heureka

A fair six-sided number cube with the digits 1-6 on its faces is rolled three times.
The positive difference between the values on the first two rolls is equal to the value of the third roll.
What is the probability that at least one 3 was rolled?
Express your answer as a common fraction.

\(\begin{array}{|c|c|c|c|} \hline \text{first roll} & \text{second roll} & \text{third roll} \\ \mathbf{ 1} & (\text{difference}) &\text{1 2 $\color{red}3$ 4 5 6} \\ \hline & 1~(0) & \text{_ _ _ _ _ _} \\ & 2~(1) & \text{$\color{blue}x$_ _ _ _ _} \\ & {\color{red}3}~(2) & \text{_ $\color{red}x$ _ _ _ _} \\ & 4~(3) & \text{_ _ $\color{red}x$ _ _ _} \\ & 5~(4) & \text{_ _ _ $\color{blue}x$ _ _} \\ & 6~(5) & \text{_ _ _ _ $\color{blue}x$ _} \\ \hline \end{array} \begin{array}{|c|c|c|c|} \hline \text{first roll} & \text{second roll} & \text{third roll} \\ \mathbf{2} & (\text{difference}) &\text{1 2 $\color{red}3$ 4 5 6} \\ \hline & 1~(1) & \text{$\color{blue}x$_ _ _ _ _} \\ & 2~(0) & \text{_ _ _ _ _ _} \\ & {\color{red}3}~(1) & \text{$\color{red}x$_ _ _ _ _} \\ & 4~(2) & \text{_ $\color{blue}x$ _ _ _ _} \\ & 5~(3) & \text{_ _ $\color{red}x$ _ _ _} \\ & 6~(4) & \text{_ _ _ $\color{blue}x$ _ _} \\ \hline \end{array}\)

\(\begin{array}{|c|c|c|c|} \hline \text{first roll} & \text{second roll} & \text{third roll} \\ \mathbf{ {\color{red}3}} & (\text{difference}) &\text{1 2 $\color{red}3$ 4 5 6} \\ \hline & 1~(2) & \text{_ $\color{red}x$ _ _ _ _} \\ & 2~(1) & \text{$\color{red}x$_ _ _ _ _} \\ & {\color{red}3}~(0) & \text{_ _ _ _ _ _} \\ & 4~(1) & \text{$\color{red}x$_ _ _ _ _} \\ & 5~(2) & \text{_ $\color{red}x$ _ _ _ _} \\ & 6~(3) & \text{_ _ $\color{red}x$ _ _ _} \\ \hline \end{array} \begin{array}{|c|c|c|c|} \hline \text{first roll} & \text{second roll} & \text{third roll} \\ \mathbf{ 4} & (\text{difference}) &\text{1 2 $\color{red}3$ 4 5 6} \\ \hline & 1~(3) & \text{_ _ $\color{red}x$ _ _ _} \\ & 2~(2) & \text{_ $\color{blue}x$ _ _ _ _} \\ & {\color{red}3}~(1) & \text{$\color{red}x$_ _ _ _ _} \\ & 4~(0) & \text{_ _ _ _ _ _} \\ & 5~(1) & \text{$\color{blue}x$_ _ _ _ _} \\ & 6~(2) & \text{_ $\color{blue}x$ _ _ _ _} \\ \hline \end{array}\)

\(\begin{array}{|c|c|c|c|} \hline \text{first roll} & \text{second roll} & \text{third roll} \\ \mathbf{ 5} & (\text{difference}) &\text{1 2 $\color{red}3$ 4 5 6} \\ \hline & 1~(4) & \text{_ _ _ $\color{blue}x$ _ _} \\ & 2~(3) & \text{_ _ $\color{red}x$ _ _ _} \\ & {\color{red}3}~(2) & \text{_ $\color{red}x$ _ _ _ _} \\ & 4~(1) & \text{$\color{blue}x$_ _ _ _ _} \\ & 5~(0) & \text{_ _ _ _ _ _} \\ & 6~(1) & \text{$\color{blue}x$_ _ _ _ _} \\ \hline \end{array} \begin{array}{|c|c|c|c|} \hline \text{first roll} & \text{second roll} & \text{third roll} \\ \mathbf{ 6} & (\text{difference}) &\text{1 2 $\color{red}3$ 4 5 6} \\ \hline & 1~(5) & \text{_ _ _ _ $\color{blue}x$ _} \\ & 2~(4) & \text{_ _ _ $\color{blue}x$ _ _} \\ & {\color{red}3}~(3) & \text{_ _ $\color{red}x$ _ _ _} \\ & 4~(2) & \text{_ $\color{blue}x$ _ _ _ _} \\ & 5~(1) & \text{$\color{blue}x$_ _ _ _ _} \\ & 6~(0) & \text{_ _ _ _ _ _} \\ \hline \end{array}\)

The probability that at least one 3 was rolled is \(\mathbf{ \dfrac{14\ {\color{red}x} }{30\ {\color{blue}x} } = \dfrac{7}{15} }\)

Find the remainder when \(24^{50} - 15^{50}\) is divided by \(13\).

\(\begin{array}{|rcll|} \hline &&\mathbf{24^{50} - 15^{50} \pmod{13}} \\\\ && \boxed{24\pmod{13}\equiv 24-13 \\ \equiv 11 \\ \equiv 11-13 \\ \equiv -2\pmod{13} \\ 15\pmod{13}\equiv 15-13 \\ \equiv 2 \pmod{13} } \\\\ &\equiv& (-2)^{50} - 2^{50} \pmod{13} \\ &\equiv & (-1)^{50}2^{50}- 2^{50} \pmod{13} \quad | \quad (-1)^{50}=1 \\ &\equiv & 1\cdot 2^{50}- 2^{50} \pmod{13} \\ &\equiv & 2^{50}- 2^{50} \pmod{13} \\ &\equiv & \mathbf{ 0 \pmod{13} } \\ \hline \end{array}\)

PSL4#18

\(\begin{array}{|c|c|c|c|c|} \hline & \text{penny} & \text{nickel} & \text{dime}& \text{quater} \\ & 1\ \text{cent} & 5\ \text{cent} & 10\ \text{cent}& 25\ \text{cent} \\ \hline &0&0&0&3 \\ &0&0&3&0 \\ &0&3&0&0 \\ &3&0&0&0 \\ \hline &1&2&0&0 \\ &1&0&2&0 \\ &1&0&0&2 \\ &0&1&2&0 \\ &0&1&0&2 \\ &0&0&1&2 \\ \hline &2&1&0&0 \\ &2&0&1&0 \\ &2&0&0&1 \\ &0&2&1&0 \\ &0&2&0&1 \\ &0&0&2&1 \\ \hline &1&1&1&0 \\ &1&1&0&1 \\ &1&0&1&1 \\ &0&1&1&1 \\ \hline sum &15&15&15&15 \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline \text{sum} &=& 15\times 1\ \text{cent}+ 15\times 5\ \text{cent} + 15\times 10\ \text{cent}+ 15\times 25\ \text{cent} \\ &=& 15 + 75 + 150 + 375 \\ &=& 615 \\ &=& \mathbf{$6.15} \\ \hline \end{array}\)

Two circles of radius 1 are centered at (4,0) and (-4,0).

How many circles are tangent to both of the given circles and also pass through the point (0,5)?

OSL4#29

\(\begin{array}{|c|c|c|c|} \hline \color{red}1 & 12 & 3 & 8 \\ \hline 6 & 9 & 14 & 11 \\ \hline 13 & \color{red}2 & 7 & 4 \\ \hline &5& 10 & \color{blue} 15 \\ \hline \end{array} \)

Thank you, Logic !

For each positive integer n,
the set of integers \(\{0,\ 1,\ 2,\ \ldots \,\ n-1 \}\) is known as the residue system modulo n.
Within the residue system modulo \(2^4\),
let A be the sum of all invertible integers modulo \(2^4\) and
let B be the sum all of non-invertible integers modulo \(2^4\).
What is A-B?

The integer s of the set is invertible, if \(gcd(2^4,s)=1\)

\(\begin{array}{|c|c|c|l|} \hline \text{set of integers} & gcd(2^4,s) & \text{not invertible} & \text{modulo inverse} \\ \hline 0 & 16 & \checkmark \\ \hline 1 & 1 && 1^{-1} \pmod{16} = 1 \quad |\quad 1\cdot 1 \equiv 1 \pmod{16} \\ \hline 2 & 2 & \checkmark \\ \hline 3 & 1 && 3^{-1} \pmod{16} = 11 \quad |\quad 3\cdot 11 \equiv 1 \pmod{16} \\ \hline 4 & 4 & \checkmark \\ \hline 5 & 1 && 5^{-1} \pmod{16} = 13 \quad |\quad 5\cdot 13 \equiv 1 \pmod{16} \\ \hline 6 & 2 & \checkmark \\ \hline 7 & 1 && 7^{-1} \pmod{16} = 7 \quad |\quad 7\cdot 7 \equiv 1 \pmod{16} \\ \hline 8 & 8 & \checkmark \\ \hline 9 & 1 && 9^{-1} \pmod{16} = 9 \quad |\quad 9\cdot 9 \equiv 1 \pmod{16} \\ \hline 10 & 2 & \checkmark \\ \hline 11 & 1 && 11^{-1} \pmod{16} = 3 \quad |\quad 11\cdot 3 \equiv 1 \pmod{16} \\ \hline 12 & 4 & \checkmark \\ \hline 13 & 1 && 13^{-1} \pmod{16} = 5 \quad |\quad 13\cdot 5 \equiv 1 \pmod{16} \\ \hline 14 & 2 & \checkmark \\ \hline 15 & 1 && 15^{-1} \pmod{16} = 15 \quad |\quad 15\cdot 15 \equiv 1 \pmod{16} \\ \hline \hline \end{array}\)

\( A = 1+3+5+7+9+11+13+15 = \mathbf{64} \\ B = 0+2+4+6+8+10+12+14 = \mathbf{56}\)

Two positive numbers \(p\) and \(q\) have the property that their sum is equal to their product.
If their difference is \(7\), what is \(\dfrac{1}{\dfrac{1}{p^2}+\dfrac{1}{q^2}}\)?

Your answer will be of the form \(\dfrac{a+b\sqrt{c}}{d}\),

where \(a\) and \(b\) don't both share the same common factor with \(d\) and \(c\) has no square as a factor.

Find \(a+b+c+d\).

\(\begin{array}{|lrcll|} \hline &\mathbf{p+q} &=& \mathbf{pq} \quad | \quad :pq \\\\ &\dfrac{1}{q} + \dfrac{1}{p} &=& 1 \\ &\left(\dfrac{1}{q} + \dfrac{1}{p}\right)^2 &=& 1^2 \\ & \dfrac{1}{q^2} + \dfrac{1}{p^2} + \dfrac{2}{pq} &=& 1 \\ (1)& \mathbf{ \dfrac{1}{q^2} + \dfrac{1}{p^2}} &=& \mathbf{ 1-\dfrac{2}{pq}} \\ \hline \end{array} \)

\(\begin{array}{|lrcll|} \hline &\mathbf{p-q} &=& \mathbf{7} \quad | \quad :pq \\\\ &\dfrac{1}{q} - \dfrac{1}{p} &=& \dfrac{7}{pq} \\ &\left(\dfrac{1}{q} - \dfrac{1}{p}\right)^2 &=& \dfrac{49}{(pq)^2} \\ & \dfrac{1}{q^2} + \dfrac{1}{p^2} - \dfrac{2}{pq} &=& \dfrac{49}{(pq)^2} \\ (2)& \mathbf{ \dfrac{1}{q^2} + \dfrac{1}{p^2}} &=& \mathbf{ \dfrac{49}{(pq)^2}+\dfrac{2}{pq}} \\ \hline \end{array} \)

\(\begin{array}{|lrcll|} \hline (1)=(2): & \dfrac{1}{q^2} + \dfrac{1}{p^2} = 1-\dfrac{2}{pq} &=& \dfrac{49}{(pq)^2}+\dfrac{2}{pq} \\\\ & 1-\dfrac{2}{pq} &=& \dfrac{49}{(pq)^2}+\dfrac{2}{pq} \\\\ & 1-\dfrac{4}{pq} &=& \dfrac{49}{(pq)^2} \quad | \quad \cdot (pq)^2 \\\\ & (pq)^2-4(pq) &=& 49 \\\\ & (pq)^2-4(pq) -49 &=& 0 \\\\ & (pq) &=& \dfrac{4 \pm \sqrt{16-4\cdot (-49) } } {2} \\ & (pq) &=& \dfrac{4 \pm \sqrt{212} } {2} \\ & (pq) &=& \dfrac{4 \pm \sqrt{53} } {2} \\ & (pq) &=&2 \pm \sqrt{53} \\\\ (3) & \mathbf{(pq)_1} &=& \mathbf{2 + \sqrt{53}} \\ (4) & \mathbf{(pq)_2} &=& \mathbf{2 - \sqrt{53}} \\ \hline \end{array}\)

\(\mathbf{(pq)_1 = 2 + \sqrt{53}}\):

\(\begin{array}{|rcll|} \hline \mathbf{ \dfrac{1}{p^2} + \dfrac{1}{q^2}} &=& \mathbf{ 1-\dfrac{2}{(pq)_1}} \\\\ \dfrac{1}{p^2} + \dfrac{1}{q^2} &=& \dfrac{(pq)_1-2}{(pq)_1} \\\\ x=\dfrac{1}{\dfrac{1}{p^2} + \dfrac{1}{q^2}} &=& \dfrac{(pq)_1}{(pq)_1-2} \\\\ x &=& \dfrac{2 + \sqrt{53}}{2 + \sqrt{53}-2} \\\\ x &=& \left( \dfrac{2 + \sqrt{53}}{ \sqrt{53} } \right)\cdot \dfrac{\sqrt{53}}{\sqrt{53}} \\\\ \mathbf{x} &=& \mathbf{\dfrac{53+2\sqrt{53}}{ 53 }} \quad | \quad \dfrac{a+b\sqrt{c}}{d} \\ && a=53,\ b=2,\ c=53,\ d=53 \\ && \mathbf{a+b+c+d} = 53+2+53+53 &=& \mathbf{ 161 } \\ \hline \end{array} \)

\(\mathbf{(pq)_2 = 2 - \sqrt{53}}\):

\(\begin{array}{|rcll|} \hline \mathbf{ \dfrac{1}{p^2} + \dfrac{1}{q^2}} &=& \mathbf{ 1-\dfrac{2}{(pq)_2}} \\\\ \dfrac{1}{p^2} + \dfrac{1}{q^2} &=& \dfrac{(pq)_2-2}{(pq)_2} \\\\ x=\dfrac{1}{\dfrac{1}{p^2} + \dfrac{1}{q^2}} &=& \dfrac{(pq)_2}{(pq)_2-2} \\\\ x &=& \dfrac{2 - \sqrt{53}}{2 - \sqrt{53}-2} \\\\ x &=& \dfrac{2 - \sqrt{53}}{ - \sqrt{53} } \\\\ x &=& \left( \dfrac{-2 + \sqrt{53}}{ \sqrt{53} } \right)\cdot \dfrac{\sqrt{53}}{\sqrt{53}} \\\\ \mathbf{x} &=& \mathbf{\dfrac{53-2\sqrt{53}}{ 53 }} \quad | \quad \dfrac{a+b\sqrt{c}}{d} \\ && a=53,\ b=-2,\ c=53,\ d=53 \\ && \mathbf{a+b+c+d} = 53-2+53+53 &=& \mathbf{ 157 } \\ \hline \end{array}\)

What is the residue of \(9^{2010}\), modulo \(17\) ?

\(\begin{array}{|rcll|} \hline && \mathbf{9^{2010} \pmod {17}} \\ &\equiv & \left(3^2 \right)^{2010} \pmod {17} \\ &\equiv & 3^{2\cdot 2010} \pmod {17} \\ &\equiv & 3^{4020} \pmod {17} \\ & & \boxed{ a^{\varphi(n)} \equiv 1 \pmod {n} \text{, if gcd(a,n)=1 } \\ 3^{\varphi(17)} \equiv 1 \pmod {17} \text{, gcd(3,17)=1 } \quad \varphi(17) = 16 \\ 3^{16} \equiv 1 \pmod {17} } \\ &\equiv & 3^{16\cdot 251+4} \pmod {17} \\ &\equiv & \left(3^{16} \right)^{251}3^4 \pmod {17} \\ &\equiv & \left(1 \right)^{251}3^4 \pmod {17} \\ &\equiv & 3^4 \pmod {17} \\ &\equiv & 81 \pmod {17} \\ &\equiv & \mathbf{13 \pmod {17}} \\ \hline \end{array}\)

Two circles of radius \(1\) are centered at \((4,0)\) and \((-4, 0)\).
How many circles are tangent to both of the given circles and also pass through the point \((0, 5)\)?

4 circles are tangent to both of the given circles and also pass through the point \((0, 5)\):\(\begin{array}{|l|lcll|} \hline 1 & x^2+\left(y-\dfrac{5}{3}\right)^2=\left(\dfrac{10}{3}\right)^2 \\ \hline 2 & \left(x+1.0328\right)^2+\left(y-1\right)^2\ =\ \left(4.13118\right)^2 \\ \hline 3 & \left(x-1.0328\right)^2+\left(y-1\right)^2=\left(4.13118\right)^2 \\ \hline 4 & x^2+y^2=5^2 \\ \hline \end{array} \)