Stanford University Mathematics Camp (SUMaC) 2014
Admission Exam
Solve as many of the following problems as you can.
Although SUMaC admission is competitive, correct answers to every problem is not required
for admission to SUMaC. Your work on these probl
USA
AMC 12/AHSME
2000
1 In the year 2001, the United States will host the International Mathematical Olympiad. Let
I, M , and O be distinct positive integers such that the product I M O = 2001. Whats the
largest possible value of the sum I + M + O?
(A) 23
2006 AMC 8
AMC 8 2006
1
Mindy made three purchases for $1.98, $5.04 and $9.89. What was her total,
to the nearest dollar?
(A) $10
(B) $15
(C) $16
(D) $17
(E) $18
2
On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions
incorrectly
2014 AMC 8
AMC 8 2014
1
Harry and Terry are each told to calculate 8 (2 + 5). Harry gets the correct
answer. Terry ignores the parentheses and calculates 82+5. If Harrys answer
is H and Terrys answer is T , what is H T ?
(A) 10
2
(D) 4
(E) 5
(B) 250
(C) 2
2013 AMC 8
AMC 8 2013
1
Danica wants to arrange her model cars in rows with exactly 6 cars in each
row. She now has 23 model cars. What is the smallest number of additional
cars she must buy in order to be able to arrange all her cars this way?
(A) 1
2
(B
2007 AMC 8
AMC 8 2007
1
Theresas parents have agreed to buy her tickets to see her favorite band if she
spends an average of 10 hours per week helping around the house for 6 weeks.
For the rst 5 weeks, she helps around the house for 8, 11, 7, 12 and 10 ho
2009 AMC 8
AMC 8 2009
1
Bridget bought a bag of apples at the grocery store. She gave half of the apples
to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many
apples did Bridget buy?
(A) 3
(B) 4
(C) 7
(D) 11
(E) 14
2
On average, fo
2012 AMC 8
AMC 8 2012
1
Rachelle uses 3 pounds of meat to make 8 hamburgers for her family. How
many pounds of meat does she need to make 24 hamburgers for a neighborhood
picnic?
2
1
(A) 6
(B) 6
(C) 7
(D) 8
(E) 9
3
2
2
In the country of East Westmore, sta
2008 AMC 8
AMC 8 2008
1
Susan had $50 to spend at the carnival. She spent $12 on food and twice as
much on rides. How many dollars did she have left to spend?
(A) 12
2
(B) 14
(B) 8672
(D) 9782
(E) 9872
(B) Monday
(C) Wednesday
(D) Thursday
(E) Saturday
(B
2011 IMO Shortlist
IMO Shortlist 2011
Algebra
1
Given any set A = cfw_a1 , a2 , a3 , a4 of four distinct positive integers, we denote
the sum a1 + a2 + a3 + a4 by sA . Let nA denote the number of pairs (i, j) with
1 i < j 4 for which ai + aj divides sA .
2013 IMO Shortlist
IMO Shortlist 2013
Algebra
A1
Let n be a positive integer and let a1 , . . . , an1 be arbitrary real numbers. Dene
the sequences u0 , . . . , un and v0 , . . . , vn inductively by u0 = u1 = v0 = v1 = 1,
and uk+1 = uk + ak uk1 , vk+1 = v
2012 IMO Shortlist
IMO Shortlist 2012
Algebra
A1
Find all functions f : Z Z such that, for all integers a, b, c that satisfy
a + b + c = 0, the following equality holds:
f (a)2 + f (b)2 + f (c)2 = 2f (a)f (b) + 2f (b)f (c) + 2f (c)f (a).
(Here Z denotes t
2010 IMO Shortlist
IMO Shortlist 2010
Algebra
1
Find all function f : R R such that for all x, y R the following equality
holds
f (x y) = f (x) f (y)
where a is greatest integer not greater than a.
Proposed by Pierre Bornsztein, France
2
Let the real numb
2009 IMO Shortlist
IMO Shortlist 2009
Algebra
1
Find the largest possible integer k, such that the following statement is true:
Let 2009 arbitrary non-degenerated triangles be given. In every triangle the
three sides are coloured, such that one is blue, o
1998 IMO Shortlist
IMO Shortlist 1998
Geometry
1
A convex quadrilateral ABCD has perpendicular diagonals. The perpendicular
bisectors of the sides AB and CD meet at a unique point P inside ABCD.
Prove that the quadrilateral ABCD is cyclic if and only if t
2008 IMO Shortlist
IMO Shortlist 2008
Algebra
1
Find all functions f : (0, ) (0, ) (so f is a function from the positive real
numbers) such that
w 2 + x2
(f (w)2 + (f (x)2
= 2
f (y 2 ) + f (z 2 )
y + z2
for all positive real numbes w, x, y, z, satisfying
2005 IMO Shortlist
IMO Shortlist 2005
Algebra
1
Find all pairs of integers a, b for which there exists a polynomial P (x) Z[X]
such that product (x2 + ax + b) P (x) is a polynomial of a form
xn + cn1 xn1 + + c1 x + c0
where each of c0 , c1 , . . . , cn1 i
2002 IMO Shortlist
IMO Shortlist 2002
Geometry
1
Let B be a point on a circle S1 , and let A be a point distinct from B on the
tangent at B to S1 . Let C be a point not on S1 such that the line segment AC
meets S1 at two distinct points. Let S2 be the cir
2001 IMO Shortlist
IMO Shortlist 2001
Geometry
1
Let A1 be the center of the square inscribed in acute triangle ABC with two
vertices of the square on side BC. Thus one of the two remaining vertices of
the square is on side AB and the other is on AC. Poin
1996 IMO Shortlist
IMO Shortlist 1996
Algebra
1
Suppose that a, b, c > 0 such that abc = 1. Prove that
ab
bc
ca
+
+
1.
ab + a5 + b5 bc + b5 + c5 ca + c5 + a5
2
Let a1 a2 . . . an be real numbers such that for all integers k > 0,
ak + ak + . . . + ak 0.
1
1997 IMO Shortlist
IMO Shortlist 1997
1
In the plane the points with integer coordinates are the vertices of unit squares.
The squares are coloured alternately black and white (as on a chessboard). For
any pair of positive integers m and n, consider a rig
1999 IMO Shortlist
IMO Shortlist 1999
Geometry
1
Let ABC be a triangle and M be an interior point. Prove that
mincfw_M A, M B, M C + M A + M B + M C < AB + AC + BC.
2
A circle is called a separator for a set of ve points in a plane if it passes
through th
2000 IMO Shortlist
IMO Shortlist 2000
Algebra
1
Let a, b, c be positive real numbers so that abc = 1. Prove that
a1+
1
b
b1+
1
c
c1+
1
a
1.
2
Let a, b, c be positive integers satisfying the conditions b > 2a and c > 2b. Show
that there exists a real numb
USA
AIME
2000
I
1 Find the least positive integer n such that no matter how 10n is expressed as the product of
any two positive integers, at least one of these two integers contains the digit 0.
2 Let u and v be integers satisfying 0 < v < u. Let A = (u,
USA
USAMO
2000
Day 1 - 02 May 2000
1 Call a real-valued function f very convex if
f (x) + f (y)
2
f
x+y
2
+ |x
y|
holds for all real numbers x and y. Prove that no very convex function exists.
2 Let S be the set of all triangles ABC for which
1
1
1
3
6
5
USA
USAJMO
2010
Day 1 - 27 April 2010
1 A permutation of the set of positive integers [n] = 1, 2, ., n is a sequence (a1 , a2 , . . . , an ) such
that each element of [n] appears precisely one time as a term of the sequence. For example,
(3, 5, 1, 2, 4) i
Qualifying Quiz
2015 Quiz
Background Reading
Past Quizzes
Solutions
Policy on Getting Help
Tips on Writing Proofs
Using Computers
LaTeX Tutorial
PDF Tutorial
Ask a Question
Back to Application
Welcome to the 2015 Qualifying Quiz!
What is the Quiz?
The Qu
Art of Problem Solvi
www.aops.com
Preparing students for success since i 993.
AMC 10/12 preparation
materials for
students and teachers
Proud sponsor of the
American Mathematics Competitions
MATHEMATICAL ASSOCIATION OF AMERICA :Textbooks
The Art of Pr
AwesomeMath Admission Test Cover Sheet
Your Name
Last Name
Admission Test
Contact Information
Please Print
First Name
A
Phone Number
Email
Number of pages (not including this cover sheet)
B
C
Check one
Awesome Math Test A
March 16 - April 6, 2012
Do not
2008 State of Utah Senior Ma thematics Contest 1
State Senior Mathematics Contest
Spring 2008
1. Points A, B, C lie on a circle with radius r centered at (3. Segment DE
has length r.
1fm(< 80(3) 2 30°, then M(< BBC) =
(a) 15°
g) 12"
{£2399 100
(d) 20°
(