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Olympiad Training for Individual Study

Thirty mock IMO tests

IMO 2001 through IMO 2010

Evan Chen《陳誼廷》
5 May 2024

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MOCK-IMO

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By popular request, for each year between 2001 and 2010, inclusive, I created three

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days of “mock IMO” exams. If you have a little extra time (or a lot of it) and want to
work through some additional practice tests, this is for you! Of course this is totally

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optional. It is also not graded.

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In general I tried to mostly pick problems which I had not done before, and to decrease
the chance of using a problem which appears in OTIS. Thus difficulty may be wildly off

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as I mostly went by gut impression and shortlist numbers1 .

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Corrections, or suggestions for changes are welcome!

Contents
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1 Mock IMO 2001 — Washington, United States of America 3

v
1(i) Mock IMO 2001, Day I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

E S
1(ii) Mock IMO 2001, Day II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

I
1(iii) Mock IMO 2001, Day III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

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2 Mock IMO 2002 — Glasgow, United
2(i) Mock IMO 2002, Day I . . . . . .
2(ii) Mock IMO 2002, Day II . . . . .
2(iii) Mock IMO 2002, Day III . . . . .
Kingdom
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
6
6
7
8

3 Mock IMO 2003 — Tokyo, Japan 9

3(i) Mock IMO 2003, Day I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3(ii) Mock IMO 2003, Day II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3(iii) Mock IMO 2003, Day III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Mock IMO 2004 — Athens, Greece 12

4(i) Mock IMO 2004, Day I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4(ii) Mock IMO 2004, Day II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4(iii) Mock IMO 2004, Day III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Mock IMO 2005 — Mérida, Mexico 15

5(i) Mock IMO 2005, Day I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1
Much like the actual IMO jury.

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

5(ii) Mock IMO 2005, Day II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5(iii) Mock IMO 2005, Day III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Mock IMO 2006 — Ljubljana, Slovenia 18

6(i) Mock IMO 2006, Day I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6(ii) Mock IMO 2006, Day II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6(iii) Mock IMO 2006, Day III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7 Mock IMO 2007 — Hanoi, Vietnam 21

7(i) Mock IMO 2007, Day I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
7(ii) Mock IMO 2007, Day II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
7(iii) Mock IMO 2007, Day III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

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8 Mock IMO 2008 — Madrid, Spain 24
8(i) Mock IMO 2008, Day I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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8(ii) Mock IMO 2008, Day II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
8(iii) Mock IMO 2008, Day III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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9 Mock IMO 2009 — Bremen, Germany 27

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9(i) Mock IMO 2009, Day I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
9(ii) Mock IMO 2009, Day II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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9(iii) Mock IMO 2009, Day III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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10 Mock IMO 2010 — Astana, Kazakhstan 30

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10(i) Mock IMO 2010, Day I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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10(ii)Mock IMO 2010, Day II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

C t
10(iii)Mock IMO 2010, Day III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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A Shortlist numbers 33

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§1 Mock IMO 2001 — Washington, United States of America

§1(i) Mock IMO 2001, Day I
1. Let ABC be a triangle with centroid G. Determine, with proof, the position of the
point P in the plane of ABC such that AP ·AG + BP ·BG + CP ·CG is a minimum,
and express this minimum value in terms of the side lengths of ABC.

2. Let a1 = 1111 , a2 = 1212 , a3 = 1313 , and an = |an−1 − an−2 | + |an−2 − an−3 |, n ≥ 4.

Determine a1414 .

3. Is it possible to find 100 positive integers not exceeding 25, 000, such that all

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pairwise sums of them are different?

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§1(ii) Mock IMO 2001, Day II

4. 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. Points B1 , C1 are defined in a similar way for
inscribed squares with two vertices on sides AC and AB, respectively. Prove that
lines AA1 , BB1 , CC1 are concurrent.

5. Let x1 , x2 , . . . , xn be arbitrary real numbers. Prove the inequality

x1 x2 xn √
2 + 2 2 + ··· + 2 2
< n.
1 + x1 1 + x1 + x2 1 + x1 + · · · + xn

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6. For a positive integer n define a sequence of zeros and ones to be balanced if it

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contains n zeros and n ones. Two balanced sequences a and b are neighbors if you
can move one of the 2n symbols of a to another position to form b. For instance,

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when n = 4, the balanced sequences 01101001 and 00110101 are neighbors because
the third (or fourth) zero in the first sequence can be moved to the first or second

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position to form the second sequence. Prove that there is a set S of at most n+1
1 2n
n
balanced sequences such that every balanced sequence is equal to or is a neighbor

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of at least one sequence in S.

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§1(iii) Mock IMO 2001, Day III

7. Define a k-clique to be a set of k people such that every pair of them are acquainted
with each other. At a certain party, every pair of 3-cliques has at least one person
in common, and there are no 5-cliques. Prove that there are two or fewer people at
the party whose departure leaves no 3-clique remaining.

8. Let p ≥ 5 be a prime number. Prove that there exists an integer a with 1 ≤ a ≤ p−2
such that neither ap−1 − 1 nor (a + 1)p−1 − 1 is divisible by p2 .

9. Find all positive integers a1 , a2 , . . . , an such that

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99 a0 a1 an−1
= + + ··· + ,
100 a1 a2 an

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where a0 = 1 and (ak+1 − 1)ak−1 ≥ a2k (ak − 1) for k = 1, 2, . . . , n − 1.

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§2 Mock IMO 2002 — Glasgow, United Kingdom

§2(i) Mock IMO 2002, Day I
1. Let p1 , p2 , . . . , pn be distinct primes greater than 3. Show that 2p1 p2 ···pn + 1 has at
least 4n divisors.

2. Let T be the set of ordered triples (x, y, z), where x, y, z are integers with 0 ≤
x, y, z ≤ 9. Players A and B play the following guessing game. Player A chooses
a triple (x, y, z) in T , and Player B has to discover A’s triple in as few moves as
possible. A move consists of the following: B gives A a triple (a, b, c) in T , and A
replies by giving B the number |x + y − a − b| + |y + z − b − c| + |z + x − c − a|.
Find the minimum number of moves that B needs to be sure of determining A’s

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triple.

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3. Let A be a non-empty set of positive integers. Suppose that there are positive

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integers b1 , . . . bn and c1 , . . . , cn such that

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• for each i the set bi A + ci = {bi a + ci : a ∈ A} is a subset of A, and
• the sets bi A + ci and bj A + cj are disjoint whenever i 6= j

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Prove that

n
1 1

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+ ··· + ≤ 1.
b1 bn

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§2(ii) Mock IMO 2002, Day II

4. Let ABC be a triangle for which there exists an interior point F such that ∠AF B =
∠BF C = ∠CF A. Let the lines BF and CF meet the sides AC and AB at D and
E respectively. Prove that
AB + AC ≥ 4DE.

5. Let P be a cubic polynomial given by P (x) = ax3 + bx2 + cx + d, where a, b, c, d

are integers and a 6= 0. Suppose that xP (x) = yP (y) for infinitely many pairs x, y
of integers with x 6= y. Prove that the equation P (x) = 0 has an integer root.

6. Among a group of 120 people, some pairs are friends. A weak quartet is a set of

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four people containing exactly one pair of friends. What is the maximum possible
number of weak quartets?

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§2(iii) Mock IMO 2002, Day III

7. For n an odd positive integer, the unit squares of an n × n chessboard are coloured
alternately black and white, with the four corners coloured black. A it tromino is
an L-shape formed by three connected unit squares. For which values of n is it
possible to cover all the black squares with non-overlapping trominos? When it is
possible, what is the minimum number of trominos needed?

8. Circles S1 and S2 intersect at points P and Q. Distinct points A1 and B1 (not at

P or Q) are selected on S1 . The lines A1 P and B1 P meet S2 again at A2 and B2
respectively, and the lines A1 B1 and A2 B2 meet at C. Prove that, as A1 and B1
vary, the circumcentres of triangles A1 A2 C all lie on one fixed circle.

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9. Let n be a positive integer that is not a perfect cube. Define real numbers a, b, c by

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√ 1 1

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a= n, b= , c= .
a − bac b − bbc

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Prove that there are infinitely many such integers n with the property that there
exist integers r, s, t, not all zero, such that ra + sb + tc = 0.

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§3 Mock IMO 2003 — Tokyo, Japan

§3(i) Mock IMO 2003, Day I
1. Let aij , for i = 1, 2, 3 and j = 1, 2, 3, be real numbers such that aij is positive for
i = j and negative for i 6= j. Prove the existence of positive real numbers c1 , c2 , c3
such that the numbers

a11 c1 + a12 c2 + a13 c3 , a21 c1 + a22 c2 + a23 c3 , a31 c1 + a32 c2 + a33 c3

are either all negative, all positive, or all zero.

2. Let n ≥ 5 be a given integer. Determine the greatest integer k for which there

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exists a polygon with n vertices (convex or not, with non-self-intersecting boundary)
having k internal right angles.

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3. Let ABC be a triangle with semiperimeter s and inradius r. The semicircles with
diameters BC, CA, AB are drawn on the outside of the triangle ABC. The circle

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tangent to all of these three semicircles has radius t. Prove that

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√ !
s s 3

n
<t≤ + 1− r.
2 2 2

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§3(ii) Mock IMO 2003, Day II

4. Let ABC be a triangle and let P be a point in its interior. Denote by D, E, F the
feet of the perpendiculars from P to the lines BC, CA, AB, respectively. Suppose
that
AP 2 + P D2 = BP 2 + P E 2 = CP 2 + P F 2 .
Denote by IA , IB , IC the excenters of the triangle ABC. Prove that P is the
circumcenter of the triangle IA IB IC .

5. An integer n is said to be good if |n| is not the square of an integer. Determine all
integers m with the following property: m can be represented, in infinitely many
ways, as a sum of three distinct good integers whose product is the square of an

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odd integer.

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6. Every point with integer coordinates in the plane is the center of a disk with radius

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1/1000.
(1) Prove that there exists an equilateral triangle whose vertices lie in different

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discs.

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(2) Prove that every equilateral triangle with vertices in different discs has side-

n
length greater than 96.

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§3(iii) Mock IMO 2003, Day III

7. Let D1 , D2 , . . . , Dn be closed discs in the plane. (A closed disc is the region limited
by a circle, taken jointly with this circle.) Suppose that every point in the plane
is contained in at most 2003 discs Di . Prove that there exists a disc Dk which
intersects at most 7 · 2003 − 1 = 14020 other discs Di .

8. Consider pairs of the sequences of positive real numbers

a1 ≥ a2 ≥ a3 ≥ · · · , b1 ≥ b2 ≥ b3 ≥ · · ·

and the sums

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An = a1 + · · · + an , Bn = b1 + · · · + bn ; n = 1, 2, . . . .

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For any pair define cn = min{ai , bi } and Cn = c1 + · · · + cn , n = 1, 2, . . ..
(1) Does there exist a pair (ai )i≥1 , (bi )i≥1 such that the sequences (An )n≥1 and

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(Bn )n≥1 are unbounded while the sequence (Cn )n≥1 is bounded?

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(2) Does the answer to question (1) change by assuming additionally that bi = 1/i,
i = 1, 2, . . .?

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9. Let p be a prime number and let A be a set of positive integers that satisfies the

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following conditions:

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(i) the set of prime divisors of the elements in A consists of p − 1 elements;

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(ii) for any nonempty subset of A, the product of its elements is not a perfect

n I n
p-th power.

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§4 Mock IMO 2004 — Athens, Greece

§4(i) Mock IMO 2004, Day I
1. The function f from the set N of positive integers into itself is defined by the
equality
n
gcd(k, n),
X
f (n) = n ∈ N.
k=1

(a) Prove that f (mn) = f (m)f (n) for every two relatively prime m, n ∈ N.
(b) Prove that for each a ∈ N the equation f (x) = ax has a solution.
(c) Find all a ∈ N such that the equation f (x) = ax has a unique solution.

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2. Let O be the circumcenter of an acute-angled triangle ABC with ∠B < ∠C. The

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line AO meets the side BC at D. The circumcenters of the triangles ABD and
ACD are E and F , respectively. Extend the sides BA and CA beyond A, and

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choose on the respective extensions points G and H such that AG = AC and
AH = AB. Prove that the quadrilateral EF GH is a rectangle if and only if

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∠ACB − ∠ABC = 60◦ .

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3. For an n × n matrix A, let Xi be the set of entries in row i, and Yj the set of entries

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in column j, 1 ≤ i, j ≤ n. We say that A is golden if X1 , . . . , Xn , Y1 , . . . , Yn are

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distinct sets. Find the least integer n such that there exists a 2004 × 2004 golden

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matrix with entries in the set {1, 2, . . . , n}.

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§4(ii) Mock IMO 2004, Day II

4. There are 10001 students at an university. Some students join together to form
several clubs (a student may belong to different clubs). Some clubs join together to
form several societies (a club may belong to different societies). There are a total
of k societies. Suppose that the following conditions hold:
(i) Each pair of students are in exactly one club.
(ii) For each student and each society, the student is in exactly one club of the
society.
(iii) Each club has an odd number of students. In addition, a club with 2m + 1
students (m is a positive integer) is in exactly m societies.
Find all possible values of k.

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5. Let k be a fixed integer greater than 1, and let m = 4k 2 − 5. Show that there exist
positive integers a and b such that the sequence (xn ) defined by

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has all of its terms relatively prime to m.

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6. Let a1 , a2 , . . . , an be positive real numbers, n > 1. Denote by gn their geometric

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mean, and by A1 , A2 , . . . , An the sequence of arithmetic means defined by

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a1 + a2 + · · · + ak

t
Ak = , k = 1, 2, . . . , n.

n n
k

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Let Gn be the geometric mean of A1 , A2 , . . . , An . Prove the inequality

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r
Gn gn

I
nn + ≤n+1
An Gn

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and establish the cases of equality.

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§4(iii) Mock IMO 2004, Day III

7. Given an integer n > 1, denote by Pn the product of all positive integers x less
than n and such that n divides x2 − 1. For each n > 1, find the remainder of Pn
on division by n.

8. Consider a matrix of size n × n whose entries are real numbers of absolute value
not exceeding 1. The sum of all entries of the matrix is 0. Let n be an even positive
integer. Determine the least number C such that every such matrix necessarily has
a row or a column with the sum of its entries not exceeding C in absolute value.

9. Let P be a convex polygon. Prove that there exists a convex hexagon that is

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contained in P and whose area is at least 34 of the area of the polygon P .

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§5 Mock IMO 2005 — Mérida, Mexico

§5(i) Mock IMO 2005, Day I
1. Four real numbers p, q, r, s satisfy p + q + r + s = 9 and p2 + q 2 + r2 + s2 = 21.
Prove that there exists a permutation (a, b, c, d) of (p, q, r, s) such that ab − cd ≥ 2.

2. Let ABCD be a parallelogram. A variable line g through the vertex A intersects

the rays BC and DC at the points X and Y , respectively. Let K and L be
the A-excenters of the triangles ABX and ADY . Show that the angle ∠KCL is
independent of the line g.

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3. Suppose we have a n-gon. Some n − 3 diagonals are coloured black and some other
n − 3 diagonals are coloured red (a side is not a diagonal), so that no two diagonals

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of the same colour can intersect strictly inside the polygon, although they can
share a vertex. Find the maximum number of intersection points between diagonals

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coloured differently strictly inside the polygon, in terms of n.

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§5(ii) Mock IMO 2005, Day II

4. A house has an even number of lamps distributed among its rooms in such a way
that there are at least three lamps in every room. Each lamp shares a switch with
exactly one other lamp, not necessarily from the same room. Each change in the
switch shared by two lamps changes their states simultaneously. Prove that for
every initial state of the lamps there exists a sequence of changes in some of the
switches at the end of which each room contains lamps which are on as well as
lamps which are off.

5. Denote by d(n) the number of divisors of the positive integer n. A positive integer
n is called highly divisible if d(n) > d(m) for all positive integers m < n. Two
highly divisible integers m and n with m < n are called consecutive if there exists

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no highly divisible integer s satisfying m < s < n.

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(a) Show that there are only finitely many pairs of consecutive highly divisible
integers of the form (a, b) with a | b.

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(b) Show that for every prime number p there exist infinitely many positive highly
divisible integers r such that pr is also highly divisible.

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6. In an acute triangle ABC, let D, E, F be the feet of the perpendiculars from

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the points A, B, C to the lines BC, CA, AB, respectively, and let P , Q, R be

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the feet of the perpendiculars from the points A, B, C to the lines EF , F D, DE,

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respectively.

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Prove that p (ABC) p (P QR) ≥ (p (DEF ))2 , where p (T ) denotes the perimeter of

n n
triangle T .

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§5(iii) Mock IMO 2005, Day III

7. Given a triangle ABC satisfying AC + BC = 3 · AB. The incircle of triangle ABC
has center I and touches the sides BC and CA at the points D and E, respectively.
Let K and L be the reflections of the points D and E with respect to I. Prove
that the points A, B, K, L lie on one circle.

8. Find all functions f : R → R such that f (x + y) + f (x)f (y) = f (xy) + 2xy + 1 for
all real numbers x and y..

9. Let P (x) = an xn + an−1 xn−1 + . . . + a0 , where a0 , . . . , an are integers, an > 0,

n ≥ 2. Prove that there exists a positive integer m such that P (m!) is a composite

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number.

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§6 Mock IMO 2006 — Ljubljana, Slovenia

§6(i) Mock IMO 2006, Day I
1. Let ABCD be a trapezoid with parallel sides AB > CD. Points K and L lie on
the line segments AB and CD, respectively, so that AK/KB = DL/LC. Suppose
that there are points P and Q on the line segment KL satisfying

∠AP B = ∠BCD and ∠CQD = ∠ABC.

Prove that the points P , Q, B and C are concyclic.

2. We define a sequence (a1 , a2 , a3 , . . .) by

》
1 j n k j n k j n k
an = + + ··· + ,

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n 1 2 n
where bxc denotes the integer part of x.

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(a) Prove that an+1 > an infinitely often.

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(b) Prove that an+1 < an infinitely often.

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3. Consider a convex polyhedron without parallel edges and without an edge parallel

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to any face other than the two faces adjacent to it. Call a pair of points of the

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polyhedron antipodal if there exist two parallel planes passing through these points

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and such that the polyhedron is contained between these planes. Let A be the

t
number of antipodal pairs of vertices, and let B be the number of antipodal pairs

n n
of midpoint edges. Determine the difference A − B in terms of the numbers of

a , I
vertices, edges, and faces.

Ev I S
y
B O T

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§6(ii) Mock IMO 2006, Day II

4. For x ∈ (0, 1) let y ∈ (0, 1) be the number whose n-th digit after the decimal point
is the 2n -th digit after the decimal point of x. Show that if x is rational then so is
y.

5. Prove the inequality:

X ai aj n X
≤ · ai aj
ai + aj 2(a1 + a2 + · · · + an )
i<j i<j

for positive reals a1 , a2 , . . . , an .

》
6. Let ABCD be a convex quadrilateral. A circle passing through the points A and

廷
D and a circle passing through the points B and C are externally tangent at a
point P inside the quadrilateral. Suppose that

誼 se
∠P AB + ∠P DC ≤ 90◦ and ∠P BA + ∠P CD ≤ 90◦ .

陳 U
Prove that AB + CD ≥ BC + AD.

n《 al
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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§6(iii) Mock IMO 2006, Day III

7. The sequence of real numbers a0 , a1 , a2 , . . . is defined recursively by
n
an−k
for n ≥ 1.
X
a0 = −1, =0
k+1
k=0

Show that an > 0 for all n ≥ 1.

8. A cake has the form of an n × n square composed of n2 unit squares. Strawberries

lie on some of the unit squares so that each row or column contains exactly one
strawberry; call this arrangement A.

》
Let B be another such arrangement. Suppose that every grid rectangle with
one vertex at the top left corner of the cake contains no fewer strawberries of

廷
arrangement B than of arrangement A. Prove that arrangement B can be obtained
from A by performing a number of switches, defined as follows:

誼 se
A switch consists in selecting a grid rectangle with only two strawberries, situated
at its top right corner and bottom left corner, and moving these two strawberries

陳 U
to the other two corners of that rectangle.

n《 al
9. A holey triangle is an upward equilateral triangle of side length n with n upward

e rn
unit triangular holes cut out. A diamond is a 60◦ − 120◦ unit rhombus.

h
Prove that a holey triangle T can be tiled with diamonds if and only if the following

C t e
condition holds: Every upward equilateral triangle of side length k in T contains at

n n
most k holes, for 1 ≤ k ≤ n.

va , I
E I S
y
B O T

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§7 Mock IMO 2007 — Hanoi, Vietnam

§7(i) Mock IMO 2007, Day I
1. Find all pairs of natural numbers (a, b) such that 7a − 3b divides a4 + b2 .

2. In the Cartesian coordinate plane define the strips Sn = {(x, y)|n ≤ x < n + 1} for
every integer n. Assume each strip Sn is colored either red or blue, and let a and
b be two distinct positive integers. Prove that there exists a rectangle with side
length a and b such that its vertices have the same color.

3. Let a1 , a2 , . . . , a100 be nonnegative real numbers such that a21 + a22 + . . . + a2100 = 1.

》
Prove that
12
a21 · a2 + a22 · a3 + . . . + a2100 · a1 < .

廷
25

誼 se
陳 U
n《 al
h e rn
C nt e
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a , I
Ev I S
y
B O T

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§7(ii) Mock IMO 2007, Day II

4. Denote by M midpoint of side BC in an isosceles triangle 4ABC with AC = AB.
Take a point X on a smaller arc M
‘ A of circumcircle of triangle 4ABM . Denote
by T point inside of angle BM A such that ∠T M X = 90 and T X = BX.
Prove that ∠M T B − ∠CT M does not depend on choice of X.

5. Let n be a positive integer, and let x and y be a positive real number such that
xn + y n = 1. Prove that
n n
! !
X 1 + x2k X 1 + y 2k 1
4k
· 4k
< .
1+x 1+y (1 − x) · (1 − y)

》
k=1 k=1

廷
6. Let P be a convex polygon with n vertices. A triangle whose vertices lie on vertices
of P is called good if all its sides are unit length. Prove that there are at most 2n

誼 se
3
good triangles.

陳 U
n《 al
h e rn
C nt e
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a , I
Ev I S
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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§7(iii) Mock IMO 2007, Day III

7. Consider those functions f : N 7→ N which satisfy the condition

f (m + n) ≥ f (m) + f (f (n)) − 1

for all m, n ∈ N. Find all possible values of f (2007).

8. Find all surjective functions f : N → N such that for every m, n ∈ N and every
prime p, the number f (m + n) is divisible by p if and only if f (m) + f (n) is divisible
by p.

》
9. Point P lies on side AB of a convex quadrilateral ABCD. Let ω be the incircle of
triangle CP D, and let I be its incenter. Suppose that ω is tangent to the incircles

廷
of triangles AP D and BP C at points K and L, respectively. Let lines AC and
BD meet at E, and let lines AK and BL meet at F . Prove that points E, I, and

誼 se
F are collinear.

陳 U
n《 al
h e rn
C nt e
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a , I
Ev I S
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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§8 Mock IMO 2008 — Madrid, Spain

§8(i) Mock IMO 2008, Day I
1. Let S ⊆ R be a set of real numbers. We say that a pair (f, g) of functions from S
into S is a Spanish Couple on S, if they satisfy the following conditions:
(i) Both functions are strictly increasing, i.e. f (x) < f (y) and g(x) < g(y) for all
x, y ∈ S with x < y;
(ii) The inequality f (g (g (x))) < g (f (x)) holds for all x ∈ S.
Decide whether there exists a Spanish Couple on the set S = N of positive integers,
and whether there exists a Spanish Couple on the set S = {a − 1b : a, b ∈ N}

》
2. Let S = {x1 , x2 , . . . , xk+l } be a (k + l)-element set of real numbers contained in
the interval [0, 1]; k and l are positive integers. A k-element subset A ⊂ S is called

廷
nice if

誼 se
1 X 1 X k+l
xi − xj ≤
k l 2kl

陳 U
xi ∈A xj ∈S\A
 
2 k+l
Prove that the number of nice subsets is at least .

《 al
k+l k

n
e rn
h
3. Let n be a positive integer. Show that the numbers

C t e
 n   n   n   n 
2 −1 2 −1 2 −1 2 −1
, , , ...,

n n
0 1 2 2n−1 − 1

a , I
are congruent modulo 2n to 1, 3, 5, . . ., 2n − 1 in some order.

Ev I S
y
B O T

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§8(ii) Mock IMO 2008, Day II

4. In the coordinate plane consider the set S of all points with integer coordinates.
For a positive integer k, two distinct points a, B ∈ S will be called k-friends if
there is a point C ∈ S such that the area of the triangle ABC is equal to k. A set
T ⊂ S will be called k-clique if every two points in T are k-friends. Find the least
positive integer k for which there exits a k-clique with more than 200 elements.

5. Let k and n be integers with 0 ≤ k ≤ n − 2. Consider a set L of n lines in the

plane such that no two of them are parallel and no three have a common point.
Denote by I the set of intersections of lines in L. Let O be a point in the plane not
lying on any line of L. A point X ∈ I is colored red if the open line segment OX
intersects at most k lines in L. Prove that I contains at least 12 (k + 1)(k + 2) red

》
points.

廷
6. Let f : R → N be a function which satisfies

誼 se
   

陳 U
1 1
f x+ =f y+
f (y) f (x)

《 al
for all x, y ∈ R. Prove that there is a positive integer which is not a value of f .

n
e rn
C h e
n I nt
va ,
E I S
y
B O T

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§8(iii) Mock IMO 2008, Day III

7. Given trapezoid ABCD with parallel sides AB and CD, assume that there exist
points E on line BC outside segment BC, and F inside segment AD such that
∠DAE = ∠CBF . Denote by I the point of intersection of CD and EF , and by J
the point of intersection of AB and EF . Let K be the midpoint of segment EF ,
assume it does not lie on line AB. Prove that I belongs to the circumcircle of
ABK if and only if K belongs to the circumcircle of CDJ.

8. For an integer m, denote by t(m) the unique number in {1, 2, 3} such that m + t(m)
is a multiple of 3. A function f : Z → Z satisfies f (−1) = 0, f (0) = 1, f (1) = −1
and f (2n + m) = f (2n − t(m)) − f (m) for all integers m, n ≥ 0 with 2n > m.
Prove that f (3p) ≥ 0 holds for all integers p ≥ 0.

》
廷
9. For every n ∈ N let d(n) denote the number of (positive) divisors of n. Find all
functions f : N → N with the following properties:

誼 se
• d (f (x)) = x for all x ∈ N.

陳 U
• f (xy) divides (x − 1)y xy−1 f (x) for all x, y ∈ N.

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§9 Mock IMO 2009 — Bremen, Germany

§9(i) Mock IMO 2009, Day I
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, one is red and one is white. Now, for every
colour separately, let us sort the lengths of the sides. We obtain

b1 ≤ b2 ≤ . . . ≤ b2009 the lengths of the blue sides

r1 ≤ r2 ≤ . . . ≤ r2009 the lengths of the red sides
and w1 ≤ w2 ≤ . . . ≤ w2009 the lengths of the white sides

Then there exist k indices j such that we can form a non-degenerated triangle with

》
side lengths bj , rj , wj .

廷
2. On a 999 × 999 board a limp rook can move in the following way: From any square

誼 se
it can move to any of its adjacent squares, i.e. a square having a common side with
it, and every move must be a turn, i.e. the directions of any two consecutive moves

陳 U
must be perpendicular. A non-intersecting route of the limp rook consists of a
sequence of pairwise different squares that the limp rook can visit in that order

《 al
by an admissible sequence of moves. Such a non-intersecting route is called cyclic,

n
if the limp rook can, after reaching the last square of the route, move directly to

e rn
the first square of the route and start over. How many squares does the longest

h
possible cyclic, non-intersecting route of a limp rook visit?

C nt e
n
3. Let P be a polygon that is convex and symmetric to some point O. Prove that for

a , I
some parallelogram R satisfying P ⊂ R we have

Ev S
|R| √

I
≤ 2
|P |

y T
where |R| and |P | denote the area of the sets R and P , respectively.

B O

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§9(ii) Mock IMO 2009, Day II

4. Consider 2009 cards, each having one gold side and one black side, lying on parallel
on a long table. Initially all cards show their gold sides. Two player, standing
by the same long side of the table, play a game with alternating moves. Each
move consists of choosing a block of 50 consecutive cards, the leftmost of which
is showing gold, and turning them all over, so those which showed gold now show
black and vice versa. The last player who can make a legal move wins.
(a) Does the game necessarily end?
(b) Does there exist a winning strategy for the starting player?

5. For an integer m ≥ 1, we consider partitions of a 2m × 2m chessboard into rectangles

》
consisting of cells of chessboard, in which each of the 2m cells along one diagonal
forms a separate rectangle of side length 1. Determine the smallest possible sum of

廷
rectangle perimeters in such a partition.

誼 se
6. Let ABC be a triangle with incenter I and let X, Y and Z be the incenters of the

陳 U
triangles BIC, CIA and AIB, respectively. Let the triangle XY Z be equilateral.
Prove that ABC is equilateral too.

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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§9(iii) Mock IMO 2009, Day III

7. Let a, b, c be positive real numbers such that ab + bc + ca ≤ 3abc. Prove that
r r r
a2 + b2 b2 + c2 c2 + a2 √ √ √ √ 
+ + +3≤ 2 a+b+ b+c+ c+a
a+b b+c c+a

8. Find all positive integers n such that there exists a sequence of positive integers a1 ,
a2 ,. . ., an satisfying:
a2 + 1
ak+1 = k −1
ak−1 + 1
for every k with 2 ≤ k ≤ n − 1.

》
9. For any integer n ≥ 2, we compute the integer h(n) by applying the following

廷
procedure to its decimal representation. Let r be the rightmost digit of n.

誼 se
• If r = 0, then the decimal representation of h(n) results from the decimal
representation of n by removing this rightmost digit 0.

陳 U
• If 1 ≤ r ≤ 9 we split the decimal representation of n into a maximal right
part R that solely consists of digits not less than r and into a left part L

《 al
that either is empty or ends with a digit strictly smaller than r. Then the

n
decimal representation of h(n) consists of the decimal representation of L,

e rn
followed by two copies of the decimal representation of R − 1. For instance,

h
for the number 17, 151, 345, 543, we will have L = 17, 151, R = 345, 543 and

C t e
h(n) = 17, 151, 345, 542, 345, 542.

n n
Prove that, starting with an arbitrary integer n ≥ 2, iterated application of h

a , I
produces the integer 1 after finitely many steps.

Ev I S
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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§10 Mock IMO 2010 — Astana, Kazakhstan

§10(i) Mock IMO 2010, Day I
1. Let ABC be an acute triangle with D, E, F the feet of the altitudes lying on
BC, CA, AB respectively. One of the intersection points of the line EF and the
circumcircle is P. The lines BP and DF meet at point Q. Prove that AP = AQ.

2. Let a, b be integers, and let P (x) = ax3 + bx. For any positive integer n we say that
the pair (a, b) is n-good if n|P (m) − P (k) implies n|m − k for all integers m, k. We
say that (a, b) is very good if (a, b) is n-good for infinitely many positive integers n.
(a) Find a pair (a, b) which is 51-good, but not very good.

》
(b) Show that all 2010-good pairs are very good.

廷
3. Given a positive integer k and other two integers b > w > 1. There are two strings

誼 se
of pearls, a string of b black pearls and a string of w white pearls. The length of a
string is the number of pearls on it. One cuts these strings in some steps by the

陳 U
following rules. In each step:
(i) The strings are ordered by their lengths in a non-increasing order. If there

《 al
are some strings of equal lengths, then the white ones precede the black ones.

n
Then k first ones (if they consist of more than one pearl) are chosen; if there

e rn
are less than k strings longer than 1, then one chooses all of them.

h e
(ii) Next, one cuts each chosen string into two parts differing in length by at most

C t
one. (For instance, if there are strings of 5, 4, 4, 2 black pearls, strings of 8, 4, 3

n n
white pearls and k = 4, then the strings of 8 white, 5 black, 4 white and 4

a , I
black pearls are cut into the parts (4, 4), (3, 2), (2, 2) and (2, 2) respectively.)

v
The process stops immediately after the step when a first isolated white pearl

E S
appears.

I
Prove that at this stage, there will still exist a string of at least two black pearls.

y
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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§10(ii) Mock IMO 2010, Day II

4. Find the least positive integer n for which there exists a set {s1 , s2 , . . . , sn } consisting
of n distinct positive integers such that
    
1 1 1 51
1− 1− ··· 1 − = .
s1 s2 sn 2010

5. Let A1 A2 . . . An be a convex polygon. Point P inside this polygon is chosen so that

its projections P1 , . . . , Pn onto lines A1 A2 , . . . , An A1 respectively lie on the sides of
the polygon. Prove that for arbitrary points X1 , . . . , Xn on sides A1 A2 , . . . , An A1
respectively,  
X1 X2 Xn X1
max

》
,..., ≥ 1.
P1 P2 Pn P1

廷
6. Let P1 , . . . , Ps be arithmetic progressions of integers, such that (i) each integer

誼 se
belongs to at least one of them; (ii) each progression contains a number which does
not belong to other progressions. Denote by n the least common multiple of the

陳 U
ratios of these progressions; let n = pα1 1 · · · pαk k its prime factorization. Prove that

《 al
k

n
X
s≥1+ αi (pi − 1).

e rn
i=1

C h e
n I nt
va ,
E I S
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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§10(iii) Mock IMO 2010, Day III

7. 2500 chess kings have to be placed on a 100 × 100 chessboard so that (i) no king
can capture any other one (i.e. no two kings are placed in two squares sharing a
common vertex); (ii) each row and each column contains exactly 25 kings. Find
the number of such arrangements. (Two arrangements differing by rotation or
symmetry are supposed to be different.)

8. A sequence x1 , x2 , . . . is defined by x1 = 1 and x2k = −xk , x2k−1 = (−1)k+1 xk for

all k ≥ 1. Prove that ∀n ≥ 1 x1 + x2 + . . . + xn ≥ 0.

9. The vertices X, Y, Z of an equilateral triangle XY Z lie respectively on the sides

》
BC, CA, AB of an acute-angled triangle ABC. Prove that the incenter of triangle
ABC lies inside triangle XY Z.

廷
誼 se
陳 U
n《 al
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 Evan Chen《陳誼廷》 (OTIS, updated 2024-05-05) Thirty mock IMO tests

§A Shortlist numbers
I P1 P2 P3 II P4 P5 P6 III P7 P8 P9
2001 G3 N3 N6 G1 A3 C6 C3 N4 A5
2002 N3 C4 A6 G2 A3 C7 C2 G4 A5
2003 A1 C3 G7 G3 N5 C5 C2 A3 N8
2004 N2 G3 C6 C1 N4 A7 N6 C4 G6
2005 A3 G3 C8 C1 N5 G7 G1 A4 N7
2006 G2 N3 C7 N2 A4 G8 A2 C4 C6
2007 N1 C5 A6 G2 A3 C8 A2 N5 G8
2008 A3 C5 N4 C3 G5 A6 G2 A4 N5
2009 A1 C6 G5 C1 C4 G7 A4 N4 C8

》
2010 G1 N4 C6 N1 G3 C7 C3 A4 G6

廷
誼 se
陳 U
n《 al
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C nt e
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a , I
Ev I S
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33