Scribd
Upload
282 views2 pages

Egmo 1

The document outlines problems from the 2020 EGMO, including sequences of positive integers, conditions for non-negative real numbers, properties of a convex hexagon, fresh permutations of integers, and geometric properties of triangles. Each problem presents a unique mathematical challenge requiring proofs or determinations. The document serves as a resource for problem solvers in the AoPS community.

Uploaded by

a.ellamrhari
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
282 views2 pages

Egmo 1

The document outlines problems from the 2020 EGMO, including sequences of positive integers, conditions for non-negative real numbers, properties of a convex hexagon, fresh permutations of integers, and geometric properties of triangles. Each problem presents a unique mathematical challenge requiring proofs or determinations. The document serves as a resource for problem solvers in the AoPS community.

Uploaded by

a.ellamrhari
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 2

AoPS Community 2020 EGMO

www.artofproblemsolving.com/community/c1131451
by djmathman, alifenix-

– Day 1

1 The positive integers a0 , a1 , a2 , . . . , a3030 satisfy

2an+2 = an+1 + 4an for n = 0, 1, 2, . . . , 3028.

Prove that at least one of the numbers a0 , a1 , a2 , . . . , a3030 is divisible by 22020 .

2 Find all lists (x1 , x2 , . . . , x2020 ) of non-negative real numbers such that the following three con-
ditions are all satisfied:

- x1 ≤ x2 ≤ . . . ≤ x2020 ;
- x2020 ≤ x1 + 1;
- there is a permutation (y1 , y2 , . . . , y2020 ) of (x1 , x2 , . . . , x2020 ) such that
2020
X 2020
X
((xi + 1)(yi + 1))2 = 8 x3i .
i=1 i=1

[i]A permutation of a list is a list of the same length, with the same entries, but the entries are
allowed to be in any order. For example, (2, 1, 2) is a permutation of (1, 2, 2), and they are both
permutations of (2, 2, 1). Note that any list is a permutation of itself.[/i]

3 Let ABCDEF be a convex hexagon such that ∠A = ∠C = ∠E and ∠B = ∠D = ∠F and the


(interior) angle bisectors of ∠A, ∠C, and ∠E are concurrent.
Prove that the (interior) angle bisectors of ∠B, ∠D, and ∠F must also be concurrent.
[i]Note that ∠A = ∠F AB. The other interior angles of the hexagon are similarly described.[/i]

– Day 2

4 A permutation of the integers 1, 2, . . . , m is called fresh if there exists no positive integer k < m
such that the first k numbers in the permutation are 1, 2, . . . , k in some order. Let fm be the
number of fresh permutations of the integers 1, 2, . . . , m.
Prove that fn ≥ n · fn−1 for all n ≥ 3.

© 2020 AoPS Incorporated 1


AoPS Community 2020 EGMO

[i]For example, if m = 4, then the permutation (3, 1, 4, 2) is fresh, whereas the permutation
(2, 3, 1, 4) is not.[/i]

5 Consider the triangle ABC with ∠BCA > 90◦ . The circumcircle Γ of ABC has radius R. There
is a point P in the interior of the line segment AB such that P B = P C and the length of P A
is R. The perpendicular bisector of P B intersects Γ at the points D and E.
Prove P is the incentre of triangle CDE.

6 Let m > 1 be an integer. A sequence a1 , a2 , a3 , . . . is defined by a1 = a2 = 1, a3 = 4, and for all


n ≥ 4,
an = m(an−1 + an−2 ) − an−3 .

Determine all integers m such that every term of the sequence is a square.

© 2020 AoPS Incorporated 2


Art of Problem Solving is an ACS WASC Accredited School.

You might also like