ASIA TEAM Problems

Problem 1

Kelvin the frog is hopping along lily pads numbered with natural numbers. If he is at lily pad number $x$, he jumps to lily pad $\tfrac x2$ if $x$ is even and lily pad $x+1$ if $x$ is odd.

Let the order of a lily pad number $x$, denoted $o(x)$, be the minimum number $r$ such that Kelvin will reach $1$ for the first time in $r$ jumps. In particular, $o(1)=0$ because Kelvin is already at lily pad $1$.

How many of the lily pads numbered from $1$ to $1000$ inclusive have an even order?

Solution

Problem 2

Kelvin the frog is catching flies for dinner. Being adept at fly-catching, he can catch a fly every minute. He has $1337$ lilypads in front of him, labeled $a_1,a_2,\ldots,a_{1337}$. When he catches his first fly, he places it on $a_1$. After that, when he catches a fly, he places it on $a_i$ for which $i$ is the least such number satisfying the following rules:

0) Let the amount of flies on $a_i$ at a given time be $S_i$.

1) $S_i<4$

2) If $S_1=S_2=S_3=\ldots=S_{i-1}=4$ but $S_i\neq 4$, then he eats all the flies on $a_1$ through $a_{i-1}$ and then puts his newly caught fly on $a_i$.

For example, after $4$ minutes, $S_1=4$ while all other $S$s equal $0$. At $5$ minutes, $S_1=0$ while $S_2=1$. After $2013$ minutes, there are exactly $x$ flies in total on his lilypads. Find $x$.

Solution

Problem 3

Kelvin the frog creates an infinite sequence of rational numbers. He chooses two starting terms $a_1=\tfrac{n_1}{d_1}$ and $a_2=\tfrac{n_2}{d_2}$, and then defines $a_k=\tfrac{n_{k-1}+n_{k-2}}{d_{k-1}+d_{k-2}}$. For example, if Kelvin begins with the numbers $\tfrac12$ and $\tfrac23$, his sequence will continue $\tfrac35,\tfrac58,\tfrac8{13},\ldots$. Kelvin begins his sequence with the numbers $\tfrac a{671}$ and $\tfrac{1006}{2013}$. He then realizes that at least $1$ of the first $2013$ terms is equal to $\tfrac a{671}+\tfrac{1006}{2013}$. Find the sum of all possible nonnegative integer values of $a$.

Solution

Problem 4

Kelvin the frog lives at point $A$ at the origin on the coordinate plane. Point $B$ lies in the first quadrant such that $AB = 1$, point $C$ lies in the second quadrant such that $AC = 4$, point $D$ lies in the 3rd quadrant such that $D, A, B$ are collinear and $AD = 24$, and point $E$ lies in the 4th quadrant such that $E, A, C$ are collinear and $AE = 6$. If $BC$ is an integer, then the length of $CD$ can be written as $a\sqrt{b}$, where $a$ and $b$ are positive integers greater than $1$ and $b$ has no perfect square factors other than $1$. Compute $a + b$.

Solution

Problem 5

Kelvin the frog wants to find all positive integers $n$ not over $9000$ such that the order of $n \pmod{1000}$ is $5$. Let $N$ be the sum of all such $n$. Find the remainder when n is divided by $1000$.

Remark: The order of n is the smallest positive integer k such that $n^k \equiv 1 \pmod{1000}$. Hence order is only defined for numbers relatively prime to the modulo.

Solution

Problem 6

Kelvin the frog chooses real numbers $x,y,z>0$, such that $x+y+z=6$ and $9yz+4zx+xy=6xyz$. Find the sum of all possible values of $xyz$.

Solution

Problem 7

Kelvin the frog and $2012$ of his relatives are in a line. Each frog either feels ambivalent, happy, or sad. If a frog feels happy, the frog behind them cannot be sad. If a frog feels sad, the frog behind them cannot feel happy. If a frog and the frog behind him both feel ambivalent, then the next frog in line will also feel ambivalent. Find the last three digits of the amount of ways for $2013$ frogs to line up following these rules.

Solution

Problem 8

Kelvin the frog lives in a circle with center $O$. One day, he builds a fence $AB$ such that $AB$ is a non-diameter chord of the circle and $M$ is the midpoint of $AB$. Point $P$ lies outside the circle and on the perpendicular bisector of $AB$ (the line going through $M$ perpendicular to $AM$), $C$ is the intersection of $PM$ with circle $O$ such that $C$ lies between $P$ and $M$, and $\angle ABC = \angle APC$. Let $AP$ intersect the circle again at $D$ other than $A$ such that $AB = 24$ and $CM = 5$. The radius of the circle going through $P$, $C$, and $D$ can then be expressed as $\tfrac ab$, where a and $b$ are relatively prime positive integers such that $b$, possibly $1$, is as small as possible. Compute the remainder when $a + b$ is divided by $1000$.

Solution

Problem 9

Kelvin the frog starts writing all the positive integer squares down in a list. However, he soon realizes that he only likes odd numbers - so he multiplies each number by $2$ and adds $1$ (thus, his list now reads $3, 9, 19$, etc.). But Kelvin is also very picky. So he eats all the numbers in his list which are divisible by $737$, but not by $3$. What are the last three digits of the $2013$th number he eats?

Solution

Problem 10

Kelvin the frog is standing on one vertex of a regular $2013$-gon. Every minute, he randomly decides to move to an adjacent vertex, each with probability $\tfrac12$. Let $N$ be the expected number of minutes before Kelvin returns to his original vertex. Find the remainder when $[N]$ is divided by $1000$.

Remark: $[x]$ is defined to be the greatest positive integer that is less than $x$, so for example $[\pi ] = 3$.

Solution

Problem 11

Kelvin the frog is bored, so he generates two infinite sequences $a_n$ and $b_n$ of real, positive numbers and notices that the sequences $a_n$ and $b_n$ satisfy

$a_1^2+b_1^2=1$

$a_n=2a_{n-1}b_{n-1}$

$b_n=(b_{n-1}-a_{n-1})(a_{n-1}+b_{n-1})$.

Let $x$ be the minimum possible value of $a_1$ such that $a_1=a_{13}$ abd let $y$ be the minimum possible value of $b_1$ such that $b_1=b_{13}$. Compute the smallest positive integer $n$ such that $(y+ix)^n$ is real.

Solution

Problem 12

Kelvin the frog generates a polynomial $P(x)$ of degree $2012$ such that $P(k)=2^{2012-k}3^k$ for $k=1,2,3,\ldots,2012$. Compute the remainder when $P(2013)$ is divided by $1000$.

Solution

Problem 13

Kelvin the frog chooses integers $x,y,z$ such that $(x-y)(y-z)(z-x)=x+y+x$ and the sum $x+y+z$ is positive. Find the minimum possible value of $x+y+z$.

Solution

Problem 14

Kelvin the frog's home lily pad is a triangle $ABC$, with $\overline{AB}=32$, $\overline{AC}=24$, $\overline{BC}=28$. Points $D$ and $E$ lie on $\overline{BC}$ such that $\overline{BD}=4$, $\overline{DE}=3$, $D$ lies between $B$ and $E$, and $D$ lies between $D$ and $C$. Point $F$ is chosen on $\overline{AC}$ such that $\overline{EF}=\overline{AC}$ and $\angle EFC$ is acute. Line $FE$ is extended through $E$ to meet $AD$ at $G$. Let $CG$ intersect $AB$ at $H$. Then $CH$ can be written as $m\sqrt n$, where $m$ and $n$ are positive integers and $n$ has no perfect square factors except for $1$. Find $m+n$.

Solution

Problem 15

Kelvin the frog likes the number $46$ because $\varphi(46)+6\tau(46)=46$. Find the sum of all positive integers $n$ that Kelvin the frog likes, i.e. such that $\varphi(n)+6\tau(n)=n$.

Remark: $\varphi(n)$ denotes the number of integers between $1$ and $n$ (inclusive) which are relatively prime to $n$, and $\tau(n)$ denotes the number of integers between $1$ and $n$ (inclusive) which divide $n$.

Solution

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