Found problems: 146
2024 MMATHS, 2
Consider the recursive sequence defined by $a_{n+1}=a_n^n+1,$ with $a_1=0.$ What is the last digit of $a_{2024}$?
2019 MMATHS, Mixer Round
[b]p1.[/b] An ant starts at the top vertex of a triangular pyramid (tetrahedron). Each day, the ant randomly chooses an adjacent vertex to move to. What is the probability that it is back at the top vertex after three days?
[b]p2.[/b] A square “rolls” inside a circle of area $\pi$ in the obvious way. That is, when the square has one corner on the circumference of the circle, it is rotated clockwise around that corner until a new corner touches the circumference, then it is rotated around that corner, and so on. The square goes all the way around the circle and returns to its starting position after rotating exactly $720^o$. What is the area of the square?
[b]p3.[/b] How many ways are there to fill a $3\times 3$ grid with the integers $1$ through $9$ such that every row is increasing left-to-right and every column is increasing top-to-bottom?
[b]p4.[/b] Noah has an old-style M&M machine. Each time he puts a coin into the machine, he is equally likely to get $1$ M&M or $2$ M&M’s. He continues putting coins into the machine and collecting M&M’s until he has at least $6$ M&M’s. What is the probability that he actually ends up with $7$ M&M’s?
[b]p5.[/b] Erik wants to divide the integers $1$ through $6$ into nonempty sets $A$ and $B$ such that no (nonempty) sum of elements in $A$ is a multiple of $7$ and no (nonempty) sum of elements in $B$ is a multiple of $7$. How many ways can he do this? (Interchanging $A$ and $B$ counts as a different solution.)
[b]p6.[/b] A subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$ of size $3$ is called special if whenever $a$ and $b$ are in the set, the remainder when $a + b$ is divided by $8$ is not in the set. ($a$ and $b$ can be the same.) How many special subsets exist?
[b]p7.[/b] Let $F_1 = F_2 = 1$, and let $F_n = F_{n-1} + F_{n-2}$ for all $n \ge 3$. For each positive integer $n$, let $g(n)$ be the minimum possible value of $$|a_1F_1 + a_2F_2 + ...+ a_nF_n|,$$ where each $a_i$ is either $1$ or $-1$. Find $g(1) + g(2) +...+ g(100)$.
[b]p8.[/b] Find the smallest positive integer $n$ with base-$10$ representation $\overline{1a_1a_2... a_k}$ such that $3n = \overline{a_1a_2 a_k1}$.
[b]p9.[/b] How many ways are there to tile a $4 \times 6$ grid with $L$-shaped triominoes? (A triomino consists of three connected $1\times 1$ squares not all in a line.)
[b]p10.[/b] Three friends want to share five (identical) muffins so that each friend ends up with the same total amount of muffin. Nobody likes small pieces of muffin, so the friends cut up and distribute the muffins in such a way that they maximize the size of the smallest muffin piece. What is the size of this smallest piece?
[u]Numerical tiebreaker problems:[/u]
[b]p11.[/b] $S$ is a set of positive integers with the following properties:
(a) There are exactly 3 positive integers missing from $S$.
(b) If $a$ and $b$ are elements of $S$, then $a + b$ is an element of $S$. (We allow $a$ and $b$ to be the same.)
How many possibilities are there for the set $S$?
[b]p12.[/b] In the trapezoid $ABCD$, both $\angle B$ and $\angle C$ are right angles, and all four sides of the trapezoid are tangent to the same circle. If $\overline{AB} = 13$ and $\overline{CD} = 33$, find the area of $ABCD$.
[b]p13.[/b] Alice wishes to walk from the point $(0, 0)$ to the point $(6, 4)$ in increments of $(1, 0)$ and $(0, 1)$, and Bob wishes to walk from the point $(0, 1)$ to the point $(6, 5)$ in increments of $(1, 0)$ and $(0,1)$. How many ways are there for Alice and Bob to get to their destinations if their paths never pass through the same point (even at different times)?
[b]p14.[/b] The continuous function $f(x)$ satisfies $9f(x + y) = f(x)f(y)$ for all real numbers $x$ and $y$. If $f(1) = 3$, what is $f(-3)$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2024 MMATHS, 3
Alice picks a random three-digit number, from $100$ to $999,$ inclusive. The probability that her first digit is larger than the sum of her other two digits can be expressed as a common fraction $\tfrac{a}{b}.$ Find $a+b.$
MMATHS Mathathon Rounds, 2019
[u]Round 5 [/u]
[b]p13.[/b] Suppose $\vartriangle ABC$ is an isosceles triangle with $\overline{AB} = \overline{BC}$, and $X$ is a point in the interior of $\vartriangle ABC$. If $m \angle ABC = 94^o$, $m\angle ABX = 17^o$, and $m\angle BAX = 13^o$, then what is $m\angle BXC$ (in degrees)?
[b]p14.[/b] Find the remainder when $\sum^{2019}_{n=1} 1 + 2n + 4n^2 + 8n^3$ is divided by $2019$.
[b]p15.[/b] How many ways can you assign the integers $1$ through $10$ to the variables $a, b, c, d, e, f, g, h, i$, and $j$ in some order such that $a < b < c < d < e, f < g < h < i$, $a < g, b < h, c < i$, $f < b, g < c$, and $h < d$?
[u]Round 6 [/u]
[b]p16.[/b] Call an integer $n$ equi-powerful if $n$ and $n^2$ leave the same remainder when divided by 1320. How many integers between $1$ and $1320$ (inclusive) are equi-powerful?
[b]p17.[/b] There exists a unique positive integer $j \le 10$ and unique positive integers $n_j$ , $n_{j+1}$, $...$, $n_{10}$ such that $$j \le n_j < n_{j+1} < ... < n_{10}$$ and $${n_{10} \choose 10}+ {n_9 \choose 9}+ ... + {n_j \choose j}= 2019.$$ Find $n_j + n_{j+1} + ... + n_{10}$.
[b]p18.[/b] If $n$ is a randomly chosen integer between $1$ and $390$ (inclusive), what is the probability that $26n$ has more positive factors than $6n$?
[u]Round 7[/u]
[b]p19.[/b] Suppose $S$ is an $n$-element subset of $\{1, 2, 3, ..., 2019\}$. What is the largest possible value of $n$ such that for every $2 \le k \le n$, $k$ divides exactly $n - 1$ of the elements of $S$?
[b]p20.[/b] For each positive integer $n$, let $f(n)$ be the fewest number of terms needed to write $n$ as a sum of factorials. For example, $f(28) = 3$ because $4! + 2! + 2! = 28$ and 28 cannot be written as the sum of fewer than $3$ factorials. Evaluate $f(1) + f(2) + ... + f(720)$.
[b]p21.[/b] Evaluate $\sum_{n=1}^{\infty}\frac{\phi (n)}{101^n-1}$ , where $\phi (n)$ is the number of positive integers less than or equal to n that are relatively prime to $n$.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2788993p24519281]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2021 MMATHS, 4
Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!"
Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I'd know for certain what it is?"
Cat says, "Nope! However, if I now told you the units digit of my favorite number, you'd know which one it is!"
Claire says, "Now I know your favorite number!" What is Cat's favorite number?
[i]Proposed by Andrew Wu[/i]
2022 MMATHS, 10
Define a function $f$ on the positive integers as follows: $f(n) = m$, where $m$ is the least positive integer such that $n$ is a factor of $m^2$. Find the smallest integer $M$ such that $\sqrt{M}$ is both a product of prime numbers, of which there are at least $3$, and a factor of $$\sum_{ d|M} f(d),$$ the sum of $f(d)$ for all positive integers $d$ that divide $M$.
2021 MMATHS, 11
If $\displaystyle\prod_{i=6}^{2021} (1-\tan^2((2^i)^\circ))$ can be written in the form $a^b$ for positive integers $a,b$ with $a$ squarefree, find $a+b$.
[i]Proposed by Deyuan Li and Andrew Milas[/i]
2024 MMATHS, 1
Let $ab^2=126, bc^2=14, cd^2=128, da^2=12.$ Find $\tfrac{bd}{ac}.$
2024 MMATHS, 2
Define the [i]factorial function[/i] of $n,$ denoted $\partial (n),$ as the sum of the factorials of the digits of $n.$ For example, $\partial(2024)=2!+0!+2!+4!=29.$ There are four positive integers such that $\partial(\partial(n))=n$ and $\partial(n) \neq n.$ Given that $n=871$ is one of them, compute the sum of the other three.
2022 MMATHS, 4
How many ways are there to choose three digits $A,B,C$ with $1 \le A \le 9$ and $0 \le B,C \le 9$ such that $\overline{ABC}_b$ is even for all choices of base $b$ with $b \ge 10$?
2023 MMATHS, 4
Let $A$ and $B$ be unit hexagons that share a center. Then, let $\mathcal{P}$ be the set of points contained in at least one of the hexagons. If the maximum possible area of $\mathcal{P}$ is $X$ and the minimum possible area of $\mathcal{P}$ is $Y,$ then the value of $Y-X$ can be expressed as $\tfrac{a\sqrt{b}-c}{d},$ where $a,b,c,d$ are positive integers such that $b$ is square-free and $\gcd(a,c,d)=1.$ Find $a+b+c+d.$
2021 MMATHS, 10
Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$, and suppose that $OI$ meets $AB$ and $AC$ at $P$ and $Q$, respectively. There exists a point $R$ on arc $\widehat{BAC}$ such that the circumcircles of triangles $PQR$ and $ABC$ are tangent. Given that $AB = 14$, $BC = 20$, and $CA = 26$, find $\frac{RC}{RB}$.
[i]Proposed by Andrew Wu[/i]
2018 MMATHS, 3
Suppose $n$ points are uniformly chosen at random on the circumference of the unit circle and that they are then connected with line segments to form an $n$-gon. What is the probability that the origin is contained in the interior of this $n$-gon? Give your answer in terms of $n$, and consider only $n \ge 3$.
2020 MMATHS, 4
Define the function $f(n)$ for positive integers $n$ as follows: if $n$ is prime, then $f(n) = 1$; and $f(ab) =
a \cdot f(b)+f(a)\cdot b$ for all positive integers $a$ and $b$. How many positive integers $n$ less than $5^{50}$ have the property that $f(n) = n$?
2022 MMATHS, 7
Katherine makes Benj play a game called $50$ Cent. Benj starts with $\$0.50$, and every century thereafter has a $50\%$ chance of doubling his money and a $50\%$ chance of having his money reset to $\$0.50$. What is the expected value of the amount of money Benj will have, in dollars, after $50$ centuries?
MMATHS Mathathon Rounds, 2014
[u]Round 1[/u]
[b]p1.[/b] A circle is inscribed inside a square such that the cube of the radius of the circle is numerically equal to the perimeter of the square. What is the area of the circle?
[b]p2.[/b] If the coefficient of $z^ky^k$ is $252$ in the expression $(z + y)^{2k}$, find $k$.
[b]p3.[/b] Let $f(x) = \frac{4x^4-2x^3-x^2-3x-2}{x^4-x^3+x^2-x-1}$ be a function defined on the real numbers where the denominator is not zero. The graph of $f$ has a horizontal asymptote. Compute the sum of the x-coordinates of the points where the graph of $f$ intersects this horizontal asymptote. If the graph of f does not intersect the asymptote, write $0$.
[u]Round 2 [/u]
[b]p4.[/b] How many $5$-digit numbers have strictly increasing digits? For example, $23789$ has strictly increasing digits, but $23889$ and $23869$ do not.
[b]p5.[/b] Let
$$y =\frac{1}{1 +\frac{1}{9 +\frac{1}{5 +\frac{1}{9 +\frac{1}{5 +...}}}}}$$ If $y$ can be represented as $\frac{a\sqrt{b} + c}{d}$ , where $b$ is not divisible by any squares, and the greatest common divisor of $a$ and $d$ is $1$, find the sum $a + b + c + d$.
[b]p6.[/b] “Counting” is defined as listing positive integers, each one greater than the previous, up to (and including) an integer $n$. In terms of $n$, write the number of ways to count to $n$.
[u]Round 3 [/u]
[b]p7.[/b] Suppose $p$, $q$, $2p^2 + q^2$, and $p^2 + q^2$ are all prime numbers. Find the sum of all possible values of $p$.
[b]p8.[/b] Let $r(d)$ be a function that reverses the digits of the $2$-digit integer $d$. What is the smallest $2$-digit positive integer $N$ such that for some $2$-digit positive integer $n$ and $2$-digit positive integer $r(n)$, $N$ is divisible by $n$ and $r(n)$, but not by $11$?
[b]p9.[/b] What is the period of the function $y = (\sin(3\theta) + 6)^2 - 10(sin(3\theta) + 7) + 13$?
[u]Round 4 [/u]
[b]p10.[/b] Three numbers $a, b, c$ are given by $a = 2^2 (\sum_{i=0}^2 2^i)$, $b = 2^4(\sum_{i=0}^4 2^i)$, and $c = 2^6(\sum_{i=0}^6 2^i)$ . $u, v, w$ are the sum of the divisors of a, b, c respectively, yet excluding the original number itself. What is the value of $a + b + c -u - v - w$?
[b]p11.[/b] Compute $\sqrt{6- \sqrt{11}} - \sqrt{6+ \sqrt{11}}$.
[b]p12.[/b] Let $a_0, a_1,..., a_n$ be such that $a_n\ne 0$ and $$(1 + x + x^3)^{341}(1 + 2x + x^2 + 2x^3 + 2x^4 + x^6)^{342} =\sum_{i=0}^n a_ix^i.$$ Find the number of odd numbers in the sequence $a_0, a_1,..., a_n$.
PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2781343p24424617]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMATHS Mathathon Rounds, 2019
[u]Round 1 [/u]
[b]p1.[/b] A small pizza costs $\$4$ and has $6$ slices. A large pizza costs $\$9$ and has $14$ slices. If the MMATHS organizers got at least $400$ slices of pizza (having extra is okay) as cheaply as possible, how many large pizzas did they buy?
[b]p2.[/b] Rachel flips a fair coin until she gets a tails. What is the probability that she gets an even number of heads before the tails?
[b]p3.[/b] Find the unique positive integer $n$ that satisfies $n! \cdot (n + 1)! = (n + 4)!$.
[u]Round 2 [/u]
[b]p4.[/b] The Portland Malt Shoppe stocks $10$ ice cream flavors and $8$ mix-ins. A milkshake consists of exactly $1$ flavor of ice cream and between $1$ and $3$ mix-ins. (Mix-ins can be repeated, the number of each mix-in matters, and the order of the mix-ins doesn’t matter.) How many different milkshakes can be ordered?
[b]p5.[/b] Find the minimum possible value of the expression $(x)^2 + (x + 3)^4 + (x + 4)^4 + (x + 7)^2$, where $x$ is a real number.
[b]p6.[/b] Ralph has a cylinder with height $15$ and volume $\frac{960}{\pi}$ . What is the longest distance (staying on the surface) between two points of the cylinder?
[u]Round 3 [/u]
[b]p7.[/b] If there are exactly $3$ pairs $(x, y)$ satisfying $x^2 + y^2 = 8$ and $x + y = (x - y)^2 + a$, what is the value of $a$?
[b]p8.[/b] If $n$ is an integer between $4$ and $1000$, what is the largest possible power of $2$ that $n^4 - 13n^2 + 36$ could be divisible by?
(Your answer should be this power of $2$, not just the exponent.)
[b]p9.[/b] Find the sum of all positive integers $n \ge 2$ for which the following statement is true: “for any arrangement of $n$ points in three-dimensional space where the points are not all collinear, you can always find one of the points such that the $n - 1$ rays from this point through the other points are all distinct.”
[u]Round 4 [/u]
[b]p10.[/b] Donald writes the number $12121213131415$ on a piece of paper. How many ways can he rearrange these fourteen digits to make another number where the digit in every place value is different from what was there before?
[b]p11.[/b] A question on Joe’s math test asked him to compute $\frac{a}{b} +\frac34$ , where $a$ and $b$ were both integers. Because he didn’t know how to add fractions, he submitted $\frac{a+3}{b+4}$ as his answer. But it turns out that he was right for these particular values of $a$ and $b$! What is the largest possible value that a could have been?
[b]p12.[/b] Christopher has a globe with radius $r$ inches. He puts his finger on a point on the equator. He moves his finger $5\pi$ inches North, then $\pi$ inches East, then $5\pi$ inches South, then $2\pi$ inches West. If he ended where he started, what is the largest possible value of $r$?
PS. You should use hide for answers. Rounds 5-7 have be posted [url=https://artofproblemsolving.com/community/c4h2789002p24519497]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2024 MMATHS, 12
$S_1,S_2,\ldots,S_n$ are subsets of $\{1,2,\ldots,10000\}$ which satisfy that, whenever $|S_i| > |S_j|$, the sum of all elements in $S_i$ is less than the sum of all elements in $S_j$. Let $m$ be the maximum number of distinct values among $|S_1|,\ldots,|S_n|$. Find $\left\lfloor\frac{m}{100}\right\rfloor$.
2023 MMATHS, 1
Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit multiple of $7$.”
Claire asks, “If you just told me the tens digit of the number, would I know your number?”
Cat says, “No. However, without knowing that, if I told you the tens digit of $100$ minus my number, you could determine my favorite number.”
Claire says, “Now I know your favorite number!"
What is Cat’s favorite number?
2016 MMATHS, 2
Suppose we have $2016$ points in a $2$-dimensional plane such that no three lie on a line. Two quadrilaterals are not disjoint if they share an edge or vertex, or if their edges intersect. Show that there are at least $504$ quadrilaterals with vertices among these points such that any two of the quadrilaterals are disjoint.
2021 MMATHS, 4
Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}, BC = 14,$ and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$. If $XY$ can be expressed as $a\sqrt{b} - c$ for positive integers $a,b,c$ with $c$ squarefree, find $a + b + c$.
[i]Proposed by Andrew Wu[/i]
2024 MMATHS, 2
Grant has a box with $6$ red balls, $5$ blue balls, $4$ green balls, $3$ yellow balls, $2$ orange balls, and $1$ purple ball. Grant selects $6$ balls at random, without replacement. Let $P$ be the probability that Grant selects six balls of different colors, and let $Q$ be the probability that Grant selects six balls of the same color. What is $\tfrac{P}{Q}$?
2022 MMATHS, 4
Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit perfect square!”
Claire asks, “If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I’d know for certain what it is?”
Cat says, “Yes! Moreover, if I told you a number and identified it as the sum of the digits of my favorite number, or if I told you a number and identified it as the positive difference of the digits of my favorite number, you wouldn’t know my favorite number.”
Claire says, “Now I know your favorite number!” What is Cat’s favorite number?
MMATHS Mathathon Rounds, 2018
[u]Round 5 [/u]
[b]p13.[/b] Circles $\omega_1$, $\omega_2$, and $\omega_3$ have radii $8$, $5$, and $5$, respectively, and each is externally tangent to the other two. Circle $\omega_4$ is internally tangent to $\omega_1$, $\omega_2$, and $\omega_3$, and circle $\omega_5$ is externally tangent to the same three circles. Find the product of the radii of $\omega_4$ and $\omega_5$.
[b]p14.[/b] Pythagoras has a regular pentagon with area $1$. He connects each pair of non-adjacent vertices with a line segment, which divides the pentagon into ten triangular regions and one pentagonal region. He colors in all of the obtuse triangles. He then repeats this process using the smaller pentagon. If he continues this process an infinite number of times, what is the total area that he colors in? Please rationalize the denominator of your answer.
p15. Maisy arranges $61$ ordinary yellow tennis balls and $3$ special purple tennis balls into a $4 \times 4 \times 4$ cube. (All tennis balls are the same size.) If she chooses the tennis balls’ positions in the cube randomly, what is the probability that no two purple tennis balls are touching?
[u]Round 6 [/u]
[b]p16.[/b] Points $A, B, C$, and $D$ lie on a line (in that order), and $\vartriangle BCE$ is isosceles with $\overline{BE} = \overline{CE}$. Furthermore, $F$ lies on $\overline{BE}$ and $G$ lies on $\overline{CE}$ such that $\vartriangle BFD$ and $\vartriangle CGA$ are both congruent to $\vartriangle BCE$. Let $H$ be the intersection of $\overline{DF}$ and $\overline{AG}$, and let $I$ be the intersection of $\overline{BE}$ and $\overline{AG}$. If $m \angle BCE = arcsin \left( \frac{12}{13} \right)$, what is $\frac{\overline{HI}}{\overline{FI}}$ ?
[b]p17.[/b] Three states are said to form a tri-state area if each state borders the other two. What is the maximum possible number of tri-state areas in a country with fifty states? Note that states must be contiguous and that states touching only at “corners” do not count as bordering.
[b]p18.[/b] Let $a, b, c, d$, and $e$ be integers satisfying $$2(\sqrt[3]{2})^2 + \sqrt[3]{2}a + 2b + (\sqrt[3]{2})^2c +\sqrt[3]{2}d + e = 0$$ and $$25\sqrt5 i + 25a - 5\sqrt5 ib - 5c + \sqrt5 id + e = 0$$ where $i =\sqrt{-1}$. Find $|a + b + c + d + e|$.
[u]Round 7[/u]
[b]p19.[/b] What is the greatest number of regions that $100$ ellipses can divide the plane into? Include the unbounded region.
[b]p20.[/b] All of the faces of the convex polyhedron $P$ are congruent isosceles (but NOT equilateral) triangles that meet in such a way that each vertex of the polyhedron is the meeting point of either ten base angles of the faces or three vertex angles of the faces. (An isosceles triangle has two base angles and one vertex angle.) Find the sum of the numbers of faces, edges, and vertices of $P$.
[b]p21.[/b] Find the number of ordered $2018$-tuples of integers $(x_1, x_2, .... x_{2018})$, where each integer is between $-2018^2$ and $2018^2$ (inclusive), satisfying $$6(1x_1 + 2x_2 +...· + 2018x_{2018})^2 \ge (2018)(2019)(4037)(x^2_1 + x^2_2 + ... + x^2_{2018}).$$
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2784936p24472982]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMATHS Mathathon Rounds, 2017
[u]Round 1[/u]
[b]p1.[/b] Jom and Terry both flip a fair coin. What is the probability both coins show the same side?
[b]p2.[/b] Under the same standard air pressure, when measured in Fahrenheit, water boils at $212^o$ F and freezes at $32^o$ F. At thesame standard air pressure, when measured in Delisle, water boils at $0$ D and freezes at $150$ D. If x is today’s temperature in Fahrenheit and y is today’s temperature expressed in Delisle, we have $y = ax + b$. What is the value of $a + b$? (Ignore units.)
[b]p3.[/b] What are the last two digits of $5^1 + 5^2 + 5^3 + · · · + 5^{10} + 5^{11}$?
[u]Round 2[/u]
[b]p4.[/b] Compute the average of the magnitudes of the solutions to the equation $2x^4 + 6x^3 + 18x^2 + 54x + 162 = 0$.
[b]p5.[/b] How many integers between $1$ and $1000000$ inclusive are both squares and cubes?
[b]p6.[/b] Simon has a deck of $48$ cards. There are $12$ cards of each of the following $4$ suits: fire, water, earth, and air. Simon randomly selects one card from the deck, looks at the card, returns the selected card to the deck, and shuffles the deck. He repeats the process until he selects an air card. What is the probability that the process ends without Simon selecting a fire or a water card?
[u]Round 3[/u]
[b]p7.[/b] Ally, Beth, and Christine are playing soccer, and Ally has the ball. Each player has a decision: to pass the ball to a teammate or to shoot it. When a player has the ball, they have a probability $p$ of shooting, and $1 - p$ of passing the ball. If they pass the ball, it will go to one of the other two teammates with equal probability. Throughout the game, $p$ is constant. Once the ball has been shot, the game is over. What is the maximum value of $p$ that makes Christine’s total probability of shooting the ball $\frac{3}{20}$ ?
[b]p8.[/b] If $x$ and $y$ are real numbers, then what is the minimum possible value of the expression $3x^2 - 12xy + 14y^2$ given that $x - y = 3$?
[b]p9.[/b] Let $ABC$ be an equilateral triangle, let $D$ be the reflection of the incenter of triangle $ABC$ over segment $AB$, and let $E$ be the reflection of the incenter of triangle $ABD$ over segment $AD$. Suppose the circumcircle $\Omega$ of triangle $ADE$ intersects segment $AB$ again at $X$. If the length of $AB$ is $1$, find the length of $AX$.
[u]Round 4[/u]
[b]p10.[/b] Elaine has $c$ cats. If she divides $c$ by $5$, she has a remainder of $3$. If she divides $c$ by $7$, she has a remainder of $5$. If she divides $c$ by $9$, she has a remainder of $7$. What is the minimum value $c$ can be?
[b]p11.[/b] Your friend Donny offers to play one of the following games with you. In the first game, he flips a fair coin and if it is heads, then you win. In the second game, he rolls a $10$-sided die (its faces are numbered from $1$ to $10$) $x$ times. If, within those $x$ rolls, the number $10$ appears, then you win. Assuming that you like winning, what is the highest value of $x$ where you would prefer to play the coin-flipping game over the die-rolling game?
[b]p12.[/b] Let be the set $X = \{0, 1, 2, ..., 100\}$. A subset of $X$, called $N$, is defined as the set that contains every element $x$ of $X$ such that for any positive integer $n$, there exists a positive integer $k$ such that n can be expressed in the form $n = x^{a_1}+x^{a_2}+...+x^{a_k}$ , for some integers $a_1, a_2, ..., a_k$ that satisfy $0 \le a_1 \le a_2 \le ... \le a_k$. What is the sum of the elements in $N$?
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