...orm <math>0.\overline{abcd},</math> where at least one of the digits <math>a,</math> <math>b,</math> <math>c,</math> or <math>d</math> is nonzero. Let < Make cases by factors of <math>x</math>. (A venn diagram of cases would be nice here.) ...

When <math>30\%</math> of <math>x</math> is a positive perfect square integer, what is <math>min(x)</math> such that <mat <math>\textbf{(A)} ~9 \qquad\textbf{(B)} ~12 \qquad\textbf{(C)} ~30 \qquad\textbf{(D)} ~36 \ ...

...s as a Renewable Energy Engineer for the Southern Company, and Hannah runs a lab at Jupiter Falls University where she researches biomass (renewable fue When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego Zoo. ...

The next natural question is: how do we convert a number from another base into base 10? For example, what does <math>4201_5 ...der for us than converting from other bases to base 10. This shouldn't be a surprise, though. We work in base 10 ''all the time'' so we are naturally ...

...s is equal to <math>c</math>. What is the greatest possible value of <math>a + b + c</math> for which there are at least two values of <math>n</math> su <math>\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } ...

...0(b-1) \equiv 10(c-1) \equiv d-1 \pmod{7}</math>. We can casework on <math>a</math> backwards, finding the maximum value. Applying casework on <math>a</math>, we can eventually obtain a working value of <math>\overline{abcd} = 5694 \implies \boxed{699}</math>. ...

A palindrome between <math>1000</math> and <math>10,000</math> is chosen at r <math>\textbf{(A)}\ \dfrac{1}{10} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \df ...

...ath>d = 1000</math>, then <math>(a+ b+ c-d) + (a + b- c+ d) +(a-b+ c+d)+ (-a+ b+c+d)</math> is equal to <math>\textbf{(A) }1111\qquad ...

<math>\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qqu Therefore, there are <math>\boxed{\textbf{(A) } 0}</math> prime numbers in this sequence. ...

The digits of a positive integer <math> n </math> are four consecutive integers in decreasi A brute-force solution to this question is fairly quick, but we'll try someth ...

...ath>1(100)+10(99)+100(98)+1000(97)+ \cdots + 10^{100}</math>, so factoring a <math>10</math> we have <math>1(10)+99+10(98)+ \cdots + 10^{99}</math>. Now ...

<math>\textbf{(A) }-1\qquad If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is ...

<math>\textbf{(A) }2+0+1+7\qquad\textbf{(B) }2 \times 0 +1+7\qquad\textbf{(C) }2+0 \times 1 Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the vo ...

<math> \textbf{(A)}\ 16\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qqu ...begin with a <math>1</math>, and if you switch digits twice, you will have a <math>1</math> at the end. Thus, we are only concerned with the <math>49</ ...

...igits taken in the same order. Which of the following numbers must also be a factor of <math>Z</math>? ...xtbf{(B) }19\qquad\textbf{(C) }101\qquad\textbf{(D) }111\qquad\textbf{(E) }1111</math> ...

...st perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then w <math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ ...

...<math>1005</math> and <math>1231</math> have something in common: each is a <math>4</math>-digit number beginning with <math>1</math> that has exactly Consider a sequence of <math>4</math> digits instead of a <math>4</math>-digit number. Only looking at the sequences which have one d ...

...<math>AB=7</math>, <math>BC=9</math>, <math>CD=15</math>, and there exists a circle, lying inside the quadrilateral and having center <math>I</math>, th ...pposite sides of <math>QMPN</math> have equal length, <math>QMPN</math> is a [[parallelogram]]. Thus, the diagonals bisect each other, and <math>QI = IP ...

This sequence tends to a limit; call it <math>L</math>. What is the least value of <math>k</math> su <math>\textbf{(A)}\: 10\qquad\textbf{(B)}\: 87\qquad\textbf{(C)}\: 123\qquad\textbf{(D)}\: 3 ...

A repunit is a natural number whose digits are all <math>1</math>. For instance, <cmath>1,11,111,1111, \ldots</cmath> ...