IDEA Math Online Placement Test Cover Sheet
Your name (please print)
Last
First
IDEAMATH student ID number
Online Placement Test
1/2/3/4 (circle one)
Contact Information
(phone number)
(email address)
Number of pages (not including this cover sheet)
Part
Algebra
Geometry
Combinatorics and Number Sense
Problem solving
Recommended time
30 minutes
30 minutes
30 minutes
60 minutes
Actual time spent
IDEA MATH Test Instructions
• Before you take the test, make you have have already email communications with
info@ideamath.org!
• The test has four sections, and it corresponds to a pair of course series.
• You should include all significant steps in your reasoning and computation. We are interested
in your ability to present your work, so unsupported answers will receive much less credit
than well-reasoned progress towards a solution without a correct answer. Do not simply
submit a list of answers; this makes it very difficult for us to evaluate your thinking and
methods of approaching the problem.
• In this document, you will find a cover sheet and an answer sheet. Print out each one, and
make several copies of the blank answer sheet. Fill out the top of each answer sheet as you
go, and then fill out the cover sheet when you’re finished. Start each subsection on a new
answer sheet.
• All the work you present must be your own.
• Don’t be intimidated! Some of the problems involve complex mathematical ideas, but all
can be solved using only elementary techniques, admittedly combined in clever ways.
• Be patient and persistent! Learning comes more from struggling with problems than from
solving them. Problem-solving becomes easier with experience. Success is not a function of
cleverness alone.
• Make sure that the cover sheet is the first page of your submission, and that it is completely
filled out.
Please send your solutions to
info@ideamath.org
E-mailed solutions may be written and scanned or typed in TeX. They should be sent as an
attachment in either .doc or .pdf format. If you write and scan your solutions, insert the
scans into a .doc or .pdf file, and send just the one file. If you have any questions, please feel
free to contact us at info@ideamath.org.
Suggestions for Writing Proofs
(By Tiankai Liu)
• All proofs should be written neatly and coherently in paragraphs of standard American English. Mathematical symbols like ≡ and ≤ should be used only in equations, not as verbs or
prepositions in a sentence. Do not write things like “all of the 4’s angles are ≤ 90◦ ”—this
should be “∠A, ∠B, ∠C ≤ 90◦ ” or “all of the triangle’s angles are at most 90◦ .” Avoid the
symbols ∧ ∨ ∵ ∴ ∀ ∃; instead, write out “and,” “or,” “because,” “therefore,” “for all,” “there
exists.” Similarly, do not use =⇒ or ⇒⇐ except as part of a sequence of equations.
• Write a statement using words rather than symbols unless this would be unnecessarily awkward. Do not invent more notation than is necessary to explain your solution.
• Write true statements. Do not write something that is only partially true, and then say how
to fix it later. If you assume something in one of your statements, say clearly what you are
assuming. Define all terms you make up. If you use figures, graphs, tables, etc., explain
thoroughly what they represent.
• Use the following formats for common proof patterns:
• Proof by induction: (1) state the claim, (2) check the base case, (3) prove the induction
step, and (4) conclude with the words “induction is complete.”
• Proof by contradiction: (1) state the claim, (2) state that you are assuming the opposite,
(3) derive a contradiction (and say why it is a contradiction), and (4) conclude that the
claim follows.
• Proof by case analysis: (1) state the claim, (2) state the various cases, (3) say why they
exhaust all possibilities, (4) analyze the cases one by one, and (5) conclude that the
claim follows.
• Draw accurate diagrams with compass and straightedge for geometry problems. This is for
your own good as well as the graders’.
IDEA Math Online Placement Test Solution Sheet
Your name (please print)
Online Placement Test
Last
First
1/2/3/4 (circle one)
Page
of
Write neatly! Write all work inside the box. Do NOT write on the back of the page.
Please go the next page for the problems.
IDEA Math Online Placement Test 1: PS1 and PS2
Algebra
A1.1. Find all solutions (x, y) to the following equations: x + y = 4, x − 2y = 10.
A1.2. In the currency of Mathamania, 5 wampas are equal to 3 kabobs and 10 kabobs are equal to
2 jambas. How many jambas are equal to twenty-five wampas?
A1.3. How many integer solutions are there to the equation |x − 5| < 4?
A1.4. Mary’s secret garden contains clones of Homer Simpson and WALL-E. A WALL-E clone has 4
legs. Meanwhile, Homer Simpson clones are human and therefore have 2 legs each. A Homer
Simpson clone always has 5 donuts, while a WALL-E clone has 2. In Mary’s secret garden,
there are 184 donuts and 128 legs. How many WALL-E clones are there?
A1.5. For what real numbers x is it true that x2 − 5x + 6 > 0?
Geometry
G1.1. The three angles of a triangle form an arithmetic sequence. Find the measure of the middle
angle.
G1.2. A square is changed into a rectangle by increasing two of its sides by p% and decreasing the
two other sides by p%. The area is then reduced by 1%. What is the value of p?
G1.3. Two angles of an isosceles triangle measure 80◦ and x◦ . What is the sum of all the possible
values of x?
G1.4. What is the maximum number of right angles that a (not necessarily convex) hexagon may
have? Sketch a rough figure of such a hexagon.
1
AD
=
G1.5. Points D and L are located on segment AK, with D between A and L, such that
DK
3
DL
5
DL
and
= . What is
?
LK
9
AK
Combinatorics and Number Sense
C1.1. What is the positive difference between the largest two prime factors of 159137?
C1.2. Noah has three green books and two red books. In how many ways could he arrange them in
a row, if books of the same color are not distinguishable?
C1.3. Bob tries to remember his 2-digit extension number. He knows that the number is divisible
by 5 and that the first digit is odd. How many possibilities are there for this number?
C1.4. Elmo has a dime which lands heads with probability 34 and a quarter which lands heads with
probability 23 . If he flips both coins, what is the probability that one lands heads and the
other tails?
C1.5. Find the number of two-digit integers that are divisible by each of their digits.
Problem Solving
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P1.1. Evaluate the sum 1 − 2 + 3 − 4 + 5 − · · · + 25.
P1.2. John is entering the following summation 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 in his
calculator. However, he accidently leaves out a plus sign and the answer becomes 3582. What
is the number that comes before the missing plus sign?
P1.3. Bob bought at least 10 blueberry bushes for his garden. His garden can hold at most 50
bushes. He realized that if he planted them in rows of 5, then he had a bush left over. He
also could not plant them evenly in rows of 8. However, he then realized that he could plant
the bushes in a square. How many bushes did he buy?
P1.4. Before his lunch break, Max drives at 50 mph. After taking his lunch break, he drives at
30 mph. He starts a 300-mile journey at 10 AM and wants to arrive by 7 PM. If he takes a
1-hour long lunch break, at what time should he begin his lunch break?
P1.5. One segment with length 6 and some segments with lengths 10, 8, and 2 form the three letters
in the diagram shown below. Compute the sum of the perimeters of the three figures.
P1.6. The cookie monster has an equilateral triangular cake and wants to frost each of the 3 vertical
sides either red, green, or blue. How many ways could he frost his cake? Note that if he can
rotate his cake from one frosting pattern to another, then those two patterns are counted as
the same. (He cannot, of course, reflect his pattern by flipping the cake over, since that would
smash the decorations on top. It is also not necessary for him to use each color of frosting.)
P1.7. The side length of square ABCD is 8 centimeters. Points M, N, P , and Q are the midpoints
of sides AB, BC, CD, and DA, respectively. How many centimeters are in the sum of the
lengths of all diagonals of hexagon QM BN P D?
P1.8. Compute the 2011th smallest positive integer N that gains an extra digit when doubled.
P1.9. How many 11-letter sequences of E’s and M’s have at least as many E’s as M’s?
P1.10. Find a 5-digit number for which all of the following are true:
(a) The number’s square root is a prime palindrome. (A palindrome, such as 83438, is a
number that remains the same when its digits are reversed.);
(b) The number’s first two digits express a perfect square;
(c) The number’s middle digit is a perfect square;
(d) The number’s last two digits express a perfect square.
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