International Mathematics Olympiad Problems - 2011 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

 Page 1


Monday, July 18, 2011
Problem 1. Given any setA ={a
1
,a
2
,a
3
,a
4
} of four distinct positive integers, we denote the sum
a
1
+a
2
+a
3
+a
4
bys
A
. Let n
A
denote the number of pairs (i,j) with 1=i<j= 4 for which a
i
+a
j
divides s
A
. Find all sets A of four distinct positive integers which achieve the largest possible value
of n
A
.
Problem 2. LetS be a ?nite set of at least two points in the plane. Assume that no three points
ofS are collinear. A windmill is a process that starts with a line ` going through a single point
P?S. The line rotates clockwise about the pivot P until the ?rst time that the line meets some
other point belonging toS. This point, Q, takes over as the new pivot, and the line now rotates
clockwise about Q, until it next meets a point ofS. This process continues inde?nitely.
Show that we can choose a pointP inS and a line` going throughP such that the resulting windmill
uses each point ofS as a pivot in?nitely many times.
Problem 3. Let f : R? R be a real-valued function de?ned on the set of real numbers that
satis?es
f(x+y)=yf(x)+f(f(x))
for all real numbers x and y. Prove that f(x) = 0 for all x= 0.
Page 2


Monday, July 18, 2011
Problem 1. Given any setA ={a
1
,a
2
,a
3
,a
4
} of four distinct positive integers, we denote the sum
a
1
+a
2
+a
3
+a
4
bys
A
. Let n
A
denote the number of pairs (i,j) with 1=i<j= 4 for which a
i
+a
j
divides s
A
. Find all sets A of four distinct positive integers which achieve the largest possible value
of n
A
.
Problem 2. LetS be a ?nite set of at least two points in the plane. Assume that no three points
ofS are collinear. A windmill is a process that starts with a line ` going through a single point
P?S. The line rotates clockwise about the pivot P until the ?rst time that the line meets some
other point belonging toS. This point, Q, takes over as the new pivot, and the line now rotates
clockwise about Q, until it next meets a point ofS. This process continues inde?nitely.
Show that we can choose a pointP inS and a line` going throughP such that the resulting windmill
uses each point ofS as a pivot in?nitely many times.
Problem 3. Let f : R? R be a real-valued function de?ned on the set of real numbers that
satis?es
f(x+y)=yf(x)+f(f(x))
for all real numbers x and y. Prove that f(x) = 0 for all x= 0.
T uesday, July 19, 2011 Problem 4. Let n > 0 b e an in teger. W e are giv en a balance and n w eigh ts of w eigh t 2
0
, 2
1
,..., 2
n-1
. W e are to place eac h of the n w eigh ts on the balance, one after another, in suc h a w a y that the righ t pan is nev er hea vier than the left pan. A t eac h step w e c ho ose one of the w eigh ts that has not y et b een placed on the balance, and place it on either the left pan or the righ t pan, un til all of the w eigh ts ha v e b een placed. Determine the n um b er of w a ys in whic h this can b e done. Problem 5. Let f b e a function from the set of in tegers to the set of p ositiv e in tegers. Supp ose that, for an y t w o in tegers m and n , the diere n c e f(m)-f(n) is divisible b y f(m-n) . Pro v e that, for all in tegers m and n with f(m)=f(n) , the n um b er f(n) is divisible b y f(m) . Problem 6. Let ABC b e an acute triangle with circumcircle G . Let ` b e a tangen t line t o G , and let `
a
, `
b
and `
c
b e the lines obtained b y reecting ` in the lines BC , CA and AB , resp ectiv ely . Sho w that the circumcircle of t he triangle determined b y the lines `
a
, `
b
and `
c
is tangen t to the circle G .