Found problems: 71
2014 CHMMC (Fall), 4
If $f(i, j, k) = f(i - 1, j + k , 2i - 1)$ and $f(0, j, k) = j + k$, evaluate $f(n, 0, 0)$.
2012 CHMMC Fall, Mixer
[b]p1.[/b] Prove that $x = 2$ is the only real number satisfying $3^x + 4^x = 5^x$.
[b]p2.[/b] Show that $\sqrt{9 + 4\sqrt5} -\sqrt{9 - 4\sqrt5}$ is an integer.
[b]p3.[/b] Two players $A$ and $B$ play a game on a round table. Each time they take turn placing a round coin on the table. The coin has a uniform size, and this size is at least $10$ times smaller than the table size. They cannot place the coin on top of any part of other coins, and the whole coin must be on the table. If a player cannot place a coin, he loses. Suppose $A$ starts first. If both of them plan their moves wisely, there will be one person who will always win. Determine whether $A$ or $B$ will win, and then determine his winning strategy.
[b]p4.[/b] Suppose you are given $4$ pegs arranged in a square on a board. A “move” consists of picking up a peg, reflecting it through any other peg, and placing it down on the board. For how many integers $1 \le n \le 2013$ is it possible to arrange the $4$ pegs into a [i]larger [/i] square using exactly $n$ moves? Justify your answers.
[b]p5.[/b] Find smallest positive integer that has a remainder of $1$ when divided by $2$, a remainder of $2$ when divided by $3$, a remainder of $3$ when divided by $5$, and a remainder of $5$ when divided by $7$.
[b]p6.[/b] Find the value of $$\sum_{m|496,m>0} \frac{1}{m},$$
where $m|496$ means $496$ is divisible by $m$.
[b]p7.[/b] What is the value of
$${100 \choose 0}+{100 \choose 4}+{100 \choose 8}+ ... +{100 \choose 100}?$$
[b]p8.[/b] An $n$-term sequence $a_0, a_1, ...,a_n$ will be called [i]sweet [/i] if, for each $0 \le i \le n -1$, $a_i$ is the number of times that the number $i$ appears in the sequence. For example, $1, 2, 1,0$ is a sweet sequence with $4$ terms. Given that $a_0$, $a_1$, $...$, $a_{2013}$ is a sweet sequence, find the value of $a^2_0+ a^2_1+ ... + a^2_{2013}.$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2012 CHMMC Spring, 3
Three different faces of a regular dodecahedron are selected at random and painted. What is the probability that there is at least one pair of painted faces that share an edge?
2015 CHMMC (Fall), 1
Call a positive integer $x$ $n$-[i]cube-invariant[/i] if the last $n$ digits of $x$ are equal to the last $n$ digits of $x^3$. For example, $1$ is $n$-cube invariant for any integer $n$. How many $2015$-cube-invariant numbers $x$ are there such that $x < 10^{2015}$?
2012 CHMMC Spring, 2
A convex octahedron in Cartesian space contains the origin in its interior. Two of its vertices are on the $x$-axis, two are on the $y$-axis, and two are on the $z$-axis. One triangular face $F$ has side lengths $\sqrt{17}$, $\sqrt{37}$, $\sqrt{52}$. A second triangular face $F_0$ has side lengths $\sqrt{13}$, $\sqrt{29}$, $\sqrt{34}$. What is the minimum possible volume of the octahedron?
2010 CHMMC Fall, 3
Talithia throws a party on the fifth Saturday of every month that has five Saturdays. That is, if a month has five Saturdays, Talithia has a party on the fifth Saturday of that month, and if a month has four Saturdays, then Talithia does not have a party that month. Given that January $1$, $2010$ was a Friday, compute the number of parties Talithia will have in $2010$.
2013 CHMMC (Fall), 1
In the diagram below, point $A$ lies on the circle centered at $O$. $AB$ is tangent to circle $O$ with $\overline{AB} = 6$. Point $C$ is $\frac{2\pi}{3}$ radians away from point $A$ on the circle, with $BC$ intersecting circle $O$ at point $D$. The length of $BD$ is $3$. Compute the radius of the circle.
[img]https://cdn.artofproblemsolving.com/attachments/7/8/baa528c776eb50455f31ae50a3ec28efc291e8.png[/img]
2010 CHMMC Fall, 11
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 $6$.
2012 CHMMC Spring, Individual
[b]p1.[/b] A robot is at position $0$ on a number line. Each second, it randomly moves either one unit in the positive direction or one unit in the negative direction, with probability $\frac12$ of doing each. Find the probability that after $4$ seconds, the robot has returned to position $0$.
[b]p2.[/b] How many positive integers $n \le 20$ are such that the greatest common divisor of $n$ and $20$ is a prime number?
[b]p3.[/b] A sequence of points $A_1$, $A_2$, $A_3$, $...$, $A_7$ is shown in the diagram below, with $A_1A_2$ parallel to $A_6A_7$. We have $\angle A_2A_3A_4 = 113^o$, $\angle A_3A_4A_5 = 100^o$, and $\angle A_4A_5A_6 = 122^o$. Find the degree measure of $\angle A_1A_2A_3 + \angle A_5A_6A_7$.
[center][img]https://cdn.artofproblemsolving.com/attachments/d/a/75b06a6663b2f4258e35ef0f68fcfbfaa903f7.png[/img][/center]
[b]p4.[/b] Compute
$$\log_3 \left( \frac{\log_3 3^{3^{3^3}}}{\log_{3^3} 3^{3^3}} \right)$$
[b]p5.[/b] In an $8\times 8$ chessboard, a pawn has been placed on the third column and fourth row, and all the other squares are empty. It is possible to place nine rooks on this board such that no two rooks attack each other. How many ways can this be done? (Recall that a rook can attack any square in its row or column provided all the squares in between are empty.)
[b]p6.[/b] Suppose that $a, b$ are positive real numbers with $a > b$ and $ab = 8$. Find the minimum value of $\frac{a^2+b^2}{a-b} $.
[b]p7.[/b] A cone of radius $4$ and height $7$ has $A$ as its apex and $B$ as the center of its base. A second cone of radius $3$ and height $7$ has $B$ as its apex and $A$ as the center of its base. What is the volume of the region contained in both cones?
[b]p8.[/b] Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$ be a permutation of the numbers $1$, $2$, $3$, $4$, $5$, $6$. We say $a_i$ is visible if $a_i$ is greater than any number that comes before it; that is, $a_j < a_i$ for all $j < i$. For example, the permutation $2$, $4$, $1$, $3$, $6$, $5$ has three visible elements: $2$, $4$, $6$. How many such permutations have exactly two visible elements?
[b]p9.[/b] Let $f(x) = x+2x^2 +3x^3 +4x^4 +5x^5 +6x^6$, and let $S = [f(6)]^5 +[f(10)]^3 +[f(15)]^2$. Compute the remainder when $S$ is divided by $30$.
[b]p10.[/b] In triangle $ABC$, the angle bisector from $A$ and the perpendicular bisector of $BC$ meet at point $D$, the angle bisector from $B$ and the perpendicular bisector of $AC$ meet at point $E$, and the perpendicular bisectors of $BC$ and $AC$ meet at point $F$. Given that $\angle ADF = 5^o$, $\angle BEF = 10^o$, and $AC = 3$, find the length of $DF$.
[img]https://cdn.artofproblemsolving.com/attachments/6/d/6bb8409678a4c44135d393b9b942f8defb198e.png[/img]
[b]p11.[/b] Let $F_0 = 0$, $F_1 = 1$, and $F_n = F_{n-1} + F_{n-2}$. How many subsets $S$ of $\{1, 2,..., 2011\}$ are there such that $$F_{2012} - 1 =\sum_{i \in S}F_i?$$
[b]p12.[/b] Let $a_k$ be the number of perfect squares $m$ such that $k^3 \le m < (k + 1)^3$. For example, $a_2 = 3$ since three squares $m$ satisfy $2^3 \le m < 3^3$, namely $9$, $16$, and $25$. Compute$$ \sum^{99}_{k=0} \lfloor \sqrt{k}\rfloor a_k, $$ where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.
[b]p13.[/b] Suppose that $a, b, c, d, e, f$ are real numbers such that
$$a + b + c + d + e + f = 0,$$
$$a + 2b + 3c + 4d + 2e + 2f = 0,$$
$$a + 3b + 6c + 9d + 4e + 6f = 0,$$
$$a + 4b + 10c + 16d + 8e + 24f = 0,$$
$$a + 5b + 15c + 25d + 16e + 120f = 42.$$
Compute $a + 6b + 21c + 36d + 32e + 720f.$
[b]p14.[/b] In Cartesian space, three spheres centered at $(-2, 5, 4)$, $(2, 1, 4)$, and $(4, 7, 5)$ are all tangent to the $xy$-plane. The $xy$-plane is one of two planes tangent to all three spheres; the second plane can be written as the equation $ax + by + cz = d$ for some real numbers $a$, $b$, $c$, $d$. Find $\frac{c}{a}$ .
[b]p15.[/b] Find the number of pairs of positive integers $a$, $b$, with $a \le 125$ and $b \le 100$, such that $a^b - 1$ is divisible by $125$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2010 CHMMC Fall, 15
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.
2016 CHMMC (Fall), 1
We say that the string $d_kd_{k-1} \cdots d_1d_0$ represents a number $n$ in base $-2$ if each $d_i$ is either $0$ or $1$,
and $n = d_k(-2)^k + d_{k-1}(-2)^{k-1} + \cdots + d_1(-2) + d_0$. For example, $110_{-2}$ represents the number $2$. What string represents $2016$ in base $-2$?
2022 CHMMC Winter (2022-23), 2
Jonathan and Eric are standing one kilometer apart on a large, flat, empty field. Jonathan rotates an angle of $\theta = 120^o$ counterclockwise around Eric, then Eric moves half of the distance to Jonathan. They keep repeating the previous two movements in this order. After a very long time, their locations approach a point $P$ on the field. What is the distance, in kilometers, from Jonathan’s starting location to $P$?
2017 CHMMC (Fall), Individual
[b]p1.[/b] A dog on a $10$ meter long leash is tied to a $10$ meter long, infinitely thin section of fence. What is the minimum area over which the dog will be able to roam freely on the leash, given that we can fix the position of the leash anywhere on the fence?
[b]p2.[/b] Suppose that the equation $$\begin{tabular}{cccccc}
&\underline{C} &\underline{H} &\underline{M}& \underline{M}& \underline{C}\\
+& &\underline{H}& \underline{M}& \underline{M} & \underline{T}\\
\hline
&\underline{P} &\underline{U} &\underline{M} &\underline{A} &\underline{C}\\
\end{tabular}$$
holds true, where each letter represents a single nonnegative digit, and distinct letters represent different digits (so that $\underline{C}\, \underline{H}\, \underline{ M}\, \underline{ M}\, \underline{ C}$ and $ \underline{P}\, \underline{U}\, \underline{M}\, \underline{A}\, \underline{C}$ are both five digit positive integers, and the number $\underline{H }\, \underline{M}\, \underline{M}\, \underline{T}$ is a four digit positive integer). What is the largest possible value of the five digit positive integer$\underline{C}\, \underline{H}\, \underline{ M}\, \underline{ M}\, \underline{ C}$ ?
[b]p3.[/b] Square $ABCD$ has side length $4$, and $E$ is a point on segment $BC$ such that $CE = 1$. Let $C_1$ be the circle tangent to segments $AB$, $BE$, and $EA$, and $C_2$ be the circle tangent to segments $CD$, $DA$, and $AE$. What is the sum of the radii of circles $C_1$ and $C_2$?
[b]p4.[/b] A finite set $S$ of points in the plane is called tri-separable if for every subset $A \subseteq S$ of the points in the given set, we can find a triangle $T$ such that
(i) every point of $A$ is inside $T$ , and
(ii) every point of $S$ that is not in $A$ is outside$ T$ .
What is the smallest positive integer $n$ such that no set of $n$ distinct points is tri-separable?
[b]p5.[/b] The unit $100$-dimensional hypercube $H$ is the set of points $(x_1, x_2,..., x_{100})$ in $R^{100}$ such that $x_i \in \{0, 1\}$ for $i = 1$, $2$, $...$, $100$. We say that the center of $H$ is the point
$$\left( \frac12,\frac12, ..., \frac12 \right)$$
in $R^{100}$, all of whose coordinates are equal to $1/2$.
For any point $P \in R^{100}$ and positive real number $r$, the hypersphere centered at $P$ with radius $r$ is defined to be the set of all points in $R^{100}$ that are a distance $r$ away from $P$. Suppose we place hyperspheres of radius $1/2$ at each of the vertices of the $100$-dimensional unit hypercube $H$. What is the smallest real number $R$, such that a hypersphere of radius $R$ placed at the center of $H$ will intersect the hyperspheres at the corners of $H$?
[b]p6.[/b] Greg has a $9\times 9$ grid of unit squares. In each square of the grid, he writes down a single nonzero digit. Let $N$ be the number of ways Greg can write down these digits, so that each of the nine nine-digit numbers formed by the rows of the grid (reading the digits in a row left to right) and each of the nine nine-digit numbers formed by the columns (reading the digits in a column top to bottom) are multiples of $3$. What is the number of positive integer divisors of $N$?
[b]p7.[/b] Find the largest positive integer $n$ for which there exists positive integers $x$, $y$, and $z$ satisfying
$$n \cdot gcd(x, y, z) = gcd(x + 2y, y + 2z, z + 2x).$$
[b]p8.[/b] Suppose $ABCDEFGH$ is a cube of side length $1$, one of whose faces is the unit square $ABCD$. Point $X$ is the center of square $ABCD$, and $P$ and $Q$ are two other points allowed to range on the surface of cube $ABCDEFHG$. Find the largest possible volume of tetrahedron $AXPQ$.
[b]p9.[/b] Deep writes down the numbers $1, 2, 3, ... , 8$ on a blackboard. Each minute after writing down the numbers, he uniformly at random picks some number $m$ written on the blackboard, erases that number from the blackboard, and increases the values of all the other numbers on the blackboard by $m$. After seven minutes, Deep is left with only one number on the black board. What is the expected value of the number Deep ends up with after seven minutes?
[b]p10.[/b] Find the number of ordered tuples $(x_1, x_2, x_3, x_4, x_5)$ of positive integers such that $x_k \le 6$ for each index $k = 1$, $2$, $... $,$ 5$, and the sum $$x_1 + x_2 +... + x_5$$ is $1$ more than an integer multiple of $7$.
[b]p11.[/b] The equation $$\left( x- \sqrt[3]{13}\right)\left( x- \sqrt[3]{53}\right)\left( x- \sqrt[3]{103}\right)=\frac13$$ has three distinct real solutions $r$, $s$, and $t$ for $x$. Calculate the value of $$r^3 + s^3 + t^3.$$
[b]p12.[/b] Suppose $a$, $b$, and $c$ are real numbers such that
$$\frac{ac}{a + b}+\frac{ba}{b + c}+\frac{cb}{c + a}= -9$$
and
$$\frac{bc}{a + b}+\frac{ca}{b+c}+\frac{ab}{c + a}= 10.$$
Compute the value of
$$\frac{b}{a + b}+\frac{c}{b + c}+\frac{a}{c + a}.$$
[b]p13.[/b] The complex numbers $w$ and $z$ satisfy the equations $|w| = 5$, $|z| = 13$, and $$52w - 20z = 3(4 + 7i).$$ Find the value of the product $wz$.
[b]p14.[/b] For $i = 1, 2, 3, 4$, we choose a real number $x_i$ uniformly at random from the closed interval $[0, i]$. What is the probability that $x_1 < x_2 < x_3 < x_4$ ?
[b]p15.[/b] The terms of the infinite sequence of rational numbers $a_0$, $a_1$, $a_2$, $...$ satisfy the equation $$a_{n+1} + a_{n-2} = a_na_{n-1}$$ for all integers $n\ge 2$. Moreover, the values of the initial terms of the sequence are $a_0 =\frac52$, $a_1 = 2$ and} $a_2 =\frac52.$ Call a nonnegative integer $m$ lucky if when we write $a_m =\frac{p}{q}$ for some relatively prime positive integers $p$ and $q$, the integer $p + q$ is divisible by $13$. What is the $101^{st}$ smallest lucky number?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2010 CHMMC Fall, 14
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.
2012 CHMMC Spring, 4
The expression below has six empty boxes. Each box is to be fi lled in with a number from $1$ to $6$, where all six numbers are used exactly once, and then the expression is evaluated. What is the maximum possible final result that can be achieved?
$$\dfrac{\frac{\square}{\square}+\frac{\square}{\square}}{\frac{\square}{\square}}$$
2014 CHMMC (Fall), 1
For $a_1,..., a_5 \in R$, $$\frac{a_1}{k^2 + 1}+ ... +\frac{a_5}{k^2 + 5}=\frac{1}{k^2}$$ for all $k \in \{2, 3, 4, 5, 6\}$. Calculate $$\frac{a_1}{2}+... +\frac{a_5}{6}.$$
2017 CHMMC (Fall), 3
You are playing a game called "Hovse."
Initially you have the number $0$ on a blackboard.
If at any moment the number $x$ is written on the board, you can either:
$\bullet$ replace $x$ with $3x + 1$
$\bullet$ replace $x$ with $9x + 1$
$\bullet$ replace $x$ with $27x + 3$
$\bullet$ or replace $x$ with $\left \lfloor \frac{x}{3} \right \rfloor $.
However, you are not allowed to write a number greater than $2017$ on the board. How many positive numbers can you make with the game of "Hovse?"
2017 CHMMC (Fall), 8
Let $P(x)$ be the polynomial of degree at most $6$ which satisfies $P(k)=k!$ for $k=0,1,2,3,4,5,6$. Compute the value of $P(7)$.
2018 CHMMC (Fall), 5
Let $a,b, c, d,e$ be the roots of $p(x) = 2x^5 - 3x^3 + 2x -7$. Find the value of
$$(a^3 - 1)(b^3 - 1)(c^3 - 1)(d^3 - 1)(e^3 - 1).$$
2016 CHMMC (Fall), 3
Two towns, $A$ and $B$, are $100$ miles apart. Every $20$ minutes, (starting at midnight), a bus traveling at $60$ mph leaves town $A$ for town $B$, and every $30$ minutes (starting at midnight) a bus traveling at $20$ mph leaves town $B$ for town $A$. Dirk starts in Town $A$ and gets on a bus leaving for town $B$ at noon. However, Dirk is always afraid he has boarded a bus going in the wrong direction, so each time the bus he is in passes another bus, he gets out and transfers to that other bus. How many hours pass before Dirk finally reaches Town $B$?
2015 CHMMC (Fall), 2
Let $a_1 = 1$, $a_2 = 1$, and for $n \ge 2$, let $$a_{n+1} =\frac{1}{n} a_n + a_{n-1}.$$ What is $a_{12}$?