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Difference between revisions of "2020 INMO Problems"

(Created page with "{{INMO Problems|year=2020|n=I}} == Problem 1 == Let <math>\Gamma_1</math> and <math>\Gamma_2</math> be two circles of unequal radii, with centres <math>O_1</math> and <math>O...")
 
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{{INMO Problems|year=2020|n=I}}
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== Problem 1 ==
 
== Problem 1 ==
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<math>\emph{Proposed by Prithwijit De}</math>
 
<math>\emph{Proposed by Prithwijit De}</math>
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[[2020 INMO Problems/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
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<math>\emph{Proposed by C.R. Pranesacher}</math>
 
<math>\emph{Proposed by C.R. Pranesacher}</math>
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[[2020 INMO Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
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Let <math>S</math> be a subset of <math>\{0,1,2,\cdots ,9\}</math>. Suppose there is a positive integer <math>N</math> such that for any integer <math>n>N</math>, one can find positive integers <math>a,b</math> so that <math>n=a+b</math> and all the digits in the decimal representations of <math>a,b</math> (expressed without leading zeros) are in <math>S</math>. Find the smallest possible value of <math>|S|</math>.
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Let <math>S</math> be a subset of <math>\{0,1,2,\cdots ,9\}</math>. Suppose there is a positive integer <math>N</math> such that for any integer <math>n>N</math>, one can find positive integers <math>a,b</math> so that <math>n=a+b</math> and all the digits in the decimal(Base 10) representations of <math>a,b</math> (expressed without leading zeros) are in <math>S</math>. Find the smallest possible value of <math>|S|</math>.
  
 
<math>\emph{Proposed by Sutanay Bhattacharya}</math>
 
<math>\emph{Proposed by Sutanay Bhattacharya}</math>
  
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[[2020 INMO Problems/Problem 3|Solution]]
  
  
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Let <math>n \geqslant 2</math> be an integer and let <math>1<a_1 \le a_2 \le \dots \le a_n</math> be <math>n</math> real numbers such that <math>a_1+a_2+\dots+a_n=2n</math>. Prove that<cmath>a_1a_2\dots a_{n-1}+a_1a_2\dots a_{n-2}+\dots+a_1a_2+a_1+2 \leqslant a_1a_2\dots a_n.</cmath>
 
Let <math>n \geqslant 2</math> be an integer and let <math>1<a_1 \le a_2 \le \dots \le a_n</math> be <math>n</math> real numbers such that <math>a_1+a_2+\dots+a_n=2n</math>. Prove that<cmath>a_1a_2\dots a_{n-1}+a_1a_2\dots a_{n-2}+\dots+a_1a_2+a_1+2 \leqslant a_1a_2\dots a_n.</cmath>
  
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[[2020 INMO Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
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(c) Determine whether <math>5</math> is frameable.
 
(c) Determine whether <math>5</math> is frameable.
  
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[[2020 INMO Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
 
A stromino is a <math>3 \times 1</math> rectangle. Show that a <math>5 \times 5</math> board divided into twenty-five <math>1 \times 1</math> squares cannot be covered by <math>16</math> strominos such that each stromino covers exactly three squares of the board, and every square is covered by one or two strominos. (A stromino can be placed either horizontally or vertically on the board.)
 
A stromino is a <math>3 \times 1</math> rectangle. Show that a <math>5 \times 5</math> board divided into twenty-five <math>1 \times 1</math> squares cannot be covered by <math>16</math> strominos such that each stromino covers exactly three squares of the board, and every square is covered by one or two strominos. (A stromino can be placed either horizontally or vertically on the board.)
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[[2020 INMO Problems/Problem 6|Solution]]

Latest revision as of 04:43, 25 September 2023


Problem 1

Let $\Gamma_1$ and $\Gamma_2$ be two circles of unequal radii, with centres $O_1$ and $O_2$ respectively, intersecting in two distinct points $A$ and $B$. Assume that the centre of each circle is outside the other circle. The tangent to $\Gamma_1$ at $B$ intersects $\Gamma_2$ again in $C$, different from $B$; the tangent to $\Gamma_2$ at $B$ intersects $\Gamma_1$ again at $D$, different from $B$. The bisectors of $\angle DAB$ and $\angle CAB$ meet $\Gamma_1$ and $\Gamma_2$ again in $X$ and $Y$, respectively. Let $P$ and $Q$ be the circumcentres of triangles $ACD$ and $XAY$, respectively. Prove that $PQ$ is the perpendicular bisector of the line segment $O_1O_2$.

$\emph{Proposed by Prithwijit De}$

Solution

Problem 2

Suppose $P(x)$ is a polynomial with real coefficients, satisfying the condition $P(\cos \theta+\sin \theta)=P(\cos \theta-\sin \theta)$, for every real $\theta$. Prove that $P(x)$ can be expressed in the form\[P(x)=a_0+a_1(1-x^2)^2+a_2(1-x^2)^4+\dots+a_n(1-x^2)^{2n}\]for some real numbers $a_0, a_1, \dots, a_n$ and non-negative integer $n$.

$\emph{Proposed by C.R. Pranesacher}$

Solution

Problem 3

Let $S$ be a subset of $\{0,1,2,\cdots ,9\}$. Suppose there is a positive integer $N$ such that for any integer $n>N$, one can find positive integers $a,b$ so that $n=a+b$ and all the digits in the decimal(Base 10) representations of $a,b$ (expressed without leading zeros) are in $S$. Find the smallest possible value of $|S|$.

$\emph{Proposed by Sutanay Bhattacharya}$

Solution


Problem 4

Let $n \geqslant 2$ be an integer and let $1<a_1 \le a_2 \le \dots \le a_n$ be $n$ real numbers such that $a_1+a_2+\dots+a_n=2n$. Prove that\[a_1a_2\dots a_{n-1}+a_1a_2\dots a_{n-2}+\dots+a_1a_2+a_1+2 \leqslant a_1a_2\dots a_n.\]

Solution

Problem 5

Infinitely many equidistant parallel lines are drawn in the plane. A positive integer $n \geqslant 3$ is called frameable if it is possible to draw a regular polygon with $n$ sides all whose vertices lie on these lines, and no line contains more than one vertex of the polygon.

(a) Show that $3, 4, 6$ are frameable.

(b) Show that any integer $n \geqslant 7$ is not frameable.

(c) Determine whether $5$ is frameable.

Solution

Problem 6

A stromino is a $3 \times 1$ rectangle. Show that a $5 \times 5$ board divided into twenty-five $1 \times 1$ squares cannot be covered by $16$ strominos such that each stromino covers exactly three squares of the board, and every square is covered by one or two strominos. (A stromino can be placed either horizontally or vertically on the board.)

Solution

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