1. (2 points) Solve for x: x = 2 +
2. (2 points) What is 3(2 log4 (2(2 log3 9)?
3. (3 points) Given the sequence 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, . . ., nd n such that the sum
of the rst n terms is 2008 or 2009.
4. (3 points
1. (2 points) Sarah buys 3 gumballs from a gumball machine that contains 10 orange, 6 green, and 9
yellow gumballs. What is the probability that the rst gumball is orange, the second is green or
yellow, and the third is also orange?
1. (2 points) If a rectangles length is increased by 30% and its width is decreased by 30%, by what
percentage does its area change? State whether the area increases or decreases.
2. (2 points) What is the area of a circle with a circumference of
Number Theory B
1. (2 points)What is the remainder, in base 10, when 247 + 3647 + 437 + 127 + 37 + 17 is divided by 6?
2. (2 points)How many zeros are there at the end of 792! when written in base 10?
3. (3 points)Find all integral solutions to xy y x = 1
1. (2 points) Calculate
6 + . +
2. (2 points) Find log2 3 log3 4 log4 5 . . . log62 63 log63 64.
3. (3 points) What is the smallest positive integer value of x for which x 4 (mod 9) and x 7 (mod
4. (3 points) What is the dier
Individual Finals B
1. Find all pairs of positive real numbers (a, b) such that
for all positive integes
2. Let P be a convex polygon, and let n 3 be a positive integer. On each side of P, erect a regular
n-gon that shares that side of
1. (2 points) Given the sequence 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, . . ., nd n such that the sum
of the rst n terms is 2008 or 2009.
2. (2 points) What is the polynomial of smallest degree that passes through (2, 2), (1, 1), (0, 2),
1. (2 points) How many 3-digit numbers contain the digit 7 exactly once?
2. (2 points) Draw a regular hexagon. Then make a square from each edge of the hexagon. Then form
equilateral triangles by drawing an edge between every pair of neigh
1. (2 points) What is the area of a circle with a circumference of 8?
(2 points) Consider a convex polygon P in space with perimeter 20 and area 30. What is the
volume of the locus of points that are at most 1 unit away from some point in the
1. (2 points) What is the dierence between the median and the mean of the following data set: 12,
41, 44, 48, 47, 53, 60, 62, 56, 32, 23, 25, 31?
2. (2 points) Quadrilateral ABCD has both an inscribed and a circumscribed circle and sidelengths
Number Theory A
1. (2 points)How many zeros are there at the end of 792! when written in base 10?
2. (3 points)Find all integral solutions to xy y x = 1.
3. (3 points) Find the largest integer n, where 2009n divides 20082009
4. (3 p
Individual Finals A
Find all positive real numbers b for which there exists a positive real number k such that
n k bn n for all positive integers n.
2. A hypergraph consists of a set of vertices V and a set of subsets of those vertices, each of which i
1. Let the operation
be dened by x y = y x xy. Calculate (3 4) (4 3).
2. Let p(x) = x2 + x + 1. Find the fourth smallest prime q such that p(n) is divisble by q for
some integer n.
= a + b 2 + c 4 + d 8 + e 16, with a, b, c, d, and e i
1. The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play.
If the Band cant play any two tiring songs in a row, how many ways can the band play its 8
2. PUMaCDonalds, a newly-opened fast foo
1. In a polygon, every external angle is one sixth of its corresponding internal angle. How many
sides does the polygon have?
2. On rectangular coordinates, point A = (1, 2), B = (3, 4). P = (a, 0) is on x-axis. Given that
P is chosen such that
Number Theory B
1. Find the positive integer less than 18 with the most positive divisors.
2. Let f (n) be the sum of the digits of n. Find
3. Find the smallest positive integer n such that n4 + (n + 1)4 is composite.
4. Find the sum of the
Number Theory A
1. Find the smallest positive integer n such that n4 + (n + 1)4 is composite.
2. Find the largest positive integer n such that (n) = 28, where (n) is the sum of the divisors
of n, including n.
3. Find the sum of the rst 5 positive integers
1. Find the sum of the coecients of the polynomial (63x 61)4 .
2010 1 where x is the largest integer less than or equal to x.
3. Let S be the sum of all real x such that 4x = x4 . Find the nearest integer to S.
4. Dene f (x) =
1. PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the rst 4 customers each choose one menu item at random, the probability that the 4th customer orders a
previously unordered item is m/n, where m and n are relative
1. As in the following diagram, square ABCD and square CEF G are placed side by side (i.e. C
is between B and E and G is between C and D). If CE = 14, AB > 14, compute the minimal
area of AEG.
2. In a rectangular plot of land, a man walks in a
Atomic Theory & Light
From C. Mitchell
This assessment is to help you identify what you know and, more importantly, what you DONT KNOW.
1) Which of the following statements about atoms is FALSE?
A) Atoms compose all matter.
PROBLEMS AND SOLUTIONS
Curtis Cooper Shing 8. 80
CM] Problems CMJ Solutions
Department of Mathematics and Department of Mathematics and
Computer Science Computer Science
University of Central Missouri University of Central Missouri
2004 SLC Business Calculations
Mark the correct answer on your Scantron sheet for each of the following questions.
1. The MelMar Motel estimates its costs at $25 per room. Management wants an
80% markup based on cost. Calculate the selling price per room
BUSINESS CALCULATIONS TEST
Emily bought a new car for $11,400. She used the car three years and then traded it in for
$5,100. Find the average annual depreciation.
Trisha wants to buy a condomi
WRC 2008 BUSINESS CALCULATIONS
Identify the letter of the choice that best completes the statement or answers the question.
1. Which of the following is an example of a job benefit?
a. paid holidays
c. child care
OBJECTIVE test EVENTS at COMPETENCIES'
Overview NLC Registration
These events consist of a 60-minute test administered during the Participants must be registered for the NLC and pay the national
National Leadership Conference (NL ). conference registratio
CHAPTER FOUR - COMPETITIVE EVENTS
FBLA members need to develop the proper attitude toward competitive events. Learning how to win as
well as how to lose should be an integral part of leadership development. The main value is not derived
from winning or
Online Cognitive-Behavioral Therapy Workshop
The following styles in thinking can be subtle yet very powerful in causing us to
experience needless emotional distress. Interestingly, the more distressed we become,
the more our thinking can
Korean Mathematical Olympiad
First Round 2006-2015
Translation by rkm0959, problems written by KMS.
Latex revision by Leon.
Problems 1-5 are worth 4 points, Problems 6-15 are worth 5, Problems 16-20 are worth 6.
Time limit is 4 hours.
KMO First Round 20