Difference between revisions of "2020 CAMO Problems"

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===Problem 1===
 
===Problem 1===
 
Let <math>f:\mathbb R_{>0}\to\mathbb R_{>0}</math> (meaning <math>f</math> takes positive real numbers to positive real numbers) be a nonconstant function such that for any positive real numbers <math>x</math> and <math>y</math>, <cmath>f(x)f(y)f(x+y)=f(x)+f(y)-f(x+y).</cmath>Prove that there is a constant <math>a>1</math> such that <cmath>f(x)=\frac{a^x-1}{a^x+1}</cmath>for all positive real numbers <math>x</math>.
 
Let <math>f:\mathbb R_{>0}\to\mathbb R_{>0}</math> (meaning <math>f</math> takes positive real numbers to positive real numbers) be a nonconstant function such that for any positive real numbers <math>x</math> and <math>y</math>, <cmath>f(x)f(y)f(x+y)=f(x)+f(y)-f(x+y).</cmath>Prove that there is a constant <math>a>1</math> such that <cmath>f(x)=\frac{a^x-1}{a^x+1}</cmath>for all positive real numbers <math>x</math>.
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[[2020 CAMO Problems/Problem 1|Solution]]
  
 
===Problem 2===
 
===Problem 2===
 
Let <math>k</math> be a positive integer, <math>p>3</math> a prime, and <math>n</math> an integer with <math>0\le n\le p^{k-1}</math>. Prove that <cmath>\binom{p^k}{pn}\equiv\binom{p^{k-1}}n\pmod{p^{2k+1}}.</cmath>
 
Let <math>k</math> be a positive integer, <math>p>3</math> a prime, and <math>n</math> an integer with <math>0\le n\le p^{k-1}</math>. Prove that <cmath>\binom{p^k}{pn}\equiv\binom{p^{k-1}}n\pmod{p^{2k+1}}.</cmath>
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[[2020 CAMO Problems/Problem 2|Solution]]
  
 
===Problem 3===
 
===Problem 3===
 
Let <math>ABC</math> be a triangle with incircle <math>\omega</math>, and let <math>\omega</math> touch <math>\overline{BC}</math>, <math>\overline{CA}</math>, <math>\overline{AB}</math> at <math>D</math>, <math>E</math>, <math>F</math>, respectively. Point <math>M</math> is the midpoint of <math>\overline{EF}</math>, and <math>T</math> is the point on <math>\omega</math> such that <math>\overline{DT}</math> is a diameter. Line <math>MT</math> meets the line through <math>A</math> parallel to <math>\overline{BC}</math> at <math>P</math> and <math>\omega</math> again at <math>Q</math>. Lines <math>DF</math> and <math>DE</math> intersect line <math>AP</math> at <math>X</math> and <math>Y</math> respectively. Prove that the circumcircles of <math>\triangle APQ</math> and <math>\triangle DXY</math> are tangent.
 
Let <math>ABC</math> be a triangle with incircle <math>\omega</math>, and let <math>\omega</math> touch <math>\overline{BC}</math>, <math>\overline{CA}</math>, <math>\overline{AB}</math> at <math>D</math>, <math>E</math>, <math>F</math>, respectively. Point <math>M</math> is the midpoint of <math>\overline{EF}</math>, and <math>T</math> is the point on <math>\omega</math> such that <math>\overline{DT}</math> is a diameter. Line <math>MT</math> meets the line through <math>A</math> parallel to <math>\overline{BC}</math> at <math>P</math> and <math>\omega</math> again at <math>Q</math>. Lines <math>DF</math> and <math>DE</math> intersect line <math>AP</math> at <math>X</math> and <math>Y</math> respectively. Prove that the circumcircles of <math>\triangle APQ</math> and <math>\triangle DXY</math> are tangent.
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[[2020 CAMO Problems/Problem 3|Solution]]
  
 
==Day 2==
 
==Day 2==
 
===Problem 4===
 
===Problem 4===
 
Let <math>ABC</math> be a triangle and <math>Q</math> a point on its circumcircle. Let <math>E</math> and <math>F</math> be the reflections of <math>Q</math> over <math>\overline{AB}</math> and <math>\overline{AC}</math>, respectively. Select points <math>X</math> and <math>Y</math> on line <math>EF</math> such that <math>\overline{BX}\parallel\overline{AC}</math> and <math>\overline{CY}\parallel\overline{AB}</math>, and let <math>M</math> and <math>N</math> be the reflections of <math>X</math> and <math>Y</math> over <math>B</math> and <math>C</math> respectively. Prove that <math>M</math>, <math>Q</math>, <math>N</math> are collinear.
 
Let <math>ABC</math> be a triangle and <math>Q</math> a point on its circumcircle. Let <math>E</math> and <math>F</math> be the reflections of <math>Q</math> over <math>\overline{AB}</math> and <math>\overline{AC}</math>, respectively. Select points <math>X</math> and <math>Y</math> on line <math>EF</math> such that <math>\overline{BX}\parallel\overline{AC}</math> and <math>\overline{CY}\parallel\overline{AB}</math>, and let <math>M</math> and <math>N</math> be the reflections of <math>X</math> and <math>Y</math> over <math>B</math> and <math>C</math> respectively. Prove that <math>M</math>, <math>Q</math>, <math>N</math> are collinear.
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[[2020 CAMO Problems/Problem 4|Solution]]
  
 
===Problem 5===
 
===Problem 5===
 
Let <math>f(x)=x^2-2</math>. Prove that for all positive integers <math>n</math>, the polynomial <cmath>P(x)=\underbrace{f(f(\ldots f}_{n\text{ times}}(x)\ldots))-x</cmath>can be factored into two polynomials with integer coefficients and equal degree.
 
Let <math>f(x)=x^2-2</math>. Prove that for all positive integers <math>n</math>, the polynomial <cmath>P(x)=\underbrace{f(f(\ldots f}_{n\text{ times}}(x)\ldots))-x</cmath>can be factored into two polynomials with integer coefficients and equal degree.
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[[2020 CAMO Problems/Problem 5|Solution]]
  
 
===Problem 6===
 
===Problem 6===
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Eric wins if he ever occupies the same square as the squid. Suppose the squid has the first turn, and both Eric and the squid play optimally. Both Eric and the squid always know each other's location and the colors of all the squares. Find all positive integers <math>n</math> such that Eric can win in finitely many moves.
 
Eric wins if he ever occupies the same square as the squid. Suppose the squid has the first turn, and both Eric and the squid play optimally. Both Eric and the squid always know each other's location and the colors of all the squares. Find all positive integers <math>n</math> such that Eric can win in finitely many moves.
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[[2020 CAMO Problems/Problem 6|Solution]]
  
 
==See also==
 
==See also==
 
{{CAMO newbox|year= 2020|before=First CAMO|after=[[2021 CAMO]]}}
 
{{CAMO newbox|year= 2020|before=First CAMO|after=[[2021 CAMO]]}}
 
{{MAC Notice}}
 
{{MAC Notice}}

Latest revision as of 14:09, 5 September 2020

Day 1

Problem 1

Let $f:\mathbb R_{>0}\to\mathbb R_{>0}$ (meaning $f$ takes positive real numbers to positive real numbers) be a nonconstant function such that for any positive real numbers $x$ and $y$, \[f(x)f(y)f(x+y)=f(x)+f(y)-f(x+y).\]Prove that there is a constant $a>1$ such that \[f(x)=\frac{a^x-1}{a^x+1}\]for all positive real numbers $x$.

Solution

Problem 2

Let $k$ be a positive integer, $p>3$ a prime, and $n$ an integer with $0\le n\le p^{k-1}$. Prove that \[\binom{p^k}{pn}\equiv\binom{p^{k-1}}n\pmod{p^{2k+1}}.\]

Solution

Problem 3

Let $ABC$ be a triangle with incircle $\omega$, and let $\omega$ touch $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$, respectively. Point $M$ is the midpoint of $\overline{EF}$, and $T$ is the point on $\omega$ such that $\overline{DT}$ is a diameter. Line $MT$ meets the line through $A$ parallel to $\overline{BC}$ at $P$ and $\omega$ again at $Q$. Lines $DF$ and $DE$ intersect line $AP$ at $X$ and $Y$ respectively. Prove that the circumcircles of $\triangle APQ$ and $\triangle DXY$ are tangent.

Solution

Day 2

Problem 4

Let $ABC$ be a triangle and $Q$ a point on its circumcircle. Let $E$ and $F$ be the reflections of $Q$ over $\overline{AB}$ and $\overline{AC}$, respectively. Select points $X$ and $Y$ on line $EF$ such that $\overline{BX}\parallel\overline{AC}$ and $\overline{CY}\parallel\overline{AB}$, and let $M$ and $N$ be the reflections of $X$ and $Y$ over $B$ and $C$ respectively. Prove that $M$, $Q$, $N$ are collinear.

Solution

Problem 5

Let $f(x)=x^2-2$. Prove that for all positive integers $n$, the polynomial \[P(x)=\underbrace{f(f(\ldots f}_{n\text{ times}}(x)\ldots))-x\]can be factored into two polynomials with integer coefficients and equal degree.

Solution

Problem 6

Let $n$ be a positive integer. Eric and a squid play a turn-based game on an infinite grid of unit squares. Eric's goal is to capture the squid by moving onto the same square as it.

Initially, all the squares are colored white. The squid begins on an arbitrary square in the grid, and Eric chooses a different square to start on. On the squid's turn, it performs the following action exactly $2020$ times: it chooses an adjacent unit square that is white, moves onto it, and sprays the previous unit square either black or gray. Once the squid has performed this action $2020$ times, all squares colored gray are automatically colored white again, and the squid's turn ends. If the squid is ever unable to move, then Eric automatically wins. Moreover, the squid is claustrophobic, so at no point in time is it ever surrounded by a closed loop of black or gray squares. On Eric's turn, he performs the following action at most $n$ times: he chooses an adjacent unit square that is white and moves onto it. Note that the squid can trap Eric in a closed region, so that Eric can never win.

Eric wins if he ever occupies the same square as the squid. Suppose the squid has the first turn, and both Eric and the squid play optimally. Both Eric and the squid always know each other's location and the colors of all the squares. Find all positive integers $n$ such that Eric can win in finitely many moves.

Solution

See also

2020 CAMO (ProblemsResources)
Preceded by
First CAMO
Followed by
2021 CAMO
1 2 3 4 5 6
All CAMO Problems and Solutions

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