ASIA TEAM Problems
Kelvin the frog is hopping along lily pads numbered with natural numbers. If he is at lily pad number , he jumps to lily pad if is even and lily pad if is odd.
Let the order of a lily pad number , denoted , be the minimum number such that Kelvin will reach for the first time in jumps. In particular, because Kelvin is already at lily pad .
How many of the lily pads numbered from to inclusive have an even order?
Kelvin the frog is catching ﬂies for dinner. Being adept at ﬂy-catching, he can catch a ﬂy every minute. He has lilypads in front of him, labeled . When he catches his first fly, he places it on . After that, when he catches a fly, he places it on for which is the least such number satisfying the following rules:
0) Let the amount of flies on at a given time be .
2) If but , then he eats all the flies on through and then puts his newly caught fly on .
For example, after minutes, while all other s equal . At minutes, while . After minutes, there are exactly flies in total on his lilypads. Find .
Kelvin the frog creates an infinite sequence of rational numbers. He chooses two starting terms and , and then defines . For example, if Kelvin begins with the numbers and , his sequence will continue . Kelvin begins his sequence with the numbers and . He then realizes that at least of the first terms is equal to . Find the sum of all possible nonnegative integer values of .
Kelvin the frog lives at point at the origin on the coordinate plane. Point lies in the ﬁrst quadrant such that , point lies in the second quadrant such that , point lies in the 3rd quadrant such that are collinear and , and point lies in the 4th quadrant such that are collinear and . If is an integer, then the length of can be written as , where and are positive integers greater than and has no perfect square factors other than . Compute .
Kelvin the frog wants to ﬁnd all positive integers not over such that the order of is . Let be the sum of all such . Find the remainder when n is divided by .
Remark: The order of n is the smallest positive integer k such that . Hence order is only deﬁned for numbers relatively prime to the modulo.
Kelvin the frog chooses real numbers , such that and . Find the sum of all possible values of .
Kelvin the frog and 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 frogs to line up following these rules.
Kelvin the frog lives in a circle with center . One day, he builds a fence such that is a non-diameter chord of the circle and is the midpoint of . Point lies outside the circle and on the perpendicular bisector of (the line going through perpendicular to ), is the intersection of with circle such that lies between and , and . Let intersect the circle again at other than such that and . The radius of the circle going through , , and can then be expressed as , where a and are relatively prime positive integers such that , possibly , is as small as possible. Compute the remainder when is divided by .
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 and adds (thus, his list now reads , etc.). But Kelvin is also very picky. So he eats all the numbers in his list which are divisible by , but not by . What are the last three digits of the th number he eats?
Kelvin the frog is standing on one vertex of a regular -gon. Every minute, he randomly decides to move to an adjacent vertex, each with probability . Let be the expected number of minutes before Kelvin returns to his original vertex. Find the remainder when is divided by .
Remark: is deﬁned to be the greatest positive integer that is less than , so for example .
Kelvin the frog is bored, so he generates two infinite sequences and of real, positive numbers and notices that the sequences and satisfy
Let be the minimum possible value of such that abd let be the minimum possible value of such that . Compute the smallest positive integer such that is real.
Kelvin the frog generates a polynomial of degree such that for . Compute the remainder when is divided by .
Kelvin the frog chooses integers such that and the sum is positive. Find the minimum possible value of .
Kelvin the frog's home lily pad is a triangle , with , , . Points and lie on such that , , lies between and , and lies between and . Point is chosen on such that and is acute. Line is extended through to meet at . Let intersect at . Then can be written as , where and are positive integers and has no perfect square factors except for . Find .
Kelvin the frog likes the number because . Find the sum of all positive integers that Kelvin the frog likes, i.e. such that .
Remark: denotes the number of integers between and (inclusive) which are relatively prime to , and denotes the number of integers between and (inclusive) which divide .