Found problems: 71
2010 CHMMC Winter, 4
Compute the number of positive divisors of $2010$.
2012 CHMMC Fall, 3
For a positive integer $n$, let $\sigma (n)$ be the sum of the divisors of $n$ (for example $\sigma (10) = 1 + 2 + 5 + 10 = 18$). For how many $n \in \{1, 2,. .., 100\}$, do we have $\sigma (n) < n+ \sqrt{n}$?
2018 CHMMC (Fall), 2
A cat is tied to one corner of the base of a tower. The base forms an equilateral triangle of side length $4$ m, and the cat is tied with a leash of length $8$ m. Let $A$ be the area of the region accessible to the cat. If we write $A = \frac{m}{n} k - \sqrt{\ell}$, where $m,n, k, \ell$ are positive integers such that $m$ and $n$ are relatively prime, and $\ell$ is squarefree, what is the value of $m + n + k + \ell$ ?
2010 CHMMC Fall, 4
Suppose $a$ is a real number such that $3a + 6$ is the greatest integer less than or equal to $a$ and $4a + 9$ is the least integer greater than or equal to $a$. Compute $a$.
2013 CHMMC (Fall), Individual
[b]p1.[/b] Compute
$$\sqrt{(\sqrt{63} +\sqrt{112} +\sqrt{175})(-\sqrt{63} +\sqrt{112} +\sqrt{175})(\sqrt{63}-\sqrt{112} +\sqrt{175})(\sqrt{63} +\sqrt{112} -\sqrt{175})}$$
[b]p2.[/b] Consider the set $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many distinct $3$-element subsets are there such that the sum of the elements in each subset is divisible by $3$?
[b]p3.[/b] Let $a^2$ and $b^2$ be two integers. Consider the triangle with one vertex at the origin, and the other two at the intersections of the circle $x^2 + y^2 = a^2 + b^2$ with the graph $ay = b|x|$. If the area of the triangle is numerically equal to the radius of the circle, what is this area?
[b]p4.[/b] Suppose $f(x) = x^3 + x - 1$ has roots $a$, $b$ and $c$. What is $$\frac{a^3}{1-a}+\frac{b^3}{1-b}+\frac{c^3}{1-c} ?$$
[b]p5.[/b] Lisa has a $2D$ rectangular box that is $48$ units long and $126$ units wide. She shines a laser beam into the box through one of the corners such that the beam is at a $45^o$ angle with respect to the sides of the box. Whenever the laser beam hits a side of the box, it is reflected perfectly, again at a $45^o$ angle. Compute the distance the laser beam travels until it hits one of the four corners of the box.
[b]p6.[/b] How many ways can we form a group with an odd number of members (plural) from $99$ people total?
Express your answer in the form $a^b + c$, where $a$, $b$, and $c$ are integers, and $a$ is prime.
[b]p7.[/b] Let $$S = \log_2 9 \log_3 16 \log_4 25 ...\log_{999} 1000000.$$ Compute the greatest integer less than or equal to $\log_2 S$.
[b]p8.[/b] A prison, housing exactly four hundred prisoners in four hundred cells numbered $1$-$400$, has a really messed-up warden. One night, when all the prisoners are asleep and all of their doors are locked, the warden toggles the locks on all of their doors (that is, if the door is locked, he unlocks the door, and if the door is unlocked, he locks it again), starting at door $1$ and ending at door $400$. The warden then toggles the lock on every other door starting at door $2$ ($2$, $4$, $6$, etc). After he has toggled the lock on every other door, the warden then toggles every third door (doors $3$, $6$, $9$, etc.), then every fourth door, etc., finishing by toggling every $400$th door (consisting of only the $400$th door). He then collapses in exhaustion.
Compute the number of prisoners who go free (that is, the number of unlocked doors) when they wake up the next morning.
[b]p9.[/b] Let $A$ and $B$ be fixed points on a $2$-dimensional plane with distance $AB = 1$. An ant walks on a straight line from point $A$ to some point $C$ on the same plane and finds that the distance from itself to $B$ always decreases at any time during this walk. Compute the area of the locus of points where point $C$ could possibly be located.
[b]p10.[/b] A robot starts in the bottom left corner of a $4 \times 4$ grid of squares. How many ways can it travel to each square exactly once and then return to its start if it is only allowed to move to an adjacent (not diagonal) square at each step?
[b]p11.[/b] Assuming real values for $p$, $q$, $r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $4 + 7i$, and the product of the other two roots is $3 - 4i$. Find $q$.
[b]p12.[/b] A cube is inscribed in a right circular cone such that one face of the cube lies on the base of the cone. If the ratio of the height of the cone to the radius of the cone is $2 : 1$, what fraction of the cone's volume does the cube take up? Express your answer in simplest radical form.
[b]p13.[/b] Let $$y =\dfrac{1}{1 +\dfrac{1}{9 +\dfrac{1}{5 +\dfrac{1}{9 +\dfrac{1}{5 +...}}}}}$$
If $y$ can be represented as $\frac{a\sqrt{b} + c}{d}$, where $b$ is not divisible by the square of any prime, and the greatest common divisor of $a$ and $d$ is $1$, find the sum $a + b + c + d$.
[b]p14.[/b] Alice wants to paint each face of an octahedron either red or blue. She can paint any number of faces a particular color, including zero. Compute the number of ways in which she can do this. Two ways of painting the octahedron are considered the same if you can rotate the octahedron to get from one to the other.
[b]p15.[/b] Find $n$ in the equation $$133^5 + 110^5 + 84^5 + 27^5 = n^5,$$ where $n$ is an integer less than $170$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2013 CHMMC (Fall), 3
Bill plays a game in which he rolls two fair standard six-sided dice with sides labeled one through six. He wins if the number on one of the dice is three times the number on the other die. If Bill plays this game three times, compute the probability that he wins at least once.
2022 CHMMC Winter (2022-23), 3
Suppose that $a,b,c$ are complex numbers with $a+b+c = 0$, $|abc| = 1$, $|b| = |c|$, and $$\frac{9-\sqrt{33}}{48}
\le \cos^2 \left( arg \left( \frac{b}{a} \right) \right)\le \frac{9+\sqrt{33}}{48} .$$
Find the maximum possible value of $|-a^6+b^6+c^6|$.
2017 CHMMC (Fall), 5
Find the number of primes $p$ such that $p! + 25p$ is a perfect square.
2010 CHMMC Fall, Individual
[b]p1.[/b] Susan plays a game in which she rolls two fair standard six-sided dice with sides labeled one through six. She wins if the number on one of the dice is three times the number on the other die. If Susan plays this game three times, compute the probability that she wins at least once.
[b]p2.[/b] In triangles $\vartriangle ABC$ and $\vartriangle DEF$, $DE = 4AB$, $EF = 4BC$, and $FD = 4CA$. The area of $\vartriangle DEF$ is $360$ units more than the area of $\vartriangle ABC$. Compute the area of $\vartriangle ABC$.
[b]p3.[/b] Andy has $2010$ square tiles, each of which has a side length of one unit. He plans to arrange the tiles in an $m\times n$ rectangle, where $mn = 2010$. Compute the sum of the perimeters of all of the different possible rectangles he can make. Two rectangles are considered to be the same if one can be rotated to become the other, so, for instance, a $1\times 2010$ rectangle is considered to be the same as a $2010\times 1$ rectangle.
[b]p4.[/b] Let $$S = \log_2 9 \log_3 16 \log_4 25 ... \log_{999} 1000000.$$
Compute the greatest integer less than or equal to $\log_2 S$.
[b]p5.[/b] Let $A$ and $B$ be fixed points in the plane with distance $AB = 1$. An ant walks on a straight line from point $A$ to some point $C$ in the plane and notices that the distance from itself to B always decreases at any time during this walk. Compute the area of the region in the plane containing all points where point $C$ could possibly be located.
[b]p6.[/b] Lisette notices that $2^{10} = 1024$ and $2^{20} = 1 048 576$. Based on these facts, she claims that every number of the form $2^{10k}$ begins with the digit $1$, where k is a positive integer. Compute the smallest $k$ such that Lisette's claim is false. You may or may not find it helpful to know that $ln 2 \approx 0.69314718$, $ln 5 \approx 1.60943791$, and $log_{10} 2 \approx 0:30103000$.
[b]p7.[/b] Let $S$ be the set of all positive integers relatively prime to $6$. Find the value of $\sum_{k\in S}\frac{1}{2^k}$ .
[b]p8.[/b] Euclid's algorithm is a way of computing the greatest common divisor of two positive integers $a$ and $b$ with $a > b$. The algorithm works by writing a sequence of pairs of integers as follows.
1. Write down $(a, b)$.
2. Look at the last pair of integers you wrote down, and call it $(c, d)$.
$\bullet$ If $d \ne 0$, let r be the remainder when c is divided by d. Write down $(d, r)$.
$\bullet$ If $d = 0$, then write down c. Once this happens, you're done, and the number you just wrote down is the greatest common divisor of a and b.
3. Repeat step 2 until you're done.
For example, with $a = 7$ and $b = 4$, Euclid's algorithm computes the greatest common divisor in $4$ steps:
$$(7, 4) \to (4, 3) \to (3, 1) \to (1, 0) \to 1$$
For $a > b > 0,$ compute the least value of a such that Euclid's algorithm takes $10$ steps to compute the greatest common divisor of $a$ and $b$.
[b]p9.[/b] Let $ABCD$ be a square of unit side length. Inscribe a circle $C_0$ tangent to all of the sides of the square. For each positive integer $n$, draw a circle Cn that is externally tangent to $C_{n-1}$ and also tangent to sides $AB$ and $AD$. Suppose $r_i$ is the radius of circle $C_i$ for every nonnegative integer $i$. Compute $\sqrt[200]{r_0/r_{100}}$.
[b]p10.[/b] Rachel and Mike are playing a game. They start at $0$ on the number line. At each positive integer on the number line, there is a carrot. At the beginning of the game, Mike picks a positive integer $n$ other than $30$. Every minute, Rachel moves to the next multiple of $30$ on the number line that has a carrot on it and eats that carrot. At the same time, every minute, Mike moves to the next multiple of $n$ on the number line that has a carrot on it and eats that carrot. Mike wants to pick an $n$ such that, as the game goes on, he is always within $1000$ units of Rachel. Compute the average (arithmetic mean) of all such $n$.
[b]p11.[/b] Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$. He knows the die is weighted so that one face comes up with probability $1/2$ and the other five faces have equal probability of coming up. He unfortunately does not know which side is weighted, but he knows each face is equally likely to be the weighted one. He rolls the die 5 times and gets a $1$, $2$, $3$, $4$ and $5$ in some unspecified order. Compute the probability that his next roll is a $6$.
[b]p12.[/b] Let $F_0 = 1$, $F_1 = 1$ and $F_k = F_{k-1} + F_{k-2}$. Let $P(x) =\sum^{99}_{k=0} x^{F_k}$ . The remainder when $P(x)$ is divided by $x^3 - 1$ can be expressed as $ax^2 + bx + c$. Find $2a + b$.
[b]p13.[/b] Let $\theta \ne 0$ be the smallest acute angle for which $\sin \theta$, $\sin (2\theta)$, $\sin (3\theta)$, when sorted in increasing order, form an arithmetic progression. Compute $\cos (\theta/2)$.
[b]p14.[/b] A $4$-dimensional hypercube of edge length 1 is constructed in $4$-space with its edges parallel to the coordinate axes and one vertex at the origin. The coordinates of its sixteen vertices are given by $(a, b, c, d)$, where each of $a$, $b$, $c$, and $d$ is either $0$ or $1$. The $3$-dimensional hyperplane given by $x + y + z + w = 2$ intersects the hypercube at $6$ of its vertices. Compute the $3$-dimensional volume of the solid formed by the intersection.
[b]p15.[/b] A student puts $2010$ red balls and $1957$ blue balls into a box. Weiqing draws randomly from the box one ball at a time without replacement. She wins if, at anytime, the total number of blue balls drawn is more than the total number of red balls drawn. Assuming Weiqing keeps drawing balls until she either wins or runs out, ompute the probability that she eventually wins.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2021 CHMMC Winter (2021-22), Individual
[b]p1.[/b] Fleming has a list of 8 mutually distinct integers between $90$ to $99$, inclusive. Suppose that the list has median $94$, and that it contains an even number of odd integers. If Fleming reads the numbers in the list from smallest to largest, then determine the sixth number he reads.
[b]p2.[/b] Find the number of ordered pairs $(x,y)$ of three digit base-$10$ positive integers such that $x-y$ is a positive integer, and there are no borrows in the subtraction $x-y$. For example, the subtraction on the left has a borrow at the tens digit but not at the units digit, whereas the subtraction on the right has no borrows.
$$\begin{tabular}{ccccc}
& 4 & 7 & 2 \\
- & 1 & 9 & 1\\
\hline
& 2 & 8 & 1 \\
\end{tabular}\,\,\, \,\,\, \begin{tabular}{ccccc}
& 3 & 7 & 9 \\
- & 2 & 6 & 3\\
\hline
& 1 & 1 & 6 \\
\end{tabular}$$
[b]p3.[/b] Evaluate
$$1 \cdot 2 \cdot 3-2 \cdot 3 \cdot 4+3 \cdot 4 \cdot 5- 4 \cdot 5 \cdot 6+ ... +2017 \cdot 2018 \cdot 2019 -2018 \cdot 2019 \cdot 2020+1010 \cdot 2019 \cdot 2021$$
[b]p4.[/b] Find the number of ordered pairs of integers $(a,b)$ such that $$\frac{ab+a+b}{a^2+b^2+1}$$ is an integer.
[b]p5.[/b] Lin Lin has a $4\times 4$ chessboard in which every square is initially empty. Every minute, she chooses a random square $C$ on the chessboard, and places a pawn in $C$ if it is empty. Then, regardless of whether $C$ was previously empty or not, she then immediately places pawns in all empty squares a king’s move away from $C$. The expected number of minutes before the entire chessboard is occupied with pawns equals $\frac{m}{n}$ for relatively prime positive integers $m$,$n$. Find $m+n$.
A king’s move, in chess, is one square in any direction on the chessboard: horizontally, vertically, or diagonally.
[b]p6.[/b] Let $P(x) = x^5-3x^4+2x^3-6x^2+7x+3$ and $a_1,...,a_5$ be the roots of$ P(x)$. Compute
$$\sum^5_{k=1}(a^3_k -4a^2_k +a_k +6).$$
[b]p7.[/b] Rectangle $AXCY$ with a longer length of $11$ and square $ABCD$ share the same diagonal $\overline{AC}$. Assume $B$,$X$ lie on the same side of $\overline{AC}$ such that triangle$ BXC$ and square $ABCD$ are non-overlapping. The maximum area of $BXC$ across all such configurations equals $\frac{m}{n}$ for relatively prime positive integers $m$,$n$. Compute $m+n$.
[b]p8.[/b] Earl the electron is currently at $(0,0)$ on the Cartesian plane and trying to reach his house at point $(4,4)$. Each second, he can do one of three actions: move one unit to the right, move one unit up, or teleport to the point that is the reflection of its current position across the line $y=x$. Earl cannot teleport in two consecutive seconds, and he stops taking actions once he reaches his house.
Earl visits a chronologically ordered sequence of distinct points $(0,0)$, $...$, $(4,4)$ due to his choice of actions. This is called an [i]Earl-path[/i]. How many possible such [i]Earl-paths[/i] are there?
[b]p9.[/b] Let $P(x)$ be a degree-$2022$ polynomial with leading coefficient $1$ and roots $\cos \left( \frac{2\pi k}{2023} \right)$ for $k = 1$ , $...$,$2022$ (note $P(x)$ may have repeated roots). If $P(1) =\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, then find the remainder when $m+n$ is divided by $100$.
[b]p10.[/b] A randomly shuffled standard deck of cards has $52$ cards, $13$ of each of the four suits. There are $4$ Aces and $4$ Kings, one of each of the four suits. One repeatedly draws cards from the deck until one draws an Ace. Given that the first King appears before the first Ace, the expected number of cards one draws after the first King and before the first Ace is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[b]p11.[/b] The following picture shows a beam of light (dashed line) reflecting off a mirror (solid line). The [i]angle of incidence[/i] is marked by the shaded angle; the[i] angle of reflection[/i] is marked by the unshaded angle.
[img]https://cdn.artofproblemsolving.com/attachments/9/d/d58086e5cdef12fbc27d0053532bea76cc50fd.png[/img]
The sides of a unit square $ABCD$ are magically distorted mirrors such that whenever a light beam hits any of the mirrors, the measure of the angle of incidence between the light beam and the mirror is a positive real constant $q$ degrees greater than the measure of the angle of reflection between the light beam and the mirror. A light beam emanating from $A$ strikes $\overline{CD}$ at $W_1$ such that $2DW_1 =CW_1$, reflects off of $\overline{CD}$ and then strikes $\overline{BC}$ at $W_2$ such that $2CW_2 = BW_2$, reflects off of $\overline{BC}$, etc. To this end, denote $W_i$ the $i$-th point at which the light beam strikes $ABCD$.
As $i$ grows large, the area of $W_iW_{i+1}W_{i+2}W_{i+3}$ approaches $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
[b]p12.[/b] For any positive integer $m$, define $\phi (m)$ the number of positive integers $k \le m$ such that $k$ and $m$ are relatively prime. Find the smallest positive integer $N$ such that $\sqrt{ \phi (n) }\ge 22$ for any integer $n \ge N$.
[b]p13.[/b] Let $n$ be a fixed positive integer, and let $\{a_k\}$ and $\{b_k\}$ be sequences defined recursively by
$$a_1 = b_1 = n^{-1}$$
$$a_j = j(n- j+1)a_{j-1}\,\,\, , \,\,\, j > 1$$
$$b_j = nj^2b_{j-1}+a_j\,\,\, , \,\,\, j > 1$$
When $n = 2021$, then $a_{2021} +b_{2021} = m \cdot 2017^2$ for some positive integer $m$. Find the remainder when $m$ is divided by $2017$.
[b]p14.[/b] Consider the quadratic polynomial $g(x) = x^2 +x+1020100$. A positive odd integer $n$ is called $g$-[i]friendly[/i] if and only if there exists an integer $m$ such that $n$ divides $2 \cdot g(m)+2021$. Find the number of $g$-[i]friendly[/i] positive odd integers less than $100$.
[b]p15.[/b] Let $ABC$ be a triangle with $AB < AC$, inscribed in a circle with radius $1$ and center $O$. Let $H$ be the intersection of the altitudes of $ABC$. Let lines $\overline{OH}$, $\overline{BC}$ intersect at $T$. Suppose there is a circle passing through $B$, $H$, $O$, $C$. Given $\cos (\angle ABC-\angle BCA) = \frac{11}{32}$ , then $TO = \frac{m\sqrt{p}}{n}$ for relatively prime positive integers $m$,$n$ and squarefree positive integer $p$. Find $m+n+ p$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2015 CHMMC (Fall), Individual
[b]p1.[/b] The following number is the product of the divisors of $n$.
$$2^63^3$$
What is $n$?
[b]p2.[/b] Let a right triangle have the sides $AB =\sqrt3$, $BC =\sqrt2$, and $CA = 1$. Let $D$ be a point such that $AD = BD = 1$. Let $E$ be the point on line $BD$ that is equidistant from $D$ and $A$. Find the angle $\angle AEB$.
[b]p3.[/b] There are twelve indistinguishable blackboards that are distributed to eight different schools. There must be at least one board for each school. How many ways are there of distributing the boards?
[b]p4.[/b] A Nishop is a chess piece that moves like a knight on its first turn, like a bishop on its second turn, and in general like a knight on odd-numbered turns and like a bishop on even-numbered turns. A Nishop starts in the bottom-left square of a $3\times 3$-chessboard. How many ways can it travel to touch each square of the chessboard exactly once?
[b]p5.[/b] Let a Fibonacci Spiral be a spiral constructed by the addition of quarter-circles of radius $n$, where each $n$ is a term of the Fibonacci series:
$$1, 1, 2, 3, 5, 8,...$$
(Each term in this series is the sum of the two terms that precede it.) What is the arclength of the maximum Fibonacci spiral that can be enclosed in a rectangle of area $714$, whose side lengths are terms in the Fibonacci series?
[b]p6.[/b] Suppose that $a_1 = 1$ and
$$a_{n+1} = a_n -\frac{2}{n + 2}+\frac{4}{n + 1}-\frac{2}{n}$$
What is $a_{15}$?
[b]p7.[/b] Consider $5$ points in the plane, no three of which are collinear. Let $n$ be the number of circles that can be drawn through at least three of the points. What are the possible values of $n$?
[b]p8.[/b] Find the number of positive integers $n$ satisfying $\lfloor n /2014 \rfloor =\lfloor n/2016 \rfloor$.
[b]p9.[/b] Let $f$ be a function taking real numbers to real numbers such that for all reals $x \ne 0, 1$, we have
$$f(x) + f \left( \frac{1}{1 - x}\right)= (2x - 1)^2 + f\left( 1 -\frac{1}{ x}\right)$$
Compute $f(3)$.
[b]p10.[/b] Alice and Bob split $5$ beans into piles. They take turns removing a positive number of beans from a pile of their choice. The player to take the last bean loses. Alice plays first. How many ways are there to split the piles such that Alice has a winning strategy?
[b]p11.[/b] Triangle $ABC$ is an equilateral triangle of side length $1$. Let point $M$ be the midpoint of side $AC$. Another equilateral triangle $DEF$, also of side length $1$, is drawn such that the circumcenter of $DEF$ is $M$, point $D$ rests on side $AB$. The length of $AD$ is of the form $\frac{a+\sqrt{b}}{c}$ , where $b$ is square free. What is $a + b + c$?
[b]p12.[/b] Consider the function $f(x) = \max \{-11x- 37, x - 1, 9x + 3\}$ defined for all real $x$. Let $p(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with x values $t_1$, $t_2$ and $t_3$ Compute the maximum value of $t_1 + t_2 + t_3$ over all possible $p$.
[b]p13.[/b] Circle $J_1$ of radius $77$ is centered at point $X$ and circle $J_2$ of radius $39$ is centered at point $Y$. Point $A$ lies on $J1$ and on line $XY$ , such that A and Y are on opposite sides of $X$. $\Omega$ is the unique circle simultaneously tangent to the tangent segments from point $A$ to $J_2$ and internally tangent to $J_1$. If $XY = 157$, what is the radius of $\Omega$ ?
[b]p14.[/b] Find the smallest positive integer $n$ so that for any integers $a_1, a_2,..., a_{527}$,the number
$$\left( \prod^{527}_{j=1} a_j\right) \cdot\left( \sum^{527}_{j=1} a^n_j\right)$$
is divisible by $527$.
[b]p15.[/b] A circle $\Omega$ of unit radius is inscribed in the quadrilateral $ABCD$. Let circle $\omega_A$ be the unique circle of radius $r_A$ externally tangent to $\Omega$, and also tangent to segments $AB$ and $DA$. Similarly define circles $\omega_B$, $\omega_C$, and $\omega_D$ and radii $r_B$, $r_C$, and $r_D$. Compute the smallest positive real $\lambda$ so that $r_C < \lambda$ over all such configurations with $r_A > r_B > r_C > r_D$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2022 CHMMC Winter (2022-23), Individual
[b]p1.[/b] Given any four digit number $X = \underline{ABCD}$, consider the quantity $Y(X) = 2 \cdot \underline{AB}+\underline{CD}$. For example, if $X = 1234$, then $Y(X) = 2 \cdot 12+34 = 58$. Find the sum of all natural numbers $n \le 10000$ such that over all four digit numbers $X$, the number $n$ divides $X$ if and only if it also divides $Y(X)$.
[b]p2.[/b] A sink has a red faucet, a blue faucet, and a drain. The two faucets release water into the sink at constant but different rates when turned on, and the drain removes water from the sink at a constant rate when opened. It takes $5$ minutes to fill the sink (from empty to full) when the drain is open and only the red faucet is on, it takes $10$ minutes to fill the sink when the drain is open and only the blue faucet is on, and it takes $15$ seconds to fill the sink when both faucets are on and the drain is closed. Suppose that the sink is currently one-thirds full of water, and the drain is opened. Rounded to the nearest integer, how many seconds will elapse before the sink is emptied (keeping the two faucets closed)?
[b]p3.[/b] One of the bases of a right triangular prism is a triangle $XYZ$ with side lengths $XY = 13$, $YZ = 14$, $ZX = 15$. Suppose that a sphere may be positioned to touch each of the five faces of the prism at exactly one point. A plane parallel to the rectangular face of the prism containing $\overline{YZ}$ cuts the prism and the sphere, giving rise to a cross-section of area $A$ for the prism and area $15\pi$ for the sphere. Find the sum of all possible values of $A$.
[b]p4.[/b] Albert, Brian, and Christine are hanging out by a magical tree. This tree gives each of them a stick, each of which have a non-negative real length. Say that Albert gets a branch of length $x$, Brian a branch of length $y$, and Christine a branch of length $z$, and the lengths follow the condition that $x+y+z = 2$. Let $m$ and $n$ be the minimum and maximum possible values of $xy+yz+xz-xyz$, respectively. What is $m+n$?
[b]p5.[/b] Let $S := MATHEMATICSMATHEMATICSMATHE...$ be the sequence where $7$ copies of the word $MATHEMATICS$ are concatenated together. How many ways are there to delete all but five letters of $S$ such that the resulting subsequence is $CHMMC$?
[b]p6.[/b] Consider two sequences of integers $a_n$ and $b_n$ such that $a_1 = a_2 = 1$, $b_1 = b_2 = 1$ and that the following recursive relations are satisfied for integers $n > 2$:
$$a_n = a_{n-1}a_{n-2}-b_{n-1}b_{n-2},$$
$$b_n = b_{n-1}a_{n-2}+a_{n-1}b_{n-2}.$$
Determine the value of $$\sum_{1\le n\le2023,b_n \ne 0} \frac{a_n}{b_n}.$$
[b]p7.[/b] Suppose $ABC$ is a triangle with circumcenter $O$. Let $A'$ be the reflection of $A$ across $\overline{BC}$. If $BC =12$, $\angle BAC = 60^o$, and the perimeter of $ABC$ is $30$, then find $A'O$.
[b]p8.[/b] A class of $10$ students wants to determine the class president by drawing slips of paper from a box. One of the students, Bob, puts a slip of paper with his name into the box. Each other student has a $\frac12$ probability of putting a slip of paper with their own name into the box and a $\frac12$ probability of not doing so. Later, one slip is randomly selected from the box. Given that Bob’s slip is selected, find the expected number of slips of paper in the box before the slip is selected.
[b]p9.[/b] Let $a$ and $b$ be positive integers, $a > b$, such that $6! \cdot 11$ divides $x^a -x^b$ for all positive integers $x$. What is the minimum possible value of $a+b$?
[b]p10.[/b] Find the number of pairs of positive integers $(m,n)$ such that $n < m \le 100$ and the polynomial $x^m+x^n+1$ has a root on the unit circle.
[b]p11.[/b] Let $ABC$ be a triangle and let $\omega$ be the circle passing through $A$, $B$, $C$ with center $O$. Lines $\ell_A$, $\ell_B$, $\ell_C$ are drawn tangent to $\omega$ at $A$, $B$, $C$ respectively. The intersections of these lines form a triangle $XYZ$ where $X$ is the intersection of $\ell_B$ and $\ell_C$, $Y$ is the intersection of $\ell_C$ and $\ell_A$, and $Z$ is the intersection of $\ell_A$ and $\ell_B$. Let $P$ be the intersection of lines $\overline{OX}$ and $\overline{YZ}$. Given $\angle ACB = \frac32 \angle ABC$ and $\frac{AC}{AB} = \frac{15}{16}$ , find $\frac{ZP}{YP}$.
[b]p12.[/b] Compute the remainder when $$\sum_{1\le a,k\le 2021} a^k$$ is divided by $2022$ (in the above summation $a,k$ are integers).
[b]p13.[/b] Consider a $7\times 2$ grid of squares, each of which is equally likely to be colored either red or blue. Madeline would like to visit every square on the grid exactly once, starting on one of the top two squares and ending on one of the bottom two squares. She can move between two squares if they are adjacent or diagonally adjacent. What is the probability that Madeline may visit the squares of the grid in this way such that the sequence of colors she visits is alternating (i.e., red, blue, red,... or blue, red, blue,... )?
[b]p14.[/b] Let $ABC$ be a triangle with $AB = 8$, $BC = 10$, and $CA = 12$. Denote by $\Omega_A$ the $A$-excircle of $ABC$, and suppose that $\Omega_A$ is tangent to $\overline{AB}$ and $\overline{AC}$ at $F$ and $E$, respectively. Line $\ell \ne \overline{BC}$ is tangent to $\Omega_A$ and passes through the midpoint of $\overline{BC}$. Let $T$ be the intersection of $\overline{EF}$ and $\ell$. Compute the area of triangle $ATB$.
[b]p15.[/b] For any positive integer $n$, let $D_n$ be the set of ordered pairs of positive integers $(m,d)$ such that $d$ divides $n$ and gcd$(m,n) = 1$, $1 \le m \le n$. For any positive integers $a$, $b$, let $r(a,b)$ be the non-negative remainder when $a$ is divided by $b$. Denote by $S_n$ the sum $$S_n = \sum_{(m,d)\in D_n} r(m,d).$$ Determine the value of $S_{396}$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2017 CHMMC (Fall), 1
Let $p$, $q$, $r$, and $s$ be 4 distinct primes such that $p+q+r+s$ is prime, and the numbers $p^2+qr$ and $p^2+qs$ are both perfect squares. What is the value of $p+q+r+s$?
2017 CHMMC (Fall), 2
Let $N$ be the number of sequences $a_1, a_2, . . . , a_{10}$ of ten positive integers such that
(i) the value of each term of the sequence at most $30$,
(ii) the arithmetic mean of any three consecutive terms of the sequence is an integer, and
(iii) the arithmetic mean of any fi ve consecutive terms of the sequence is an integer.
Compute $\sqrt{N}$.
2019 CHMMC (Fall), Individual
[b]p1.[/b] Consider a cube with side length $2$. Take any one of its vertices and consider the three midpoints of the three edges emanating from that vertex. What is the distance from that vertex to the plane formed by those three midpoints?
[b]p2.[/b] Digits $H$, $M$, and $C$ satisfy the following relations where $\overline{ABC}$ denotes the number whose digits in base $10$ are $A$, $B$, and $C$.
$$\overline{H}\times \overline{H} = \overline{M}\times \overline{C} + 1$$
$$\overline{HH}\times \overline{H} = \overline{MC}\times \overline{C} + 1$$
$$\overline{HHH}\times \overline{H} = \overline{MCC}\times \overline{C} + 1$$
Find $\overline{HMC}$.
[b]p3.[/b] Two players play the following game on a table with fair two-sided coins. The first player starts with one, two, or three coins on the table, each with equal probability. On each turn, the player flips all the coins on the table and counts how many coins land heads up. If this number is odd, a coin is removed from the table. If this number is even, a coin is added to the table. A player wins when he/she removes the last coin on the table. Suppose the game ends. What is the probability that the first player wins?
[b]p4.[/b] Cyclic quadrilateral $[BLUE]$ has right $\angle E$. Let $R$ be a point not in $[BLUE]$. If $[BLUR] =[BLUE]$, $\angle ELB = 45^o$, and $\overline{EU} = \overline{UR}$, find $\angle RUE$.
[b]p5.[/b] There are two tracks in the $x, y$ plane, defined by the equations
$$y =\sqrt{3 - x^2}\,\,\, \text{and} \,\,\,y =\sqrt{4- x^2}$$
A baton of length $1$ has one end attached to each track and is allowed to move freely, but no end may be picked up or go past the end of either track. What is the maximum area the baton can sweep out?
[b]p6.[/b] For integers $1 \le a \le 2$, $1 \le b \le 10$,$ 1 \le c \le 12$, $1 \le d \le 18$, let $f(a, b, c, d)$ be the unique integer between $0$ and $8150$ inclusive that leaves a remainder of a when divided by $3$, a remainder of $b$ when divided by $11$, a remainder of $c$ when divided by $13$, and a remainder of $d$ when divided by $19$. Compute $$\sum_{a+b+c+d=23}f(a, b, c, d).$$
[b]p7.[/b] Compute $\cos ( \theta)$ if $$\sum^{\infty}_{n=0} \frac{ \cos (n\theta)}{3^n} = 1.$$
[b]p8.[/b] How many solutions does this equation $$\left(\frac{a+b}{2}\right)^2=\left(\frac{b+c}{2019}\right)^2$$ have in positive integers $a, b, c$ that are all less than $2019^2$?
[b]p9.[/b] Consider a square grid with vertices labeled $1, 2, 3, 4$ clockwise in that order. Fred the frog is jumping between vertices, with the following rules: he starts at the vertex label $1$, and at any given vertex he jumps to the vertex diagonally across from him with probability $\frac12$ and the vertices adjacent to him each with probability $\frac14$ . After $2019$ jumps, suppose the probability that the sum of the labels on the last two vertices he has visited is $3$ can be written as $2^{-m} -2^{-n}$ for positive integers $m,n$. Find $m + n$.
[b]p10.[/b] The base ten numeral system uses digits $0-9$ and each place value corresponds to a power of $10$. For example, $$2019 = 2 \cdot 10^3 + 0 \cdot 10^2 + 1 \cdot 10^1 + 9 \cdot 10^0.$$
Let $\phi =\frac{1 +\sqrt5}{2}$. We can define a similar numeral system, base , where we only use digits $0$ and $1$, and each place value corresponds to a power of . For example, $$11.01 = 1 \cdot \phi^1 + 1 \cdot \phi^0 + 0 \cdot \phi^{-1} + 1 \cdot \phi^{-2}$$
Note that base representations are not unique, because, for example, $100_{\phi} = 11_{\phi}$ . Compute the base $\phi$ representation of $7$ with the fewest number of $1$s.
[b]p11.[/b] Let $ABC$ be a triangle with $\angle BAC = 60^o$ and with circumradius $1$. Let $G$ be its centroid and $D$ be the foot of the perpendicular from $A$ to $BC$. Suppose $AG =\frac{\sqrt6}{3}$ . Find $AD$.
[b]p12.[/b] Let $f(a, b)$ be a function with the following properties for all positive integers $a \ne b$:
$$f(1, 2) = f(2, 1)$$
$$f(a, b) + f(b, a) = 0$$
$$f(a + b, b) = f(b, a) + b$$
Compute: $$\sum^{2019}_{i=1} f(4^i - 1, 2^i) + f(4^i + 1, 2^i)$$
[b]p13.[/b] You and your friends have been tasked with building a cardboard castle in the two-dimensional Cartesian plane. The castle is built by the following rules:
1. There is a tower of height $2^n$ at the origin.
2. From towers of height $2^i \ge 2$, a wall of length $2^{i-1}$ can be constructed between the aforementioned tower and a new tower of height $2^{i-1}$. Walls must be parallel to a coordinate axis, and each tower must be connected to at least one other tower by a wall.
If one unit of tower height costs $\$9$ and one unit of wall length costs $\$3$ and $n = 1000$, how many distinct costs are there of castles that satisfy the above constraints? Two castles are distinct if there exists a tower or wall that is in one castle but not in the other.
[b]p14.[/b] For $n$ digits, $(a_1, a_2, ..., a_n)$ with $0 \le a_i < n$ for $i = 1, 2,..., n$ and $a_1 \ne 0$ define $(\overline{a_1a_2 ... a_n})_n$ to be the number with digits $a_1$, $a_2$, $...$, $a_n$ written in base $n$. Let $S_n = \{(a_1, a_2, a_3,..., a_n)| \,\,\, (n + 1)| (\overline{a_1a_2 ... a_n})_n, a_1 \ge 1\}$ be the set of $n$-tuples such that $(\overline{a_1a_2 ... a_n})_n$ is divisible by $n + 1$.
Find all $n > 1$ such that $n$ divides $|S_n| + 2019$.
[b]p15.[/b] Let $P$ be the set of polynomials with degree $2019$ with leading coefficient $1$ and non-leading coefficients from the set $C = \{-1, 0, 1\}$. For example, the function $f = x^{2019} - x^{42} + 1$ is in $P$, but the functions $f = x^{2020}$, $f = -x^{2019}$, and $f = x^{2019} + 2x^{21}$ are not in $P$.
Define a [i]swap [/i]on a polynomial $f$ to be changing a term $ax^n$ to $bx^n$ where $b \in C$ and there are no terms with degree smaller than $n$ with coefficients equal to $a$ or $b$. For example, a swap from $x^{2019} + x^{17} - x^{15} + x^{10}$ to $x^{2019} + x^{17} - x^{15} - x^{10}$ would be valid, but the following swaps would not be valid:
$$x^{2019} + x^3 \,\,\, \text{to} \,\,\, x^{2019}$$
$$x^{2019} + x^3 \,\,\, \text{to} \,\,\, x^{2019} + x^3 + x^2$$
$$x^{2019} + x^2 + x + 1 \,\,\, \text{to} \,\,\, x^{2019} - x^2 - x - 1$$
Let $B$ be the set of polynomials in $P$ where all non-leading terms have the same coefficient. There are $p$ polynomials that can be reached from each element of $B$ in exactly $s$ swaps, and there exist $0$ polynomials that can be reached from each element of $B$ in less than $s$ swaps.
Compute $p \cdot s$, expressing your answer as a prime factorization.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2010 CHMMC Winter, Mixer
[b]p1.[/b] Compute $x$ such that $2009^{2010} \equiv x$ (mod $2011$) and $0 \le x < 2011$.
[b]p2.[/b] Compute the number of "words" that can be formed by rearranging the letters of the word "syzygy" so that the y's are evenly spaced. (The $y$'s are evenly spaced if the number of letters (possibly zero) between the first $y$ and the second $y$ is the same as the number of letters between the second $y$ and the third $y$.)
[b]p3.[/b] Let $A$ and $B$ be subsets of the integers, and let $A + B$ be the set containing all sums of the form $a + b$, where $a$ is an element of $A$, and $b$ is an element of $B$. For example, if $A = \{0, 4, 5\}$ and $B =\{-3,-1, 2, 6\}$, then $A + B = \{-3,-1, 1, 2, 3, 4, 6, 7, 10, 11\}$. If $A$ has $1955$ elements and $B$ has $1891$ elements, compute the smallest possible number of elements in $A + B$.
[b]p4.[/b] Compute the sum of all integers of the form $p^n$ where $p$ is a prime, $n \ge 3$, and $p^n \le 1000$.
[b]p5.[/b] In a season of interhouse athletics at Caltech, each of the eight houses plays each other house in a particular sport. Suppose one of the houses has a $1/3$ chance of beating each other house. If the results of the games are independent, compute the probability that they win at least three games in a row.
[b]p6.[/b] A positive integer $n$ is special if there are exactly $2010$ positive integers smaller than $n$ and relatively prime to $n$. Compute the sum of all special numbers.
[b]p7.[/b] Eight friends are playing informal games of ultimate frisbee. For each game, they split themselves up into two teams of four. They want to arrange the teams so that, at the end of the day, each pair of players has played at least one game on the same team. Determine the smallest number of games they need to play in order to achieve this.
[b]p8.[/b] Compute the number of ways to choose five nonnegative integers $a, b, c, d$, and $e$, such that $a + b + c + d + e = 20$.
[b]p9.[/b] Is $23$ a square mod $41$? Is $15$ a square mod $41$?
[b]p10.[/b] Let $\phi (n)$ be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Compute $ \sum_{d|15015} \phi (d)$.
[b]p11.[/b] Compute the largest possible volume of an regular tetrahedron contained in a cube with volume $1$.
[b]p12.[/b] Compute the number of ways to cover a $4 \times 4$ grid with dominoes.
[b]p13.[/b] A collection of points is called mutually equidistant if the distance between any two of them is the same. For example, three mutually equidistant points form an equilateral triangle in the plane, and four mutually equidistant points form a regular tetrahedron in three-dimensional space. Let $A$, $B$, $C$, $D$, and $E$ be five mutually equidistant points in four-dimensional space. Let $P$ be a point such that $AP = BP = CP = DP = EP = 1$. Compute the side length $AB$.
[b]p14. [/b]Ten turtles live in a pond shaped like a $10$-gon. Because it's a sunny day, all the turtles are sitting in the sun, one at each vertex of the pond. David decides he wants to scare all the turtles back into the pond. When he startles a turtle, it dives into the pond. Moreover, any turtles on the two neighbouring vertices also dive into the pond. However, if the vertex opposite the startled turtle is empty, then a turtle crawls out of the pond and sits at that vertex. Compute the minimum number of times David needs to startle a turtle so that, by the end, all but one of the turtles are in the pond.
[b]p15.[/b] The game hexapawn is played on a $3 \times 3$ chessboard. Each player starts with three pawns on the row nearest him or her. The players take turns moving their pawns. Like in chess, on a player's turn he or she can either
$\bullet$ move a pawn forward one space if that square is empty, or
$\bullet$ capture an opponent's pawn by moving his or her own pawn diagonally forward one space into the opponent's pawn's square.
A player wins when either
$\bullet$ he or she moves a pawn into the last row, or
$\bullet$ his or her opponent has no legal moves.
Eve and Fred are going to play hexapawn. However, they're not very good at it. Each turn, they will pick a legal move at random with equal probability, with one exception: If some move will immediately win the game (by either of the two winning conditions), then he or she will make that move, even if other moves are available. If Eve moves first, compute the probability that she will win.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2010 CHMMC Winter, Individual
[b]p1.[/b] Compute the degree of the least common multiple of the polynomials $x - 1$, $x^2 - 1$, $x^3 - 1$,$...$, $x^{10} -1$.
[b]p2.[/b] A line in the $xy$ plane is called wholesome if its equation is $y = mx+b$ where $m$ is rational and $b$ is an integer. Given a point with integer coordinates $(x,y)$ on a wholesome line $\ell$, let $r$ be the remainder when $x$ is divided by $7$, and let $s$ be the remainder when y is divided by $7$. The pair $(r, s)$ is called an [i]ingredient[/i] of the line $\ell$. The (unordered) set of all possible ingredients of a wholesome line $\ell$ is called the [i]recipe [/i] of $\ell$. Compute the number of possible recipes of wholesome lines.
[b]p3.[/b] Let $\tau (n)$ be the number of distinct positive divisors of $n$. Compute $\sum_{d|15015} \tau (d)$, that is, the sum of $\tau (d)$ for all $d$ such that $d$ divides $15015$.
[b]p4.[/b] Suppose $2202010_b - 2202010_3 = 71813265_{10}$. Compute $b$. ($n_b$ denotes the number $n$ written in base $b$.)
[b]p5.[/b] Let $x = (3 -\sqrt5)/2$. Compute the exact value of $x^8 + 1/x^8$.
[b]p6.[/b] Compute the largest integer that has the same number of digits when written in base $5$ and when written in base $7$. Express your answer in base $10$.
[b]p7.[/b] Three circles with integer radii $a$, $b$, $c$ are mutually externally tangent, with $a \le b \le c$ and $a < 10$. The centers of the three circles form a right triangle. Compute the number of possible ordered triples $(a, b, c)$.
[b]p8.[/b] Six friends are playing informal games of soccer. For each game, they split themselves up into two teams of three. They want to arrange the teams so that, at the end of the day, each pair of players has played at least one game on the same team. Compute the smallest number of games they need to play in order to achieve this.
[b]p9.[/b] Let $A$ and $B$ be points in the plane such that $AB = 30$. A circle with integer radius passes through $A$ and $B$. A point $C$ is constructed on the circle such that $AC$ is a diameter of the circle. Compute all possible radii of the circle such that $BC$ is a positive integer.
[b]p10.[/b] Each square of a $3\times 3$ grid can be colored black or white. Two colorings are the same if you can rotate or reflect one to get the other. Compute the total number of unique colorings.
[b]p11.[/b] Compute all positive integers $n$ such that the sum of all positive integers that are less than $n$ and relatively prime to $n$ is equal to $2n$.
[b]p12.[/b] The distance between a point and a line is defined to be the smallest possible distance between the point and any point on the line. Triangle $ABC$ has $AB = 10$, $BC = 21$, and $CA = 17$. Let $P$ be a point inside the triangle. Let $x$ be the distance between $P$ and $\overleftrightarrow{BC}$, let $y$ be the distance between $P$ and $\overleftrightarrow{CA}$, and let $z$ be the distance between $P$ and $\overleftrightarrow{AB}$. Compute the largest possible value of the product $xyz$.
[b]p13.[/b] Alice, Bob, David, and Eve are sitting in a row on a couch and are passing back and forth a bag of chips. Whenever Bob gets the bag of chips, he passes the bag back to the person who gave it to him with probability $\frac13$ , and he passes it on in the same direction with probability $\frac23$ . Whenever David gets the bag of chips, he passes the bag back to the person who gave it to him with probability $\frac14$ , and he passes it on with probability $\frac34$ . Currently, Alice has the bag of chips, and she is about to pass it to Bob when Cathy sits between Bob and David. Whenever Cathy gets the bag of chips, she passes the bag back to the person who gave it to her with probability $p$, and passes it on with probability $1-p$. Alice realizes that because Cathy joined them on the couch, the probability that Alice gets the bag of chips back before Eve gets it has doubled. Compute $p$.
[b]p14.[/b] Circle $O$ is in the plane. Circles $A$, $B$, and $C$ are congruent, and are each internally tangent to circle $O$ and externally tangent to each other. Circle $X$ is internally tangent to circle $O$ and externally tangent to circles $A$ and $B$. Circle $X$ has radius $1$. Compute the radius of circle $O$.
[img]https://cdn.artofproblemsolving.com/attachments/f/d/8ddab540dca0051f660c840c0432f9aa5fe6b0.png[/img]
[b]p15.[/b] Compute the number of primes $p$ less than 100 such that $p$ divides $n^2 +n+1$ for some integer $n$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2017 CHMMC (Fall), 2
Adam the spider is sitting at the bottom left of a 4 × 4 coordinate grid, where adjacent parallel grid lines are each separated by one unit. He wants to crawl to the top right corner of the square, and starts off with 9 “crumb’s” worth of energy. Adam only walks in one-unit segments along the grid lines, and cannot walk off of the grid.
Walking one unit costs him one crumb’s worth of energy, and Adam cannot move anymore once he runs out of energy. Also, Adam stops moving once he reaches the top right corner.
There is also a single crumb on the grid located one unit to the right and one unit up from Adam’s starting position. If he goes to this point and eats the crumb, he will gain one crumb’s worth of energy.
How many paths can Adam take to get to the upper right corner of the grid? Note that Adam does not care if he has extra energy left over once he arrives at his destination.
2017 CHMMC (Fall), 1
Let $a, b$ be the roots of the quadratic polynomial $Q(x) = x^2 + x + 1$, and let $u, v$ be the roots of the quadratic polynomial $R(x) = 2x^2 + 7x + 1$.
Suppose $P$ is a cubic polynomial which satis es the equations
$$\begin{cases}
P(au) = Q(u)R(a) \\
P(bu) = Q(u)R(b) \\
P(av) = Q(v)R(a) \\
P(bv) = Q(v)R(b)
\end{cases}$$
If $M$ and$ N$ are the coeffcients of $x^2$ and $x$ respectively in $P(x)$, what is the value of $M+ N$?
2015 CHMMC (Fall), 4
The following number is the product of the divisors of $n$.
$$46, 656, 000, 000$$
What is $n$?
2015 CHMMC (Fall), 3
Defi ne an $n$-digit pair cycle to be a number with $n^2 + 1$ digits between $1$ and $n$ with every possible pair of consecutive digits. For instance, $11221$ is a 2-digit pair cycle since it contains the consecutive digits $11$, $12$, $22$, and $21$. How many $3$-digit pair cycles exist?
2012 CHMMC Fall, Individual
[b]p1.[/b] How many nonzero digits are in the number $(5^{94} + 5^{92})(2^{94} + 2^{92})$?
[b]p2.[/b] Suppose $A$ is a set of $2013$ distinct positive integers such that the arithmetic mean of any subset of $A$ is also an integer. Find an example of $A$.
[b]p3.[/b] How many minutes until the smaller angle formed by the minute and hour hands on the face of a clock is congruent to the smaller angle between the hands at $5:15$ pm? Round your answer to the nearest minute.
[b]p4.[/b] Suppose $a$ and $b$ are positive real numbers, $a + b = 1$, and $$1 +\frac{a^2 + 3b^2}{2ab}=\sqrt{4 +\frac{a}{b}+\frac{3b}{a}}.$$ Find $a$.
[b]p5.[/b] Suppose $f(x) = \frac{e^x- 12e^{-x}}{ 2}$ . Find all $x$ such that $f(x) = 2$.
[b]p6.[/b] Let $P_1$, $P_2$,$...$,$P_n$ be points equally spaced on a unit circle. For how many integer $n \in \{2, 3, ... , 2013\}$ is the product of all pairwise distances: $\prod_{1\le i<j\le n} P_iP_j$ a rational number?
Note that $\prod$ means the product. For example, $\prod_{1\le i\le 3} i = 1\cdot 2 \cdot 3 = 6$.
[b]p7.[/b] Determine the value $a$ such that the following sum converges if and only if $r \in (-\infty, a)$ :
$$\sum^{\infty}_{n=1}(\sqrt{n^4 + n^r} - n^2).$$
Note that $\sum^{\infty}_{n=1}\frac{1}{n^s}$ converges if and only if $s > 1$.
[b]p8.[/b] Find two pairs of positive integers $(a, b)$ with $a > b$ such that $a^2 + b^2 = 40501$.
[b]p9.[/b] Consider a simplified memory-knowledge model. Suppose your total knowledge level the night before you went to a college was $100$ units. Each day, when you woke up in the morning you forgot $1\%$ of what you had learned. Then, by going to lectures, working on the homework, preparing for presentations, you had learned more and so your knowledge level went up by $10$ units at the end of the day.
According to this model, how long do you need to stay in college until you reach the knowledge level of exactly $1000$?
[b]p10.[/b] Suppose $P(x) = 2x^8 + x^6 - x^4 +1$, and that $P$ has roots $a_1$, $a_2$, $...$ , $a_8$ (a complex number $z$ is a root of the polynomial $P(x)$ if $P(z) = 0$). Find the value of $$(a^2_1-2)(a^2_2-2)(a^2_3-2)...(a^2_8-2).$$
[b]p11.[/b] Find all values of $x$ satisfying $(x^2 + 2x-5)^2 = -2x^2 - 3x + 15$.
[b]p12.[/b] Suppose $x, y$ and $z$ are positive real numbers such that
$$x^2 + y^2 + xy = 9,$$
$$y^2 + z^2 + yz = 16,$$
$$x^2 + z^2 + xz = 25.$$
Find $xy + yz + xz$ (the answer is unique).
[b]p13.[/b] Suppose that $P(x)$ is a monic polynomial (i.e, the leading coefficient is $1$) with $20$ roots, each distinct and of the form $\frac{1}{3^k}$ for $k = 0,1,2,..., 19$. Find the coefficient of $x^{18}$ in $P(x)$.
[b]p14.[/b] Find the sum of the reciprocals of all perfect squares whose prime factorization contains only powers of $3$, $5$, $7$ (i.e. $\frac{1}{1} + \frac{1}{9} + \frac{1}{25} + \frac{1}{419} + \frac{1}{811} + \frac{1}{215} + \frac{1}{441} + \frac{1}{625} + ...$).
[b]p15.[/b] Find the number of integer quadruples $(a, b, c, d)$ which also satisfy the following system of equations:
$$1+b + c^2 + d^3 =0,$$ $$a + b^2 + c^3 + d^4 =0,$$ $$a^2 + b^3 + c^4 + d^5 =0,$$ $$a^3+b^4+c^5+d^6 =0.$$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2013 CHMMC (Fall), 4
Let
$$A =\frac12 +\frac13 +\frac15 +\frac19,$$
$$B =\frac{1}{2 \cdot 3}+\frac{1}{2 \cdot 5}+\frac{1}{2 \cdot 9}+\frac{1}{3 \cdot 5}+\frac{1}{3 \cdot 9}
+\frac{1}{5 \cdot 9},$$
$$C =\frac{1}{2 \cdot 3 \cdot 5} + \frac{1}{2 \cdot 3 \cdot 9} + \frac{1}{2 \cdot 5 \cdot 9}
+\frac{1}{3 \cdot 5 \cdot 9}.$$
Compute the value of $A + B + C$.
2022 CHMMC Winter (2022-23), 4
Let $ABC$ be a triangle with $AB = 4$, $BC = 5$, $CA = 6$. Triangles $APB$ and $CQA$ are erected outside $ABC$ such that $AP=PB$, $\overline{AP}\perp \overline{PB}$ and $CQ=QA$, $\overline{CQ}\perp \overline{QA}$. Pick a point $X$ uniformly at random from segment $\overline{BC}$. What is the expected value of the area of triangle $PXQ$?
2010 CHMMC Winter, 1
The monic polynomial $f$ has rational coefficients and is irreducible over the rational numbers. If $f(\sqrt5 +\sqrt2)= 0$, compute $f(f(\sqrt5 -\sqrt2))$. (A polynomial is [i]monic [/i] if its leading coeffi cient is $1$. A polynomial is [i]irreducible [/i] over the rational numbers if it cannot be expressed as a product of two polynomials with rational coefficients of positive degree. For example, $x^2 - 2$ is irreducible, but $x^2 - 1 = (x + 1)(x - 1)$ is not.)