Mathtime Version 1 Issue 2

Number Theory

Problem 1

$24_{x}=18_{10}$. What is the value of $x$? (Proposed by SP343)

Problem 2

What is the value of $320_{4}+10_{2}$ in base $8$? (Proposed by SP343)

Problem 3

$n\equiv0\pmod{7}$, $n\equiv0\pmod{17}$, and $n\equiv1\pmod{3}$. What is the smallest possible positive value of $n$? (Proposed by SP343)

Problem 4

Prove that $2012^{2011}+2011^{2012}$ is divisible by $7$. (Proposed by djmathman)

Problem 5

Determine the positive integer $n$ such that $n^2+20n+40$ is a perfect square. (Proposed by djmathman)

Problem 6

Find the smallest integer value of $x$ such that $\dfrac{2010x}{2011+2009}$ is an integer greater than $105$. (Proposed by djmathman)

Problem 7

What is the greatest possible integer value of $n$ such that $75^n$ has less than $1000$ distinct positive divisors? (Proposed by djmathman)

Problem 8

Find all integers $n$ such that $2^n+3^n+4^n$ is divisible by $3$. (Proposed by djmathman)

Problem 9

Prove that there are an infinite number of ordered integer pairs $(a,b)$ such that $a^2+b^2|(a+b)^2$. (Proposed by djmathman)

Problem 10

Let $A_{n,k}=\dfrac{n(n+1)}{k}$. For how many values of $k$ does there exist at least $2012$ values of $n$ less than $10,000$ such that $A_{n,k}$ is an integer? (Proposed by djmathman)

Problem 11

Let $S(n)$ and $P(n)$ be the sum and the product of the divisors or $n$, respectively. Calculate $S(P(S(P(350))))$. (Proposed by El-Etric)

Problem 12

What is the units digit of $16^{52}+17^{52}+18^{52}$? (Proposed by El-Etric)

Geometry

Problem 1

Supports are being placed inside a hemispherical dome. A support $24$ meters in length is $8$ meters closer to the edge of the dome than a support $20$ meters in length. Assuming that these supports have neglectible thickness and are perpendicular to the ground, find the radius of the dome. (Proposed by djmathman)

Problem 2

A circle in the $xy$ coordinate plane passes through the points ($18,15)$, $(10,-29)$, and $(-17,20)$. Given that the equation of the circle can be written in the form $(x-h)^2+(y-k)^2=r^2$ where r, k, and h are integers, find $h^2+k^2+r^2$. (Proposed by djmathman)

Problem 3

Circle $A$ is separated into $4$ quarter circles. Each quarter circle has a circle inscribed within the circle so that the inscribed circle is tangent to all $3$ sides. What is the ratio of the sum of the areas of all $4$ small circles to the area of the large circle? (Proposed by SP343)

Problem 4

$4$ circles are bound together with a tight rope. The rope is tight so that there is no extra rope left over when bound around the four circles. If the radius of a circle is $5$ and all the circles are congruent, what is the length of the rope? (Proposed by SP343)

Problem 5

Two triangles are combined together with overlap so that the Star of David is formed. This Star of David has outlying triangles that, when added together, has the same area as the inside hexagon. A circle is circumscribed around the Star of David. What is the ratio of the area of the Star of David to the area of the circle? (Proposed by SP343)

Problem 6

An average can of almonds is a cylinder container with a radius of $5$ and a height of $8$. If the Almond company decides to increase the volume by $50\%$, but also to decrease the height by $50\%$, what does the radius of the new can have to be? Express your answer in simplest radical form. (Proposed by SP343)

Problem 7

A regular hexagon is inscribed within a circle, which in turn is inscribed within a square, which is then inscribed within a final large circle. If the area of the hexagon is $54$ units, find the area of the largest circle. (Proposed by SP343)

Problem 8

What is the greatest amount of intersections a circle, a square, and $4$ straight distinct lines could create? (Proposed by SP343)

Problem 9

An astronaut is tethered to a vertex of a cube with side length of 3 m. What is the total volume of the space the astronaut is able to visit if the length of the rope is 4m? (ERRATUM: The original problem forgot to specify the length of the rope) (Proposed by tc1729)

Problem 10

Triangle $ABC$ is an isosceles triangle with $AB = BC$ and $AC = 15$. Let $D$ and $E$ be the points of intersection of the altitudes from points $A$ and $C$ respectfully. If $DE = 10$, then the length of one leg of the triangle can be written in the form $\dfrac{m\sqrt{n}}{p}$ where $m$, $n$, and $p$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n+p$. (Proposed by djmathman)

Problem 11

A circle is placed so that it touches the origin and has its center located on the y-axis. The circle is tangent to the line $x + 3y = 6$. Given that the radius of the circle can be written in the form $\dfrac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers (not necessarily positive), find $a+b+c$. (Proposed by djmathman)

Problem 12

A complex number $a+bi$ is graphed on the complex plane such that $a=1$. The line drawn from the origin to this point makes an angle $\theta$ with the real axis. As $\theta$ sweeps from $0$ to $\dfrac{\pi}{2}$, what is the probability that $b\ge 1$? (Proposed by BT)


Combinatorics

Problem 1

Two people decide to meet at a fire department building to discuss new additions to the fire crew. Both people are very busy running their own businesses, and they can only meet at sometime between $1$ PM and $3$ PM. Person $A$ has a $15$ minute lunch break to randomly spend at the fire building and Person $B$ has a $20$ minute lunch break to spend at the fire building. If they can meet each other at any time, then they can accomplish their required deed. What is the probability that they will be able to accomplish their deed? (Proposed by SP343)

Problem 2

At the county fair, a gambler offers two bags. The first bag has $3$ chocolate bars and $2$ packs of raisins. The second bag has $5$ chocolate bars and $4$ packs of raisins. The boy that is drawing next is allergic to raisins and tells the man so. He replies, "Very well. In your draw, if at any time you draw a chocolate bar, you may keep the chocolate bar and leave. However, if at anytime you draw a pack of raisins, you may place the pack of raisins within the other bag and draw from that other bag. You must start drawing from Bag $A$." Find a formula that expresses the chance that the boy will eventually get a chocolate bar in terms of the number of draws $n$.(Proposed by SP343)

Problem 3

What is the expected value to draw for a bag that has $1$ $$10$ bill, $3$ $$5$ bills, $5$ packs of $3$ $$1$ bills, $10$ $$1$ bills, and $19$ fake Monopoly $$1,000,000$ bills? (Proposed by SP343)

Problem 4

Carnegie Mellon's secret astrophysical department has $70$ students. $3$ classes are offered: Nuclear Implosion, Black Hole Prevention, and Extra-Terrestrial Life Hunt. The $70$ students are divided up so that $37$ students take Nuclear Implosion, $27$ students observe Extra-Terrestrial Life, and $31$ watch for Black Holes. If $6$ elite students take all $3$ classes and $11$ procrastinating students don't take any classes, then how many people take Nuclear Implosion and Black Hole Prevention, but not Extra-Terrestrial Life? (Proposed by SP343)

Problem 5

In an average Social Studies class at my school, there are $23$ students. If $14$ are blonde and $13$ are girls, at least how many blonde girls are there? (Proposed by SP343)

Problem 6

How many combinations of letters can be made by rearranging the letters of $MATHTIME$? (Proposed by SP343)

Problem 7

$7$ people go to a Thanksgiving feast. There, they are served $21$ glasses of red wine, $3$ glasses per person. The host of the party had placed a diamond ring in one of the glasses, to be given to the one he wished to propose to. Unfortunately, the waiter had mixed up the glasses so that he wasn't sure which glass had the ring. If the drinks were randomly served in $3$ rounds of $7$ drinks, then what is the chance that the bride would receive the right glass of wine in the $3^\text{rd}$ round, as planned? (Proposed by SP343)

Problem 8

A company wants to sell their product for a cure for Paul Bunyan's disease, a disease where a man might be transformed into a huge lumberjack with a great beard, angry at the huge amount of trees that are still standing. If an average person would look at an advertisement on a billboard $\frac{1}{3}$ of the time, and would have a $\frac{1}{6}$ chance of buying the item at first glance, which then steadily increases by $\frac{1}{12}$ for each additional glance at the advertisement, what is the chance that the person will buy the item if he is taking a stroll around a block with the advertisement on one side $5$ times? (Proposed by SP343)

Problem 9

A Native Cherokee tribe happened to come upon a magical sage. Not knowing this information, they helped her by feeding her and giving her shelter. In return, the sage offered them an opportunity to revive a random person from the grave. The tribe had recently lost their tribe leader, so they eagerly accepted. The sage warned that she only had a $\dfrac{2}{5}^\text{th}$ chance of successfully reviving a dead person, and that there was a $\dfrac{3}{5}^\text{th}$ chance that zombies would rise instead and kill them all. Additionally, the sage had only been able to revive a Cherokee $\dfrac{3}{8}^\text{th}$ of the time. In the other times, she had brought back a violent Creek who had always narrowly missed killing her. Even worse was the fact that the spell took $5$ days to prepare, then could only be done within $2$ hours after preparation was finished. The spell could only be done if the sky was completely clear. If there was a $70\%$ of a storm cloud coming in soon, then what is the chance that a Cherokee Chief was brought back from the dead, if $3$ in every $100$ Cherokees were chiefs? (Proposed by SP343)

Problem 10

Three identical white pawns and five identical black pawns are randomly ordered in a line. An example of such an arrangement is WBBWBWBB.

a. The above arrangement is an example arrangement that has at least two black pawns adjacent to each other. Prove that every possible arrangement results in at least two black pawns adjacent to each other.

b. The above arrangement is an example arrangement in which no two white pawns are next to each other. Determine the number of arrangements for which this is so.

(Proposed by djmathman)

Problem 11

I roll 4 dice with 6, 10, 12, and 20 sides each numbered 1-6, 1-10, 1-12, 1-20 respectively. What is the probability that the face-up value of the 12 sided die is greater than the sum of the values of the other dice? (Proposed by tc1729)

Problem 12

Suppose that a knight (as in the chess piece) travelled to every square on a chess board and counted the number of squares he could attack on the board. For instance, in the center of the board, the knight is able to move to eight different squares from that spot. If each of these numbers was recorded and placed in a hat, what is the expected value of drawing one number? (Proposed by djmathman)

Algebra

Problem 1

Have a root of $x^3 + 5x + 8$ be $r$. Find the value of $a$ such that $\dfrac{r^3 + 5r + 7}{r^3 + 5r + a} = -2$. (Proposed by KingofMath101)

Problem 2

Where $a$, $b$, $c$, and $d$ are positive, prove that $(a^2b^2 + c^2d^2)(a^2c^2 + b^2d^2)(a^2d^2 + b^2c^2) \geq 8abcd$ . (Proposed by KingofMath101)

Problem 3

If $d$ is a root of $2x^3 + 7x^2 - 4x + 1$, then simplify $d^7 + 4d^6 - 3d^5 + 22d^4 + 3d^3 - d^2 + 7d - 1$ to a quadratic in terms of $d$. (Proposed by KingofMath101)

Problem 4

Find $\prod_{x = 1}^{6}{\frac{x + 4}{x + 2} \cdot \frac{x + 3}{x + 5}}$. (Proposed by KingofMath101)

Problem 5

Determine the value of $\sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+\ldots}}}}$. (Proposed by KingofMath101)

Problem 6

Determine the value of $4^{3^2}-\left(4^3\right)^2$. (Proposed by SP343)

Problem 7

Find the sum of the reciprocals of $x^7+324x^6-20x^5+24x^3-6033x=-2011$.(Proposed by SP343)

Problem 8

$20$ oz of a $15\%$ acid mixture is added to a $50\%$ acid mixture. The resulting mixture is $30\%$ acid. How many oz of the $50\%$ acid was added? (Proposed by SP343)

Problem 9

$\sqrt{7x+21}-x=3$. Find the value of $x$. (Proposed by SP343)

Problem 10

The cube root of $\sqrt{1024}=16^x$. Solve for the value of $x$. (Proposed by SP343)

Problem 11

Given that $x=-3$ is a root to $x^3-5x^2-7x+51$, find the sum of the two other roots. (Proposed by SP343)

Problem 12

Let $x_1$ and $x_2$ be integers that sum to $9$. Given that $x_2>x_1$, find a closed-form expression for \[\sum_{a=x_1}^{x_2}\sum_{m=1}^{a} m\] in terms of $x_2$. (Proposed by djmathman)

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