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Difference between revisions of "2019 ELMO Problems"

(→‎Problem 3)
 
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Let <math>P(x)</math> be a polynomial with integer coefficients such that <math>P(0)=1</math>, and let <math>c > 1</math> be an integer. Define <math>x_0=0</math> and <math>x_{i+1} = P(x_i)</math> for all integers <math>i \ge 0</math>. Show that there are infinitely many positive integers <math>n</math> such that <math>\gcd (x_n, n+c)=1</math>.
 
Let <math>P(x)</math> be a polynomial with integer coefficients such that <math>P(0)=1</math>, and let <math>c > 1</math> be an integer. Define <math>x_0=0</math> and <math>x_{i+1} = P(x_i)</math> for all integers <math>i \ge 0</math>. Show that there are infinitely many positive integers <math>n</math> such that <math>\gcd (x_n, n+c)=1</math>.
  
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[[2018 USAJMO Problems/Problem 1|Solution]]
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[[2019 ELMO Problems/Problem 1|Solution]]
  
 
===Problem 2===
 
===Problem 2===
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*Here, the centroid of <math>n</math> points with coordinates <math>(x_1, y_1), \dots, (x_n, y_n)</math> is the point with coordinates <math>\left( \frac{x_1 + \dots + x_n}{n}, \frac{y_1 + \dots + y_n}{n}\right)</math>.
 
*Here, the centroid of <math>n</math> points with coordinates <math>(x_1, y_1), \dots, (x_n, y_n)</math> is the point with coordinates <math>\left( \frac{x_1 + \dots + x_n}{n}, \frac{y_1 + \dots + y_n}{n}\right)</math>.
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[[2018 USAJMO Problems/Problem 2|Solution]]
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[[2019 ELMO Problems/Problem 2|Solution]]
  
 
===Problem 3===
 
===Problem 3===
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Let <math>T_r</math> denote the configuration after <math>r</math> turns (so <math>T_0</math> is the initial configuration). Show that <math>T_r</math> is eventually periodic with period <math>n</math>, and find the smallest integer <math>m</math> for which, regardless of the initial configuration, <math>T_m=T_{m+n}</math>.
 
Let <math>T_r</math> denote the configuration after <math>r</math> turns (so <math>T_0</math> is the initial configuration). Show that <math>T_r</math> is eventually periodic with period <math>n</math>, and find the smallest integer <math>m</math> for which, regardless of the initial configuration, <math>T_m=T_{m+n}</math>.
 +
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[[2019 ELMO Problems/Problem 3|Solution]]
  
 
==Day 2==
 
==Day 2==
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Carl is given three distinct non-parallel lines <math>\ell_1, \ell_2, \ell_3</math> and a circle <math>\omega</math> in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line <math>\ell</math> and a point <math>P</math>, constructs a new line passing through <math>P</math> parallel to <math>\ell</math>. (Carl does not have a compass.) Show that Carl can construct a triangle with circumcircle <math>\omega</math> whose sides are parallel to <math>\ell_1,\ell_2,\ell_3</math> in some order.
 
Carl is given three distinct non-parallel lines <math>\ell_1, \ell_2, \ell_3</math> and a circle <math>\omega</math> in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line <math>\ell</math> and a point <math>P</math>, constructs a new line passing through <math>P</math> parallel to <math>\ell</math>. (Carl does not have a compass.) Show that Carl can construct a triangle with circumcircle <math>\omega</math> whose sides are parallel to <math>\ell_1,\ell_2,\ell_3</math> in some order.
  
−
[[2018 USAJMO Problems/Problem 4|Solution]]
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[[2019 ELMO Problems/Problem 4|Solution]]
  
 
===Problem 5===
 
===Problem 5===
 
Let <math>S</math> be a nonempty set of positive integers such that, for any (not necessarily distinct) integers <math>a</math> and <math>b</math> in <math>S</math>, the number <math>ab+1</math> is also in <math>S</math>. Show that the set of primes that do not divide any element of <math>S</math> is finite.
 
Let <math>S</math> be a nonempty set of positive integers such that, for any (not necessarily distinct) integers <math>a</math> and <math>b</math> in <math>S</math>, the number <math>ab+1</math> is also in <math>S</math>. Show that the set of primes that do not divide any element of <math>S</math> is finite.
  
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[[2018 USAJMO Problems/Problem 5|Solution]]
+
[[2019 ELMO Problems/Problem 5|Solution]]
  
 
===Problem 6===
 
===Problem 6===
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*These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer <math>i</math>, the variable <math>x_i</math> is a functional expression, and (iii) if <math>V</math> and <math>W</math> are functional expressions, then so are <math>f(V)</math>, <math>V+W</math>, <math>V-W</math>, and <math>V \cdot W</math>.
 
*These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer <math>i</math>, the variable <math>x_i</math> is a functional expression, and (iii) if <math>V</math> and <math>W</math> are functional expressions, then so are <math>f(V)</math>, <math>V+W</math>, <math>V-W</math>, and <math>V \cdot W</math>.
  
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[[2018 USAJMO Problems/Problem 6|Solution]]
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[[2019 ELMO Problems/Problem 6|Solution]]

Latest revision as of 08:50, 23 February 2023

Day 1

Problem 1

Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$.

Solution

Problem 2

Let $m, n \ge 2$ be integers. Carl is given $n$ marked points in the plane and wishes to mark their centroid.* He has no standard compass or straightedge. Instead, he has a device which, given marked points $A$ and $B$, marks the $m-1$ points that divide segment $\overline{AB}$ into $m$ congruent parts (but does not draw the segment).

For which pairs $(m,n)$ can Carl necessarily accomplish his task, regardless of which $n$ points he is given?

  • Here, the centroid of $n$ points with coordinates $(x_1, y_1), \dots, (x_n, y_n)$ is the point with coordinates $\left( \frac{x_1 + \dots + x_n}{n}, \frac{y_1 + \dots + y_n}{n}\right)$.

Solution

Problem 3

Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card.

Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$.

Solution

Day 2

Problem 4

Carl is given three distinct non-parallel lines $\ell_1, \ell_2, \ell_3$ and a circle $\omega$ in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line $\ell$ and a point $P$, constructs a new line passing through $P$ parallel to $\ell$. (Carl does not have a compass.) Show that Carl can construct a triangle with circumcircle $\omega$ whose sides are parallel to $\ell_1,\ell_2,\ell_3$ in some order.

Solution

Problem 5

Let $S$ be a nonempty set of positive integers such that, for any (not necessarily distinct) integers $a$ and $b$ in $S$, the number $ab+1$ is also in $S$. Show that the set of primes that do not divide any element of $S$ is finite.

Solution

Problem 6

Carl chooses a functional expression* $E$ which is a finite nonempty string formed from a set $x_1, x_2, \dots$ of variables and applications of a function $f$, together with addition, subtraction, multiplication (but not division), and fixed real constants. He then considers the equation $E = 0$, and lets $S$ denote the set of functions $f \colon \mathbb R \to \mathbb R$ such that the equation holds for any choices of real numbers $x_1, x_2, \dots$. (For example, if Carl chooses the functional equation

\[f(2f(x_1)+x_2) - 2f(x_1)-x_2 = 0,\] then $S$ consists of one function, the identity function.

(a) Let $X$ denote the set of functions with domain $\mathbb R$ and image exactly $\mathbb Z$. Show that Carl can choose his functional equation such that $S$ is nonempty but $S \subseteq X$.

(b) Can Carl choose his functional equation such that $|S|=1$ and $S \subseteq X$?

  • These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer $i$, the variable $x_i$ is a functional expression, and (iii) if $V$ and $W$ are functional expressions, then so are $f(V)$, $V+W$, $V-W$, and $V \cdot W$.

Solution

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