2009 PUMaC Problems
Algebra A Find the root that the following three polynomials have in common:
Given that is the least degree polynomial with rational coefficients such that find .
Let be non-negative real numbers such that . Find the maximum possible value of .
Find the smallest positive (in degrees) for which all the numbers are negative.
Find the maximal positive integer , so that for any real number we have .
Find the number of functions for which , for all integers and .
Let be a sequence of integers, such that , for , and . Let and be the minimal and maximal possible value of , respectively. Find the . Round your answer to nearest integer, if necessary.
The real numbers , , , and satisfy the following equation: Find 100 times the maximum possible value for .
Combinatorics A Find the number of subsets of that do not contain two consecutive integers.
It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has weights of integral values. What is the minimum for which there exist weights that satisfy this condition?
How many strings of ones and zeroes of length 10 are there such that there is an even number of ones, and no zero follows another zero?
We divide up the plane into disjoint regions using a circle, a rectangle and a triangle. What is the greatest number of regions that we can get?
There are players in a round-robin ping-pong tournament (i.e. every two persons will play exactly one game). After some matches have been played, it is known that the total number of matches that have been played among any people is equal to (where is a fixed integer). Find the sum of all possible values of .
We have a square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw?
We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?
Taotao wants to buy a bracelet. The bracelets have 7 different beads on them, arranged in a circle. Two bracelets are the same if one can be rotated or flipped to get the other. If she can choose the colors and placement of the beads, and the beads come in orange, white, and black, how many possible bracelets can she buy?
Geometry A A rectangular piece of paper has sides of lengths , . The rectangle is folded in half such that coincides with and is the folding line. Then fold the paper along a line such that the corner falls on line . How large, in degrees, is ?
Tetrahedron havs sides of lengths, in increasing order, If , then what is the length of ?
A polygon is called concave if it has at least one angle strictly greater than . What is the maximum number of symmetries that an 11-sided concave polygon can have?
In the following diagram (not to scale), , , , are four consecutive vertices of an 18-sided regular polygon with center . Let be the midpoint of and be the midpoint of . Find in degrees.
Lines and are perpendicular. Line partitions a convex polygon into two parts of equal area, and partitions the projection of the polygon onto into two line segments of length and respectively. Determine the maximum value of . (The floor notation denotes largest integer not exceeding )
Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the [i]square[/i] of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is , then you would submit 1734274).
You are given a convex pentagon with , , , , , . Find the area of this pentagon. Round your answer to the nearest integer if necessary.
Consider and a point in its interior so that , , and . What is ?
Number Theory A You are given that for some digits and . Find the two-digit number that is missing above.
Find the number of ordered pairs of positive integers that are solutions of the following equation:
Find the sum of all prime numbers which satisfy for some primes (not necessarily distinct) , and .
Find the sum of all integers for which there is an integer , such that .
Suppose that for some positive integer , the first two digits of and are identical. Suppose the first two digits are and in this order. Find the two-digit number .
Let denote the sum of the digits of the positive integer . Find the largest positive integer that has no digits equal to zero and satisfies the equation
Let . If the maximum element of is in reduced form, find .
Find the largest positive integer such that . ( denotes the number of positive integers that are smaller than and relatively prime to , and denotes the sum of divisors of ). As a hint, you are given that .
Individual Finals A The sequence of positive real numbers is defined recursively as follows: , and for ,
\[x_{n} = \begin{cases} nx_{n-1} & \quad \text{when }nx_{n-1}\leq 1, \\ \frac{x_{n-1}}{n} &\quad \text{otherwise}. \end{cases} \right.\] (Error compiling LaTeX. Unknown error_msg)
Show that there is an integer such that . Thus the elements of the sequence can get very close to 1 for large ; however, it is easy to see that they can never be 1 unless .
Let be a sequence of positive integers defined as follows: is a fixed six-digit number and for any , is a prime divisor of . Find .
Using one straight cut we partition a rectangular piece of paper into two pieces. We call this one "operation". Next, we cut one of the two pieces so obtained once again, to partition [i]this piece[/i] into two smaller pieces (i.e. we perform the operation on any [i]one[/i] of the pieces obtained). We continue this process, and so, after each operation we increase the number of pieces of paper by . What is the minimum number of operations needed to get pieces of -sided polygons? [obviously there will be other pieces too, but we will have at least (not necessarily [i]regular[/i]) -gons.]
Algebra B If is the Golden Ratio, we know that . Define a new positive real number, called , where (so ). Given that , positive integers, and the greatest common divisor of and is 1, find .
Let be the polynomial with leading coefficent 1 and rational coefficents, such that and with the least degree among all such polynomials. Find .
Find the root that the following three polynomials have in common:
Given that is the least degree polynomial with rational coefficients such that find .
Let be non-negative real numbers such that . Find the maximum possible value of .
Find the smallest positive (in degrees) for which all the numbers are negative.
Find the maximal positive integer , so that for any real number we have .
Find the number of functions for which , for all integers and .
Combinatorics B Three people, John, Macky, and Rik, play a game of passing a basketball from one to another. Find the number of ways of passing the ball starting with Macky and reaching Macky again at the end of the seventh pass.
Find the number of subsets of that do not contain two consecutive integers.
It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has weights of integral values. What is the minimum for which there exist weights that satisfy this condition?
How many strings of ones and zeroes of length 10 are there such that there is an even number of ones, and no zero follows another zero?
We divide up the plane into disjoint regions using a circle, a rectangle and a triangle. What is the greatest number of regions that we can get?
There are players in a round-robin ping-pong tournament (i.e. every two persons will play exactly one game). After some matches have been played, it is known that the total number of matches that have been played among any people is equal to (where is a fixed integer). Find the sum of all possible values of .
We have a square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw?
We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?
Geometry B Find 100 times the area of a regular dodecagon inscribed in a unit circle. Round your answer to the nearest integer if necessary.
A triangle has sides of lengths 5, 6, 7. What is 60 times the square of the radius of the inscribed circle?
A rectangular piece of paper has sides of lengths , . The rectangle is folded in half such that coincides with and is the folding line. Then fold the paper along a line such that the corner falls on line . How large, in degrees, is ?
Tetrahedron has sides of lengths, in increasing order, 7, 13, 18, 27, 36, 41. If , then what is the length of ?
A polygon is called concave if it has at least one angle strictly greater than . What is the maximum number of symmetries that an 11-sided concave polygon can have? (A [i]symmetry[/i] of a polygon is a way to rotate or reflect the plane that leaves the polygon unchanged.)
In the following diagram (not to scale), , , , are four consecutive vertices of an 18-sided regular polygon with center . Let be the midpoint of and be the midpoint of . Find in degrees.
Lines and are perpendicular. Line partitions a convex polygon into two parts of equal area, and partitions the projection of the polygon onto into two line segments of length and respectively. Determine the maximum value of . (The floor notation denotes largest integer not exceeding )
Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the [i]square[/i] of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is , then you would submit 1734274).
Number Theory B
Find the number of pairs of integers and such that .
Suppose you are given that for some positive integer , is a perfect square. Find the sum of all possible values of .
You are given that for some digits and . Find the two-digit number that is missing above.
Find the number of ordered pairs of positive integers that are solutions of the following equation:
Find the sum of all prime numbers which satisfy for some primes (not necessarily distinct) , and .
Find the sum of all integers for which there is an integer , such that .
Suppose that for some positive integer , the first two digits of and are identical. Suppose the first two digits are and in this order. Find the two-digit number .
Let denote the sum of the digits of the positive integer . Find the largest positive integer that has no digits equal to zero and satisfies the equation
Individual Finals B You have an unlimited supply of monominos, dominos, and L-trominos. How many ways, in terms of , can you cover a grid with these shapes? Please note that you do [i]NOT[/i] have to use all the shapes. Also, you are allowed to [i]rotate[/i] any of the pieces, so they do not have to be aligned exactly as they are in the diagram below.
For what positive integer is maximal?
Let be a sequence of positive integers defined as follows: is a fixed six-digit number and for any , is a prime divisor of . Find .
[hide="Thread links"] Algebra A:
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=427710
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427732
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427733
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427916
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427917
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427918
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=427919
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=427830
Combinatorics A:
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427735
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427736
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427738
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427739
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=428102
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=428103
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=428104
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=427831
Geometry A:
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427746
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427747
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427749
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=428158
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=428154
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=428155
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=428156
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=427832
Number Theory A:
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427748
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427751
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427750
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=428227
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=428228
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=428229
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=428230
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=427833
Individual Finals A:
Algebra B:
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=427708
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=427709
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=427710
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427732
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427733
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427916
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427917
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427918
Combinatorics B:
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427734
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427735
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427736
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427738
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427739
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=428102
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=428103
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=428104
Geometry B:
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=427711
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=427712
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427746
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427747
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427749
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=428158
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=428154
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=428155
Number Theory B:
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=427713
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=149&t=427714
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427748
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427751
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=427750
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=428227
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=150&t=428228
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=428229
Individual Finals B:
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=428245
- http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=427754
Team Round
All problems here: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=414&t=427597 (questions may depend on each other.)