Stanford University Mathematics Camp (SUMaC) 2014
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
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?
2006 AMC 8
AMC 8 2006
Mindy made three purchases for $1.98, $5.04 and $9.89. What was her total,
to the nearest dollar?
On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions
2014 AMC 8
AMC 8 2014
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 ?
2013 AMC 8
AMC 8 2013
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?
2007 AMC 8
AMC 8 2007
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
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?
On average, fo
2012 AMC 8
AMC 8 2012
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
In the country of East Westmore, sta
2008 AMC 8
AMC 8 2008
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?
2011 IMO Shortlist
IMO Shortlist 2011
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
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
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
Find all function f : R R such that for all x, y R the following equality
f (x y) = f (x) f (y)
where a is greatest integer not greater than a.
Proposed by Pierre Bornsztein, France
Let the real numb
2009 IMO Shortlist
IMO Shortlist 2009
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
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
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
f (y 2 ) + f (z 2 )
y + z2
for all positive real numbes w, x, y, z, satisfying
2005 IMO Shortlist
IMO Shortlist 2005
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
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
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
Suppose that a, b, c > 0 such that abc = 1. Prove that
ab + a5 + b5 bc + b5 + c5 ca + c5 + a5
Let a1 a2 . . . an be real numbers such that for all integers k > 0,
ak + ak + . . . + ak 0.
1997 IMO Shortlist
IMO Shortlist 1997
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
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.
A circle is called a separator for a set of ve points in a plane if it passes
2000 IMO Shortlist
IMO Shortlist 2000
Let a, b, c be positive real numbers so that abc = 1. Prove that
Let a, b, c be positive integers satisfying the conditions b > 2a and c > 2b. Show
that there exists a real numb
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,
Day 1 - 02 May 2000
1 Call a real-valued function f very convex if
f (x) + f (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
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
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2008 State of Utah Senior Ma thematics Contest 1
State Senior Mathematics Contest
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) =