 KB (667 words) - 13:58, 31 July 2020
2000 AIME I Problems/Problem 9
...y = \log 2 + 1 = \log 2 + \log 10 = \log 20,</math> so <math>y=20.</math> if <math>\pm</math> is negative then <math>\log y = 1 - \log 2 = \log 10 - \lo If <math>1-\log y=0</math> then <math>y=10</math>. Substituting into the first

4 KB (623 words) - 15:56, 8 May 2021
2000 AIME I Problems/Problem 8
...n be found by <math>V = \frac{\pi}{3}r^2h</math>. In the second container, if we let <math>h',r'</math> represent the height, radius (respectively) of th

4 KB (677 words) - 16:33, 30 December 2023
2000 AIME I Problems/Problem 7
...d, we can expand this to get <math>xyz+xy+yz+xz+x+y+z+1 = 145k</math>, and if we make a substitution for <math>xyz</math>, and rearrange the terms, we ge

5 KB (781 words) - 15:02, 20 April 2024
2000 AIME I Problems/Problem 6
Because <math>y > x</math>, we only consider <math>+2</math>. ...are not less than <math>y</math>, so <math>y</math> cannot be equal to 1. If <math>y = 2^2 = 4</math>, you get <math>(x - 8)^2 = 64</math>, which has 2

6 KB (966 words) - 21:48, 29 January 2024
2000 AIME I Problems/Problem 5
If we work with the problem for a little bit, we quickly see that there is no ...duct of the number of black marbles in each box is <math>54</math>, so the only combination that works is <math>18</math> black in first box, and <math>3</

7 KB (1,011 words) - 20:09, 4 January 2024
2000 AIME I Problems/Problem 4
...ould need to scale up by a factor of <math>2</math> to make them integers; if we started with <math>a_1 > 2</math> even, the resulting dimensions would n

3 KB (485 words) - 00:31, 19 January 2024
2000 AIME I Problems/Problem 2
...math>C = (-v,u)</math>, <math>D = (-v,-u)</math>, <math>E = (v,-u)</math>. If we graph those points, we notice that since the latter four points are all

3 KB (434 words) - 22:43, 16 May 2021
2000 AIME I Problems/Problem 1
If a factor of <math>10^{n}</math> has a <math>2</math> and a <math>5</math> i

1 KB (163 words) - 17:44, 16 December 2020
2001 AIME I Problems/Problem 15
...o notice that an octagonal path can jump from one tetrahedron to the other only along one of the long diagonals. It follows that an octagon must contain ei ...ch, because it must start and finish at specified vertices, can be done in only 2 ways. Since this counting method treats each path as different from its r

11 KB (1,837 words) - 18:53, 22 January 2024
2001 AIME I Problems/Problem 14
...digits of any <math>n</math>-digit string can't be <math>11</math>, so the only possibilities are <math>00</math>, <math>01</math>, and <math>10</math>. If an <math>n</math>-digit string ends in <math>00</math>, then the previous d

13 KB (2,298 words) - 19:46, 9 July 2020
2001 AIME I Problems/Problem 12
Additionally, if <math>(r,r,r)</math> is on the other side of <math>ABC</math>, we have <mat ...<math>+/-(11r/7-12/7)</math>. We take the negative value of this because if we plug in <math>(0,0,0)</math> to the equation of the plane we get a negat

6 KB (1,050 words) - 18:44, 27 September 2023
2014 USAJMO Problems
...is move, <math>B</math> may choose any counter on the board and remove it. If at any time there are <math>k</math> consecutive grid cells in a line all o

3 KB (600 words) - 16:42, 5 August 2023
2001 AIME I Problems/Problem 11
If we express all the <math>c_i</math> in terms of <math>N</math>, we have ...t there exists such an array satisfying the problem conditions if and only if

3 KB (493 words) - 13:51, 22 July 2020
2001 AIME I Problems/Problem 10
...rks, the two points must have this same type of classification (otherwise, if one doesn't match, the resulting sum for the coordinates will be odd at tha ...if a value is zero, then the segment will pertain to only two dimensions. If two values are zero then the line segment becomes one dimensional.

8 KB (1,187 words) - 02:40, 28 November 2020
2001 AIME I Problems/Problem 9
/* -- arbitrary values, I couldn't find nice values for pqr please replace if possible -- */ /* -- arbitrary values, I couldn't find nice values for pqr please replace if possible -- */

4 KB (673 words) - 20:15, 21 February 2024
2001 AIME I Problems/Problem 8
Call a positive integer <math>N</math> a ''7-10 double'' if the digits of the base-<math>7</math> representation of <math>N</math> form Given this is an AIME problem, <math>A<1000</math>. If we look at <math>B</math> in base <math>10</math>, it must be equal to <mat

3 KB (502 words) - 11:28, 9 December 2023
2001 AIME I Problems/Problem 6
If we take any [[combination]] of four numbers, there is only one way to order them in a non-decreasing order. It suffices now to find th *If <math>a - d = 0</math>, then the values of <math>b,\ c</math> are set, and

11 KB (1,729 words) - 20:50, 28 November 2023
2001 AIME I Problems/Problem 5
...math> to <math>(x/2, y).</math> This sends the ellipse to the unit circle. If we let <math>n</math> be one-fourth of the side length of the triangle, the

6 KB (1,043 words) - 10:09, 15 January 2024
2001 AIME I Problems/Problem 3
...h}}</math> degree polynomial. Note that there are no multiple roots. Thus, if <math>\frac{1}{2} - x</math> is a root, <math>x</math> is also a root. Thus Note that if <math>r</math> is a root, then <math>\frac{1}{2}-r</math> is a root and the

2 KB (335 words) - 18:38, 9 February 2023
2001 AIME I Problems/Problem 1
...ath>10a</math>. Thus <math>b = a, 2a,</math> or <math>5a</math> (note that if <math>b = 10a</math>, then <math>b</math> would not be a digit). If we ignore the case <math>b = 0</math> as we have been doing so far, then th

4 KB (687 words) - 18:37, 27 November 2022
2002 AIME I Problems/Problem 13
...ur answer. (One may just get the area via triangle similarity too--this is if you are tired by the end of test and just want to bash some stuff out--it m

6 KB (974 words) - 13:01, 29 September 2023
2002 AIME I Problems/Problem 11
...nates for the cube vertices. The coordinates will all involve 0's and 12's only, so that means that the X, Y, and Z distance traveled by the light must all ...r words, we can take the cube with the first photon, translate it and flip if necessary, to get the cube with the other photon.

3 KB (591 words) - 15:11, 21 August 2019
2002 AIME I Problems/Problem 9
...be <math>2</math>, or that will result in painting the third picket twice. If <math>h=3</math>, then <math>t</math> may not equal anything not divisible If <math>h</math> is <math>4</math>, then <math>t</math> must be even. The sam

4 KB (749 words) - 19:44, 25 April 2024
2002 AIME I Problems/Problem 4
*If <math>m=29t</math>, a similar argument to the one above implies <math>m=29(

2 KB (320 words) - 07:55, 4 November 2022
2002 AIME I Problems/Problem 3
...from now be <math>10a+b</math>, and let Dick's age be <math>10b+a</math>. If <math>10b+a>10a+b</math>, then <math>b>a</math>. The possible pairs of <mat

2 KB (246 words) - 17:02, 21 May 2023
2002 AIME I Problems/Problem 2
..., the height this triangle is <math>2\sqrt{3}</math>, and the shorter side if the triangle is therefore <math>2\sqrt{3}+2</math> and we use simplificatio

2 KB (287 words) - 19:54, 4 July 2013
2003 AIME I Problems/Problem 14
...ath>, and this first occurs when <math>n = \boxed{ 127 }</math> (note that if <math>4m'-n > 1</math>, then <math>n > 250</math>). Indeed, this gives <mat

3 KB (477 words) - 14:23, 4 January 2024
2003 AIME I Problems/Problem 13
...words, the length of the base-2 representation is at most <math>11</math>. If there are even digits, <math>2n</math>, then the leftmost digit is <math>1<

4 KB (651 words) - 19:42, 7 October 2023
2003 AIME I Problems/Problem 10
...math>m\angle AMC = 150^\circ</math>, <math>m\angle MCB = 83^\circ</math>. If we define <math>m\angle CMB = \theta</math> then we also have <math>m\angle

7 KB (1,058 words) - 01:41, 6 December 2022
2003 AIME I Problems/Problem 9
...<math>1000</math> and <math>9999</math>, inclusive, is called ''balanced'' if the sum of its two leftmost [[digit]]s equals the sum of its two rightmost ...This gives <math>\sum_{n = 1}^9 n(n + 1) = 330</math> balanced numbers. If the common sum of the first two and last two digits is <math>n</math>, such

4 KB (696 words) - 11:55, 10 September 2023
2003 AIME I Problems/Problem 8
...math>3|2d</math> or <math>3|2d-15</math> implies that <math>3|d</math>, so only <math>9</math> may work. Hence, the four terms are <math>18,\ 27,\ 36,\ 48< ...em yield integers for <math>a</math>, <math>d</math>, and this must be the only possible answer since this is an AIME problem. We got very lucky in this se

5 KB (921 words) - 23:21, 22 January 2023
2003 AIME I Problems/Problem 3
...ray, basically the whole <math>j = i + 1</math> shebang. Then, we see that if we set the sum of the whole array to <math>x,</math> we get out answer to b

2 KB (317 words) - 00:09, 9 January 2024
2024 AMC 8 Problems/Problem 8
How many values could be on the first day? Only <math>2</math> dollars. The second day, you can either add <math>3</math> d ...anching (multiply by <math>2</math> or adding <math>3</math>) from here is if <math>7+3=10\cdot 2</math> or <math>7+3=8\cdot 2</math> which both aren't t

2 KB (384 words) - 22:57, 17 February 2024
2003 AIME II Problems/Problem 14
...heorem, <math>b^2+2^2=y^2</math>, hence <math>b=10\sqrt{3}/3</math>. Also, if <math>C=(c,6)</math>, then <math>y^2=BC^2=4^2+(c-b)^2</math>, hence <math>c ...<math>ABCDEF</math> is twice the area of <math>ABCF</math>. Therefore, you only need to calculate the coordinates of <math>B</math>, <math>C</math>, and <m

9 KB (1,461 words) - 15:09, 18 August 2023
2003 AIME II Problems/Problem 13
...ability that 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 ...CCW \equiv 0 \pmod{3}</math>. Since <math>\#CW + \#CCW = 10</math>, it is only possible that <math>(\#CW,\, \#CCW) = (5,5), (8,2), (2,8)</math>.

15 KB (2,406 words) - 23:56, 23 November 2023
2003 AIME II Problems/Problem 12
Obviously, if some <math>v_i</math> would be <math>0</math> or <math>1</math>, the condit If for some <math>i</math> we have <math>v_i=2</math>, then from the inequalit

4 KB (759 words) - 13:00, 11 December 2022
2003 AIME II Problems/Problem 9
If we scale <math>Q(x)</math> by <math>x^2</math>, we get <math>x^6-x^5-x^4-x^

3 KB (475 words) - 21:53, 6 May 2024
2003 AIME II Problems/Problem 8
If you multiply the corresponding terms of two arithmetic sequences, you get t

5 KB (793 words) - 15:18, 14 July 2023

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