International Mathematics Olympiad Problems - 2020 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

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Monday, 21. Septemb er 2020
Problem 1. Consider the con v ex quadrilateral ABCD . The p oin t P is in the in terior of ABCD .
The follo wing ratio equalities hold:
?PAD :?PBA :?DPA = 1 : 2 : 3 =?CBP :?BAP :?BPC.
Pro v e that the follo wing three lines meet in a p oin t: the in ternal bisectors of angles ?ADP and
?PCB and the p erp endicular bisector of segmen t AB .
Problem 2. The real n um b ers a,b,c,d are suc h that a= b= c= d > 0 and a +b +c +d = 1 .
Pro v e that
(a+2b+3c+4d)a
a
b
b
c
c
d
d
< 1.
Problem 3. There are 4n p ebbles of w eigh ts 1,2,3,...,4n . Eac h p ebble is coloured in one of n
colours and there are four p ebbles of eac h colour. Sho w that w e can arrange the p ebbles in to t w o
piles so that the follo wing t w o conditions are b oth satisfied:
• The total w eigh ts of b oth piles are the same.
• Eac h pile con tains t w o p ebbles of eac h colour.
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Monday, 21. Septemb er 2020
Problem 1. Consider the con v ex quadrilateral ABCD . The p oin t P is in the in terior of ABCD .
The follo wing ratio equalities hold:
?PAD :?PBA :?DPA = 1 : 2 : 3 =?CBP :?BAP :?BPC.
Pro v e that the follo wing three lines meet in a p oin t: the in ternal bisectors of angles ?ADP and
?PCB and the p erp endicular bisector of segmen t AB .
Problem 2. The real n um b ers a,b,c,d are suc h that a= b= c= d > 0 and a +b +c +d = 1 .
Pro v e that
(a+2b+3c+4d)a
a
b
b
c
c
d
d
< 1.
Problem 3. There are 4n p ebbles of w eigh ts 1,2,3,...,4n . Eac h p ebble is coloured in one of n
colours and there are four p ebbles of eac h colour. Sho w that w e can arrange the p ebbles in to t w o
piles so that the follo wing t w o conditions are b oth satisfied:
• The total w eigh ts of b oth piles are the same.
• Eac h pile con tains t w o p ebbles of eac h colour.
T uesday, 22. Septemb er 2020
Problem 4. There is an in teger n > 1 . There are n
2
stations on a slop e of a moun tain, all at
differen t altitudes. Eac h of t w o cable car companies, A and B , op erates k cable cars; eac h cable
car pro vides a transfer from one of the stations to a higher one (with no in termediate stops). The
k cable cars of A ha v e k differen t starting p oin ts and k differen t finishing p oin ts, and a cable car
whic h s tarts higher also finishes higher. The same conditions hold for B . W e sa y that t w o stations
are link ed b y a compan y if one can start from the lo w er station and reac h the higher one b y using
one or more cars of that compan y (no other mo v emen ts b et w een stations are allo w ed).
Determine the smallest p ositiv e in teger k for whic h one can guaran tee that there are t w o stations
that are link ed b y b oth companies.
Problem 5. A dec k of n > 1 cards is giv en. A p ositiv e in teger is written on eac h card. The dec k
has the prop ert y that the arithmetic mean of the n um b ers on eac h pair of cards is also the geometric
mean of the n um b ers on some collec tion of one or more cards.
F or whic h n do es it follo w that the n um b ers on the cards are all equal?
Problem 6. Pro v e that there exists a p ositiv e constan tc suc h that the follo wing statemen t is true:
Consider an in teger n > 1 , and a set S of n p oin ts in the plane suc h that the distance b et w een
an y t w o differen t p oin ts in S is at least 1 . It follo ws that there is a line l separating S suc h that the
distance from an y p oin t of S to l is at least cn
-1/3
.
(A line l separates a set of p oin ts S if some segmen t joining t w o p oin ts in S crosses l .)
Note. W eak er results with cn
-1/3
replaced b y cn
-a
ma y b e a w arded p oin ts dep ending on the v alue
of the constan t a> 1/3 .