Well, The 2011 Harvard-MIT Mathematics Tournament was held on Saturday, February 12, 2011. All the archives (including problems and solutions) can be download here:
Showing posts with label high school math olympiad. Show all posts
Showing posts with label high school math olympiad. Show all posts
6/04/2011
Harvard–MIT Mathematics Tournament (Problem and Solution)
This high school mathematics tournament is sponsored by Harvard and MIT. HMMT is one of the largest math contests in the United States (based on opinion in the official site).
Well, The 2011 Harvard-MIT Mathematics Tournament was held on Saturday, February 12, 2011. All the archives (including problems and solutions) can be download here:
Well, The 2011 Harvard-MIT Mathematics Tournament was held on Saturday, February 12, 2011. All the archives (including problems and solutions) can be download here:
4/19/2011
Bank Problems (Geometric Series)
A bank invites term deposits under the following conditions:
1. The deposit (or initial principal) may be any amount in dollars and cents, with a
minimum of $1.
2. The yearly interest, calculated at the end of each year and added to the principal, is
one cent less than 10% of the current principal, fractions of a cent being discarded.
3.The deposit with accumulated interest is returned at end of the sixth year.
Find the smallest initial deposit which would result in no fractions of cents being discarded
in any of the six years.
Hints.
Let $\displaystyle \large {\color{Yellow} x_{n}}$ be the value of the principal at the end of the nth year.
Derive a relationship between $\displaystyle \large {\color{Yellow} x_{n}}$ and $\displaystyle \large {\color{Yellow} x_{n-1}}$
Guess a relationship between $\displaystyle \large {\color{Yellow} x_{n}}$ and $\displaystyle \large {\color{Yellow} x_{0}}$ ( $\displaystyle \large {\color{Yellow} x_{0}}$ is the initial principal). This relationship should remind you of the compound interest formula.
Prove your guess by mathematical induction.
What's the sum of a geometric series?
Now think in terms of congruences.
Source: Lecture Notes on Mathematical Olympiad Courses: For Junior Section Vol 1 (Mathematical Olympiad Series)
1. The deposit (or initial principal) may be any amount in dollars and cents, with a
minimum of $1.
2. The yearly interest, calculated at the end of each year and added to the principal, is
one cent less than 10% of the current principal, fractions of a cent being discarded.
3.The deposit with accumulated interest is returned at end of the sixth year.
Find the smallest initial deposit which would result in no fractions of cents being discarded
in any of the six years.
Hints.
Let $\displaystyle \large {\color{Yellow} x_{n}}$ be the value of the principal at the end of the nth year.
Derive a relationship between $\displaystyle \large {\color{Yellow} x_{n}}$ and $\displaystyle \large {\color{Yellow} x_{n-1}}$
Guess a relationship between $\displaystyle \large {\color{Yellow} x_{n}}$ and $\displaystyle \large {\color{Yellow} x_{0}}$ ( $\displaystyle \large {\color{Yellow} x_{0}}$ is the initial principal). This relationship should remind you of the compound interest formula.
Prove your guess by mathematical induction.
What's the sum of a geometric series?
Now think in terms of congruences.
Source: Lecture Notes on Mathematical Olympiad Courses: For Junior Section Vol 1 (Mathematical Olympiad Series)
4/16/2011
Geometry Problem for High School
1. ABCD is a square with E insides it. If ABE is a equilateral triangle, then <BEC = ....
2. Below is the picture of semicircle with diameter AC. AB is 3 units longer than BC. If BD= 5 units and perpendicular with AC, then BC = ....
4/04/2011
Geometry Problem for High School
California Geometry: Concepts, Skills, and Problem Solving
1. In circle O, <PRS= 500 and
2.
2.
From: MSA Math Competition High School
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