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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) <math> the remainders of </math>b_1, b_2, \cdots, b_n<math> on division by </math>n<math> are pairwise different; and
+
(b) the remainders of <math>b_1, b_2, \cdots, b_n</math> on division by <math>n</math> are pairwise different; and
  
−
(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>
+
(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>.)
  
−
(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 </math>2019<math> chords, the endpoints of which are all different. A point is considered marked if it is either
+
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 </math>4038<math> endpoints of a chord; or
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(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 </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).
+
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 </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.
+
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 </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$.
+
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 13:02, 24 December 2022

Day 1

Problem 1

Find all triples $(a, b, c)$ of real numbers such that $ab + bc + ca = 1$ and

\[a^2b + c = b^2c + a = c^2a + b.\]

Solution

Problem 2

Let $n$ be a positive integer. Dominoes are placed on a $2n \times 2n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$, determine the largest number of dominoes that can be placed in this way. (A domino is a tile of size $2 \times 1$ or $1 \times 2$. 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.)

Solution

Problem 3

Let $ABC$ be a triangle such that $\angle CAB > \angle ABC$, and let $I$ be its incentre. Let $D$ be the point on segment $BC$ such that $\angle CAD = \angle ABC$. Let $\omega$ be the circle tangent to $AC$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the circumcircle of $ABC$. Prove that the angle bisectors of $\angle DAB$ and $\angle CXB$ intersect at a point on line $BC$.

Solution

Day 2

Problem 4

Let $ABC$ be a triangle with incentre $I$. The circle through $B$ tangent to $AI$ at $I$ meets side $AB$ again at $P$. The circle through $C$ tangent to $AI$ at $I$ meets side $AC$ again at $Q$. Prove that $PQ$ is tangent to the incircle of $ABC.$

Solution

Problem 5

Let $n\ge 2$ be an integer, and let $a_1, a_2, \cdots , a_n$ be positive integers. Show that there exist positive integers $b_1, b_2, \cdots, b_n$ satisfying the following three conditions:

(a) $a_i\le b_i$ for $i=1, 2, \cdots , n;$

(b) the remainders of $b_1, b_2, \cdots, b_n$ on division by $n$ are pairwise different; and

(c) $b_1+b_2+\cdots b_n \le n\left(\frac{n-1}{2}+\left\lfloor \frac{a_1+a_2+\cdots a_n}{n}\right \rfloor \right)$

(Here, $\lfloor x \rfloor$ denotes the integer part of real number $x$, that is, the largest integer that does not exceed $x$.)

Solution

Problem 6

On a circle, Alina draws $2019$ chords, the endpoints of which are all different. A point is considered marked if it is either

(i) one of the $4038$ endpoints of a chord; or

(ii) an intersection point of at least two chords.

Alina labels each marked point. Of the $4038$ points meeting criterion (i), Alina labels $2019$ points with a $0$ and the other $2019$ points with a $1$. 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 $k$ marked points has $k-1$ 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 $N + 1$ yellow labels take each value $0, 1, . . . , N$ exactly once. Show that at least one blue label is a multiple of $3$. (A chord is a line segment joining two different points on a circle.)

Solution

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