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- ...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!}{72 KB (281 words) - 12:09, 5 April 2024
- .../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 di3 KB (433 words) - 19:42, 20 December 2021
- 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
- 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
- <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
- ...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
- ...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
- .... 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
- ...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
- ..., 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 (904 words) - 15:15, 23 June 2024
- ...<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
- ..., 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}</ma1 KB (195 words) - 15:33, 16 December 2021
- ...) 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> Since2 KB (356 words) - 19:15, 17 September 2023
- ...<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
- ...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 </math3 KB (489 words) - 19:22, 17 September 2023
- ...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
- ...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
- ...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 + 3k2 KB (398 words) - 09:48, 5 August 2014
- ...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
- 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
