Difference between revisions of "2019 EGMO Problems"
(Created page with "==Day 1== ===Problem 1=== Find all triples <math>(a, b, c)</math> of real numbers such that <math>ab + bc + ca = 1</math> and <cmath>a^2b + c = b^2c + a = c^2a + b.</cmath>...") |
(→Problem 5) |
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(a) <math>a_i\le b_i</math> for <math>i=1, 2, \cdots , n;</math> | (a) <math>a_i\le b_i</math> for <math>i=1, 2, \cdots , n;</math> | ||
− | (b) | + | (b) the remainders of <math>b_1, b_2, \cdots, b_n</math> on division by <math>n</math> are pairwise different; and |
− | (c) < | + | (c) <math>b_1+b_2+\cdots b_n \le n\left(\frac{n-1}{2}+\left\lfloor \frac{a_1+a_2+\cdots a_n}{n}\right \rfloor \right)</math> |
+ | |||
+ | (Here, <math>\lfloor x \rfloor</math> denotes the integer part of real number <math>x</math>, that is, the largest integer that does not exceed <math>x</math>.) | ||
− | |||
[[2019 EGMO Problems/Problem 5|Solution]] | [[2019 EGMO Problems/Problem 5|Solution]] | ||
===Problem 6=== | ===Problem 6=== | ||
− | On a circle, Alina draws < | + | On a circle, Alina draws <math>2019</math> chords, the endpoints of which are all different. A point is considered marked if it is either |
− | (i) one of the < | + | (i) one of the <math>4038</math> endpoints of a chord; or |
(ii) an intersection point of at least two chords. | (ii) an intersection point of at least two chords. | ||
− | Alina labels each marked point. Of the < | + | Alina labels each marked point. Of the <math>4038</math> points meeting criterion (i), Alina labels <math>2019</math> points with a <math>0</math> and the other <math>2019</math> points with a <math>1</math>. She labels each point meeting criterion (ii) with an arbitrary integer (not necessarily positive). |
− | Along each chord, Alina considers the segments connecting two consecutive marked points. (A chord with < | + | Along each chord, Alina considers the segments connecting two consecutive marked points. (A chord with <math>k</math> marked points has <math>k-1</math> such segments.) She labels each such segment in yellow with the sum of the labels of its two endpoints and in blue with the absolute value of their difference. |
− | Alina finds that the < | + | Alina finds that the <math>N + 1</math> yellow labels take each value <math>0, 1, . . . , N</math> exactly once. Show that at least one blue label is a multiple of <math>3</math>. |
(A chord is a line segment joining two different points on a circle.) | (A chord is a line segment joining two different points on a circle.) | ||
[[2019 EGMO Problems/Problem 6|Solution]] | [[2019 EGMO Problems/Problem 6|Solution]] |
Latest revision as of 14:02, 24 December 2022
Contents
Day 1
Problem 1
Find all triples of real numbers such that
and
Problem 2
Let be a positive integer. Dominoes are placed on a
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each
, determine the largest number of dominoes that can be placed in this way.
(A domino is a tile of size
or
. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)
Problem 3
Let be a triangle such that
, and let
be its incentre. Let
be the point on segment
such that
. Let
be the circle tangent to
at
and passing through
. Let
be the second point of intersection of
and the circumcircle of
. Prove that the angle bisectors of
and
intersect at a point on line
.
Day 2
Problem 4
Let be a triangle with incentre
. The circle through
tangent to
at
meets side
again at
. The circle through
tangent to
at
meets side
again at
. Prove that
is tangent to the incircle of
Problem 5
Let be an integer, and let
be positive integers. Show that there exist positive integers
satisfying the following three conditions:
(a) for
(b) the remainders of on division by
are pairwise different; and
(c)
(Here, denotes the integer part of real number
, that is, the largest integer that does not exceed
.)
Problem 6
On a circle, Alina draws chords, the endpoints of which are all different. A point is considered marked if it is either
(i) one of the endpoints of a chord; or
(ii) an intersection point of at least two chords.
Alina labels each marked point. Of the points meeting criterion (i), Alina labels
points with a
and the other
points with a
. She labels each point meeting criterion (ii) with an arbitrary integer (not necessarily positive).
Along each chord, Alina considers the segments connecting two consecutive marked points. (A chord with
marked points has
such segments.) She labels each such segment in yellow with the sum of the labels of its two endpoints and in blue with the absolute value of their difference.
Alina finds that the
yellow labels take each value
exactly once. Show that at least one blue label is a multiple of
.
(A chord is a line segment joining two different points on a circle.)