Difference between revisions of "2010 PUMaC Problems"
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Square <math>ABCD</math> is divided into four rectangles by <math>EF</math> and <math>GH</math>. <math>EF</math> is parallel to <math>AB</math> and <math>GH</math> parallel to <math>BC</math>. <math>\angle BAF = 18^\circ</math>. <math>EF</math> and <math>GH</math> meet at point <math>P</math>. The area of rectangle <math>PFCH</math> is twice that of rectangle <math>AGPE</math>. Given that the value of <math>\angle FAH</math> in degrees is <math>x</math>, find the nearest integer to <math>x</math>. | Square <math>ABCD</math> is divided into four rectangles by <math>EF</math> and <math>GH</math>. <math>EF</math> is parallel to <math>AB</math> and <math>GH</math> parallel to <math>BC</math>. <math>\angle BAF = 18^\circ</math>. <math>EF</math> and <math>GH</math> meet at point <math>P</math>. The area of rectangle <math>PFCH</math> is twice that of rectangle <math>AGPE</math>. Given that the value of <math>\angle FAH</math> in degrees is <math>x</math>, find the nearest integer to <math>x</math>. | ||
<asy> | <asy> | ||
− | size( | + | size(100); defaultpen(linewidth(0.7)+fontsize(10)); |
pair D2(pair P) { | pair D2(pair P) { | ||
dot(P,linewidth(3)); return P; | dot(P,linewidth(3)); return P; | ||
} | } | ||
− | + | // NOTE: I've tampered with the angles to make the diagram not-to-scale. The correct numbers should be 72 instead of 76, and 45 instead of 55. | |
− | pair A=(0,1), B=(0,0), C=(1,0), D=(1,1) | + | pair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0]; |
− | + | draw(A--B--C--D--cycle); | |
− | label("$A$",D2(A), | + | draw(F--A--H); draw(E--F); draw(G--H); |
− | label("$B$",D2(B), | + | label("$A$",D2(A),NW); |
− | label("$C$",D2(C), | + | label("$B$",D2(B),SW); |
− | label("$D$",D2(D), | + | label("$C$",D2(C),SE); |
− | + | label("$D$",D2(D),NE); | |
− | label("$F$",D2(F), | + | label("$E$",D2(E),plain.N); |
− | label("$G$",D2(G), | + | label("$F$",D2(F),S); |
− | label("$H$",D2(H), | + | label("$G$",D2(G),W); |
− | label("$P$",D2(P), | + | label("$H$",D2(H),plain.E); |
+ | label("$P$",D2(P),SE); | ||
</asy> | </asy> | ||
[[2011 PuMAC Problems/Geometry Problem A7|Solution]] | [[2011 PuMAC Problems/Geometry Problem A7|Solution]] |
Revision as of 01:35, 20 August 2011
Contents
Division A
Algebra
Find the sum of the coefficients of the polynomial .
Calculate where
is the largest integer less than or equal to
.
Let be the sum of all real
such that
. Find the nearest integer to
.
Define . Find the smallest integer
such that
.
Let . If the roots of
are given by
,
, and
, find
Assume that , and that
. Find
.
The expression is equal to
, where
is an integer. Find
.
Let be a polynomial with integer coefficients such that
,
, and
. Find an integer
such that
.
Combinatorics
PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, the probability that the 4th customer orders a previously unordered item is , where
and
are relatively prime positive integers. Find
.
Let represent the three-digit number with hundreds digit
, tens digit
, and units digit
, and similarly let
represent the two-digit number with tens digit
and units digit
. How many three-digit numbers
, none of whose digits are 0, are there such that
?
Sterling draws 6 circles on the plane, which divide the plane into regions (including the unbounded region). What is the maximum number of resulting regions?
Erick stands in the square in the 2nd row and 2nd column of a 5 by 5 chessboard. There are $1 bills in the top left and bottom right squares, and there are $5 bills in the top right and bottom left squares, as shown below.
\[\begin{tabular}{|p{1em}|p{1em}|p{1em}|p{1em}|p{1em}|} \hline $1 & & & & $5 \\ \hline & E & & &\\ \hline & & & &\\ \hline & & & &\\ \hline $5 & & & & $1 \\ \hline \end{tabular}\] (Error compiling LaTeX. Unknown error_msg)
Every second, Erick randomly chooses a square adjacent to the one he currently stands in (that is, a square sharing an edge with the one he currently stands in) and moves to that square. When Erick reaches a square with money on it, he takes it and quits. The expected value of Erick's winnings in dollars is , where
and
are relatively prime positive integers. Find
.
We say that a rook is ``attacking" another rook on a chessboard if the two rooks are in the same row or column of the chessboard and there is no piece directly between them. Let be the maximum number of rooks that can be placed on a
chessboard such that each rook is attacking at most one other. How many ways can
rooks be placed on a
chessboard such that each rook is attacking at most one other?
All the diagonals of a regular decagon are drawn. A regular decagon satisfies the property that if three diagonals concur, then one of the three diagonals is a diameter of the circumcircle of the decagon. How many distinct intersection points of diagonals are in the interior of the decagon?
Matt is asked to write the numbers from 1 to 10 in order, but he forgets how to count. He writes a permutation of the numbers across his paper such that:
- The leftmost number is 1.
- The rightmost number is 10.
- Exactly one number (not including 1 or 10) is less than both the number to its immediate left and the number to its immediate right.[/list]
How many such permutations are there?
Let be the sum of all binomial coefficients
such that
and
are nonnegative integers and
is an even integer less than 100. Find the remainder when
is divided by 144. (Note:
if
, and
.)
Geometry
As in the following diagram, square and square
are placed side by side (i.e.
is between
and
and
is between
and
). If
,
, compute the minimal area of
.
In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners , he starts at
and walks to
. Then, he walks to the midpoint of side
, say
. Then, he walks to the midpoint of side
say
, and then the midpoint of
which is
. He continues in this fashion, indefinitely. The total length of his path if
and
is of the form
. Find
.
Triangle has
,
, and
. An angle bisector is drawn from angle
, and meets
at
. What is the nearest integer to
?
In regular hexagon ,
,
are two diagonals. Points
,
are on
,
respectively and satisfy
. Suppose
are collinear, find
.
(diagram goes here)
Solution
A cuboctahedron is a solid with 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle (see picture). Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is . Find
.
(diagram goes here)
In the following diagram, a semicircle is folded along a chord and intersects its diameter
at
. Given that
and
. If
, find
.
Solution
Square is divided into four rectangles by
and
.
is parallel to
and
parallel to
.
.
and
meet at point
. The area of rectangle
is twice that of rectangle
. Given that the value of
in degrees is
, find the nearest integer to
.
Solution
There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?
Number Theory
Find the smallest positive integer such that
is composite.
Find the largest positive integer such that
, where
is the sum of the divisors of
, including
.
Find the sum of the first 5 positive integers such that
is the product of 3 distinct primes.
Find the largest positive integer such that
is a perfect square. (
is the number of integers
,
that are relatively prime to
)
Given that ,
are positive integers with
, but neither
nor
divides either of
or
, and
as small as possible, find
.
Find the numerator of when reduced.
Let be the number of polynomial functions from the integers modulo
to the integers modulo
.
can be written as
, where the
s are (not necessarily distinct) primes. Find
.
A consecutive pythagorean triple is a pythagorean triple of the form ,
and
positive integers. Given that
,
, and
form the third consecutive pythagorean triple, find
.
Division B
Algebra
Let the operation be defined by
. Calculate
.
Let . Find the fourth smallest prime
such that
is divisble by
for some integer
.
Write , with
,
,
,
, and
integers. Find
.
Let be the sum of all real
such that
. Find the nearest integer to
.
Let be a real root of the polynomial
. Find
.
Define . Find the smallest integer
such that
.
Let be a function such that
and
. Find the last two digits of
.
The expression is equal to
, where
is an integer. Find
.
Combinatorics
The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play. If the Band can't play any two tiring songs in a row, how many ways can the band play its 8 songs?
PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, the probability that the 4th customer orders a previously unordered item is , where
and
are relatively prime positive integers. Find
.
% Yeah, I stole the PUMaCDonalds joke from another round (the team round?). We can change the problem statement to "A newly-opened fast food restaurant has 5 menu items..." if that's better.
Let
represent the three-digit number with hundreds digit
, tens digit
, and units digit
, and similarly let
represent the two-digit number with tens digit
and units digit
. How many three-digit numbers
, none of whose digits are 0, are there such that
?
Sterling draws 6 circles on the plane, which divide the plane into regions (including the unbounded region). What is the maximum number of resulting regions?
people take part in a chess tournament:
girls and
boys. Each participant plays with each of the others exactly once. There were no ties and the number of games won by the girls is
the number of games won by the boys. How many people took part in the tournament?
A regular pentagon is drawn in the plane, along with all its diagonals. All its sides and diagonals are extended infinitely in both directions, dividing the plane into regions, some of which are unbounded. An ant starts in the center of the pentagon, and every second, the ant randomly chooses one of the edges of the region it's in, with an equal probability of choosing each edge, and crosses that edge into another region. If the ant enters an unbounded region, it explodes. After first leaving the central region of the pentagon, let be the expected number of times the ant re-enters the central region before it explodes. Find the closest integer to
.
We say that a rook is ``attacking" another rook on a chessboard if the two rooks are in the same row or column of the chessboard and there is no piece directly between them. Let be the maximum number of rooks that can be placed on a
chessboard such that each rook is attacking at most one other. How many ways can
rooks be placed on a
chessboard such that each rook is attacking at most one other?
Matt is asked to write the numbers from 1 to 10 in order, but he forgets how to count. He writes a permutation of the numbers across his paper such that:
- The leftmost number is 1.
- The rightmost number is 10.
- Exactly one number (not including 1 or 10) is less than both the number to its immediate left and the number to its immediate right.
How many such permutations are there?
Geometry
In a polygon, every external angle is one sixth of its corresponding internal angle. How many sides does the polygon have?
On rectangular coordinates, point ,
.
is on
-axis. Given that
is chosen such that
is minimized, compute
.
As in the following diagram, square and square
are placed side by side (i.e.
is between
and
and
is between
and
). Now
,
, compute the minimal area of
.
(diagram goes here)
Unit square is divided into four rectangles by
and
, with
.
is parallel to
and
parallel to
.
and
meet at point
. Suppose
, calculate the nearest integer to the degree of
.
(diagram goes here)
In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners , he starts at
and walks to
. Then, he walks to the midpoint of side
, say
. Then, he walks to the midpoint of side
say
, and then the midpoint of
which is
. He continues in this fashion, indefinitely. The total length of his path if
and
is of the form
. Find
.
In regular hexagon ,
,
are two diagonals. Points
,
are on
,
respectively and satisfy
. Suppose
are collinear, find
.
(diagram goes here)
A cuboctahedron is a solid with 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle (see picture). Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is . Find
.
(diagram goes here)
Point is in the interior of
. The side lengths of
are
,
,
. The three foots of perpendicular lines from
to sides
,
,
are
,
,
respectively. Suppose the minimal value of
can be written as
, where
and
is square free, calculate
.
(diagram goes here)
Number Theory
Find the positive integer less than 18 with the most positive divisors.
Let be the sum of the digits of
. Find
.
Find the smallest positive integer such that
is composite.
Find the sum of the first 5 positive integers such that
is the product of 3 distinct primes.
Given that ,
, and
are positive integers such that
. Find the sum of all possible
values.
Given that ,
are positive integers with
, but neither
nor
divides either of
or
, and
as small as possible, find
.
Find the numerator of when reduced.
Let be the number of (positive) divisors of
ending in the digit
. What is the remainder when
is divided by 2010?