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Difference between revisions of "2022 EGMO Problems"

(Created page with "==Day 1== ===Problem 1=== Let <math>ABC</math> be an acute-angled triangle in which <math>BC<AB</math> and <math>BC<CA</math>. Let point <math>P</math> lie on segment <math>AB...")
 
(→‎Problem 6)
 
Line 38: Line 38:
 
===Problem 6===
 
===Problem 6===
 
Let <math>ABCD</math> be a cyclic quadrilateral with circumcenter <math>O</math>. Let the internal angle bisectors at <math>A</math> and <math>B</math> meet at <math>X</math>, the internal angle bisectors at <math>B</math> and <math>C</math> meet at <math>Y</math>, the internal angle bisectors at <math>C</math> and <math>D</math> meet at <math>Z</math>, and the internal angle bisectors at <math>D</math> and <math>A</math> meet at <math>W</math>. Further, let <math>AC</math> and <math>BD</math> meet at <math>P</math>. Suppose that the points <math>X</math>, <math>Y</math>, <math>Z</math>, <math>W</math>, <math>O</math>, and <math>P</math> are distinct.
 
Let <math>ABCD</math> be a cyclic quadrilateral with circumcenter <math>O</math>. Let the internal angle bisectors at <math>A</math> and <math>B</math> meet at <math>X</math>, the internal angle bisectors at <math>B</math> and <math>C</math> meet at <math>Y</math>, the internal angle bisectors at <math>C</math> and <math>D</math> meet at <math>Z</math>, and the internal angle bisectors at <math>D</math> and <math>A</math> meet at <math>W</math>. Further, let <math>AC</math> and <math>BD</math> meet at <math>P</math>. Suppose that the points <math>X</math>, <math>Y</math>, <math>Z</math>, <math>W</math>, <math>O</math>, and <math>P</math> are distinct.
 +
 
Prove that <math>O</math>, <math>X</math>, <math>Y</math> <math>Z</math>, <math>W</math> lie on the same circle if and only if <math>P</math>, <math>X</math>, <math>Y</math>, <math>Z</math>, and <math>W</math> lie on the same circle.
 
Prove that <math>O</math>, <math>X</math>, <math>Y</math> <math>Z</math>, <math>W</math> lie on the same circle if and only if <math>P</math>, <math>X</math>, <math>Y</math>, <math>Z</math>, and <math>W</math> lie on the same circle.
  
 
[[2022 EGMO Problems/Problem 6|Solution]]
 
[[2022 EGMO Problems/Problem 6|Solution]]

Latest revision as of 14:13, 24 December 2022

Day 1

Problem 1

Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and $BQ = BC = CP$. Let $T$ be the circumcenter of triangle $APQ$, $H$ the orthocenter of triangle $ABC$, and $S$ the point of intersection of the lines $BQ$ and $CP$. Prove that $T$, $H$, and $S$ are collinear.

Solution

Problem 2

Let $\mathbb{N}=\{1, 2, 3, \dots\}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$, the following two conditions hold: (i) $f(ab) = f(a)f(b)$, and

(ii) at least two of the numbers $f(a)$, $f(b)$, and $f(a+b)$ are equal.

Solution

Problem 3

An infinite sequence of positive integers $a_1, a_2, \dots$ is called $good$ if

(i) $a_1$ is a perfect square, and

(ii) for any integer $n \ge 2$, $a_n$ is the smallest positive integer such that\[na_1 + (n-1)a_2 + \dots + 2a_{n-1} + a_n\]is a perfect square.

Prove that for any good sequence $a_1, a_2, \dots$, there exists a positive integer $k$ such that $a_n=a_k$ for all integers $n \ge k$.

Solution

Day 2

Problem 4

Given a positive integer $n \ge 2$, determine the largest positive integer $N$ for which there exist $N+1$ real numbers $a_0, a_1, \dots, a_N$ such that $(1) $ $a_0+a_1 = -\frac{1}{n},$ and $(2) $ $(a_k+a_{k-1})(a_k+a_{k+1})=a_{k-1}-a_{k+1}$ for $1 \le k \le N-1$.

Solution

Problem 5

For all positive integers $n$, $k$, let $f(n, 2k)$ be the number of ways an $n \times 2k$ board can be fully covered by $nk$ dominoes of size $2 \times 1$. (For example, $f(2, 2)=2$ and $f(3, 2)=3$.) Find all positive integers $n$ such that for every positive integer $k$, the number $f(n, 2k)$ is odd.

Solution

Problem 6

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. Let the internal angle bisectors at $A$ and $B$ meet at $X$, the internal angle bisectors at $B$ and $C$ meet at $Y$, the internal angle bisectors at $C$ and $D$ meet at $Z$, and the internal angle bisectors at $D$ and $A$ meet at $W$. Further, let $AC$ and $BD$ meet at $P$. Suppose that the points $X$, $Y$, $Z$, $W$, $O$, and $P$ are distinct.

Prove that $O$, $X$, $Y$ $Z$, $W$ lie on the same circle if and only if $P$, $X$, $Y$, $Z$, and $W$ lie on the same circle.

Solution

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