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2016 AMC 10A Problems/Problem 2
For what value of <math>x</math> does <math>10^{x}\cdot 100^{2x}=1000^{5}</math>? We can rewrite <math>10^{x}\cdot 100^{2x}=1000^{5}</math> as <math>10^{5x}=10^{15}</math>:

1 KB (190 words) - 10:58, 16 June 2023
ASIA TEAM Problems/Problem 1
...one more than the order of <math>2n</math>, and the answer is <math>\frac{1000}{2}=\boxed{500}</math>.

227 bytes (40 words) - 05:42, 16 February 2024
Kansas mathematics competitions
...ubjects, and a team competition, Mathletics. The 2006 competition had over 1000 participants. [http://contest.kcatm.org/ website]

3 KB (473 words) - 16:11, 16 June 2020
Chemistry competitions
...xam administered in mid or late April by ACS Local Sections. Approximately 1000 students qualify from local exams to take the USNCO.

2 KB (258 words) - 19:31, 8 March 2023
Princeton University Mathematics Competition
Princeton NJ 08544-1000 USA

2 KB (295 words) - 23:19, 5 January 2019
2006 Alabama ARML TST Problems/Problem 8
A bored [[mathematician]] has his computer calculate 1000 consecutive terms in the [[Fibonacci sequence]]. He notes that the smallest ...ce repeats every 16, and for every 8 numbers, there is one 0. <math>\dfrac{1000}{8}=125</math>, but we don't count the first one. <math>\boxed{124}</math>

605 bytes (78 words) - 16:43, 17 April 2008
Factorial
* <math>1000! = 402387260077093773543702433923003985719374864210714632543799910429938512

10 KB (809 words) - 16:40, 17 March 2024
Divisibility rules
Rule 3: Works for <math>1 \leq N \leq 1000</math>. Let <math>K = 3N</math>. If <math>K</math> is odd add 39 to <math

8 KB (1,315 words) - 18:18, 2 March 2024
Constructive counting
...i </math> is [[even integer | even]]. How many snakelike integers between 1000 and 9999 have four distinct digits?'' ...wever, this breaks our requirement that our integers must be between <math>1000</math> and <math>9999</math>, so there are no four-digit snakelike integers

12 KB (1,896 words) - 23:55, 27 December 2023
Michigan Mathematics Prize Competition
...parts. Part I is a 40 question, 100 minute multiple choice test. The top 1000 finishers of this round are selected to take Part II, which is a harder 5-q

981 bytes (143 words) - 12:06, 16 January 2012
Function
n-3 & \mbox{if }n\ge 1000 \\ f(f(n+5)) & \mbox{if }n<1000

10 KB (1,761 words) - 03:16, 12 May 2023
Composite number
Extra: A list of composite numbers from 1 to 1000: ...78 979 980 981 982 984 985 986 987 988 989 990 992 993 994 995 996 998 999 1000

6 KB (350 words) - 12:58, 26 September 2023
Base numbers
...rs to represent certain values (e.g. I=1, V=5, X=10, L=50, C=100, D=500, M=1000). Imagine how difficult it would be to multiply LXV by MDII! That's why t

4 KB (547 words) - 17:23, 30 December 2020
Fibonacci sequence
# Except for the first two terms, each term of the sequence <math>1000, x, 1000 - x,\ldots</math> is obtained by subtracting the preceding term from the on

6 KB (957 words) - 23:49, 7 March 2024
Writing scholarships
...rm.com/scholarship/ Rizio Lipinsky Lawyer Scholarship] USD 10,000. Essay (<1000 words) about why you’re inspired to become an attorney ...ww.seniorcare.com/scholarship/ SeniorCare.com Aging Matters Scholarship] (<1000 word essay)

7 KB (1,039 words) - 18:45, 18 January 2024
National Scholarships
* [[Gen and Kelly Tanabe Scholarship]] of <dollar/>1000 [http://www.gkscholarship.com/ website]

4 KB (538 words) - 00:48, 28 January 2024
Logarithm
...<math>b</math> satisfy the condition <math>\log_2(\log_{2^a}(\log_{2^b}(2^{1000})))=0.</math> Find the sum of all possible values of <math>a+b</math>.

4 KB (680 words) - 12:54, 16 October 2023
Rational number
...a finite [[decimal expansion]] is rational (say, <math>12.345=\frac{12345}{1000}</math>)

1 KB (207 words) - 15:51, 25 August 2022
Floor function
* How many of the first 1000 [[positive integer]]s can be expressed in the form

3 KB (508 words) - 21:05, 26 February 2024
Pascal's triangle
...>{(0000)}_{2} = 0</math>th, <math>{(0001)}_{2} = 2^0 = 1</math>st, <math>{(1000)}_{2} = 2^3 = 8</math>th, and <math>{(1001)}_{2} = 2^3+2^0 = 9</math>th col

5 KB (838 words) - 17:20, 3 January 2023
Modular arithmetic/Introduction
What is the last digit of <math>(...((7)^7)^7)...)^7</math> if there are 1000 7s as exponents and only one 7 in the middle? ...se 7 has a pattern of repetitive period 4 for the units digit. <math>(1)^{1000}</math> is simply 1, so therefore <math>7^1=7</math>, which really is the l

15 KB (2,396 words) - 20:24, 21 February 2024
2006 AIME I Problems
...ots99!100!. </math> Find the remainder when <math> N </math> is divided by 1000. ...be constructed. What is the remainder when <math> T </math> is divided by 1000?

7 KB (1,173 words) - 03:31, 4 January 2023
2006 AIME I Problems/Problem 15
...ill apply this to try and find some bounds. We can test if the first <math>1000</math> pairs of numbers each sum up to <math>-3</math>, and the rest form a

6 KB (910 words) - 19:31, 24 October 2023
2006 AIME I Problems/Problem 13
...^{n-1}}g(2k). </math> Find the greatest integer <math> n </math> less than 1000 such that <math> S_n </math> is a [[perfect square]]. ...rfect square. Hence <math>k</math> must be even. In particular, as <math>n<1000</math>, we have five choices for <math>k</math>, namely <math>k=0,2,4,6,8</

10 KB (1,702 words) - 00:45, 16 November 2023
2006 AIME I Problems/Problem 11
...constructed. What is the [[remainder]] when <math> T </math> is divided by 1000?

3 KB (436 words) - 05:40, 4 November 2022
2006 AIME I Problems/Problem 4
...100!. </math> Find the remainder when <math> N </math> is divided by <math>1000</math>.

2 KB (278 words) - 08:33, 4 November 2022
2006 AIME I Problems/Problem 3
...be <math>N</math> with the first digit deleted. Now, we know that <math>N<1000</math> (because this is an AIME problem). Thus, <math>N</math> has <math>1, ...six numbers was possible thanks to AIME problems having answers less than 1000).

4 KB (622 words) - 03:53, 10 December 2022
2005 AMC 12A Problems
A line passes through <math>A\ (1,1)</math> and <math>B\ (100,1000)</math>. How many other points with integer coordinates are on the line and ...numbers less than 1000. How many prime-looking numbers are there less than 1000?

13 KB (1,971 words) - 13:03, 19 February 2020
2005 AMC 12B Problems
A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell

12 KB (1,781 words) - 12:38, 14 July 2022
2006 AMC 12B Problems/Problem 25
so <math>999 = \max(a_1, a_2) \geq 1000</math>, a contradiction. Hence <math>(a_n)</math> completes at <math>i</mat

5 KB (924 words) - 12:02, 15 June 2022
2005 AMC 12B Problems/Problem 1
A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell \mbox{Expenses} &= 1000 \cdot \frac25 = 400 \\

1 KB (179 words) - 13:53, 14 December 2021
2005 AIME II Problems
For how many positive integers <math> n </math> less than or equal to 1000 is <math> (\sin t + i \cos t)^n = \sin nt + i \cos nt </math> true for all

7 KB (1,119 words) - 21:12, 28 February 2020
2005 AIME II Problems/Problem 9
For how many positive integers <math> n </math> less than or equal to <math>1000</math> is <math> (\sin t + i \cos t)^n = \sin nt + i \cos nt </math> true f ...on is equivalent to asking for how many [[positive integer]]s <math>n \leq 1000</math> we have that <math>\left(\sin\left(\frac\pi2 - u\right) + i \cos\lef

6 KB (1,154 words) - 03:30, 11 January 2024
2005 AIME I Problems
...<math> S. </math> Find the remainder when <math> 10K </math> is divided by 1000.

6 KB (983 words) - 05:06, 20 February 2019
2005 AIME I Problems/Problem 4
...ou have a LOT of time (and you've memorized all your perfect squares up to 1000). ...80</math>, and since AIME answers are nonnegative integers less than <math>1000</math>, we don't have to check any higher <math>n</math>. Also, we know tha

8 KB (1,248 words) - 11:43, 16 August 2022
2005 AIME I Problems/Problem 14
...S. </math> Find the remainder when <math> 10K </math> is divided by <math>1000</math>. ...{44^2}{10}</math>. The remainder when <math>10K</math> is divided by <math>1000</math> is <math>936</math>.

3 KB (561 words) - 14:11, 18 February 2018
2005 AIME I Problems/Problem 15
<cmath>2000 + 180 m^2 = 10(10+2\sqrt{2}m)^{2} + 1000</cmath> <cmath>1000 + 180 m^2 = 1000 + 400\sqrt{2}m + 80 m^{2}</cmath>

5 KB (906 words) - 23:15, 6 January 2024
2004 AIME I Problems/Problem 1
...underline{(n+2)}}\,{\underline{( n+1)}}\,{\underline {(n)}} </math><math>= 1000(n + 3) + 100(n + 2) + 10(n + 1) + n = 3210 + 1111n</math>, for <math>n \in ...be greater than 9), <math>n</math> is equal to <math>(d)+10(d+1)+100(d+2)+1000(d+3)</math> or <math>1111d +3210</math>. Now we try this number for <math>d

2 KB (374 words) - 14:53, 27 December 2019
2004 AIME I Problems/Problem 6
...i </math> is [[even integer | even]]. How many snakelike integers between 1000 and 9999 have four distinct digits? ...the leading digit will be a zero, which is bad because all numbers between 1000 and 9999 have nonzero leading digits. So, we need to select our 4 digits on

3 KB (562 words) - 18:12, 4 March 2022
2004 AIME I Problems/Problem 8
...are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there? Note that <math>n = 1000 = 2^{3}5^{3}.</math>

4 KB (620 words) - 21:26, 5 June 2021
2004 AIME I Problems
...} </math> if <math> i </math> is even. How many snakelike integers between 1000 and 9999 have four distinct digits? ...are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?

9 KB (1,434 words) - 13:34, 29 December 2021
2004 AIME II Problems/Problem 14
...777c = 7000</math>, or dividing by <math>7</math>, <math>a + 11b + 111c = 1000</math>. Then the question is asking for the number of values of <math>n = a ...is the number of multiples of <math>9</math> from <math>0</math> to <math>1000</math>, or <math>112</math>.

11 KB (1,857 words) - 21:55, 19 June 2023
2004 AIME II Problems/Problem 9
...a_n </math> be the greatest term in this sequence that is less than <math>1000</math>. Find <math> n+a_n. </math> ...<math>n</math> such that either <math>f(n)^2\, \mathrm{or}\, f(n)f(n-1) < 1000</math>. This happens with <math>f(7)f(8) = 29 \cdot 33 = 957</math>, and th

3 KB (538 words) - 21:33, 30 December 2023
2004 AIME II Problems/Problem 6
...l to <math>17z/9</math>. This means there are two possible solutions under 1000: 408 and 816. Trial and error can be done quickly to find the smallest poss

6 KB (950 words) - 14:18, 15 January 2024
2004 AIME II Problems/Problem 5
In order to complete a large job, <math>1000</math> workers were hired, just enough to complete the job on schedule. All ...1000</math> miles per hour and has one hour to reach its destination <math>1000</math> miles away. After <math>15</math> minutes and <math>250</math> miles

4 KB (592 words) - 19:02, 26 September 2020
2004 AIME II Problems/Problem 1
...ind the [[remainder]] when the product <math> abcdef </math> is divided by 1000.

2 KB (329 words) - 23:20, 4 July 2013
2004 AIME II Problems
...e. Find the remainder when the product <math> abcdef </math> is divided by 1000. In order to complete a large job, 1000 workers were hired, just enough to complete the job on schedule. All the wo

9 KB (1,410 words) - 05:05, 20 February 2019
1984 AIME Problems
n-3 & \mbox{if }n\ge 1000 \\ f(f(n+5)) & \mbox{if }n<1000

6 KB (933 words) - 01:15, 19 June 2022
1987 AIME Problems
...triples <math>(a,b,c)</math> of positive integers for which <math>[a,b] = 1000</math>, <math>[b,c] = 2000</math>, and <math>[c,a] = 2000</math>. ...e integer]] and <math>r</math> is a positive real number less than <math>1/1000</math>. Find <math>n</math>.

6 KB (869 words) - 15:34, 22 August 2023
1989 AIME Problems
A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most fre

7 KB (1,045 words) - 20:47, 14 December 2023

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