Difference between revisions of "2020 INMO Problems"
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<math>\emph{Proposed by Prithwijit De}</math> | <math>\emph{Proposed by Prithwijit De}</math> | ||
− | [[2020 INMO Problems/Problem 1 | + | [[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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<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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== Problem 4 == | == Problem 4 == | ||
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> | ||
+ | [[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. | ||
+ | [[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 05:43, 25 September 2023
Problem 1
Let and
be two circles of unequal radii, with centres
and
respectively, intersecting in two distinct points
and
. Assume that the centre of each circle is outside the other circle. The tangent to
at
intersects
again in
, different from
; the tangent to
at
intersects
again at
, different from
. The bisectors of
and
meet
and
again in
and
, respectively. Let
and
be the circumcentres of triangles
and
, respectively. Prove that
is the perpendicular bisector of the line segment
.
Problem 2
Suppose is a polynomial with real coefficients, satisfying the condition
, for every real
. Prove that
can be expressed in the form
for some real numbers
and non-negative integer
.
Problem 3
Let be a subset of
. Suppose there is a positive integer
such that for any integer
, one can find positive integers
so that
and all the digits in the decimal(Base 10) representations of
(expressed without leading zeros) are in
. Find the smallest possible value of
.
Problem 4
Let be an integer and let
be
real numbers such that
. Prove that
Problem 5
Infinitely many equidistant parallel lines are drawn in the plane. A positive integer is called frameable if it is possible to draw a regular polygon with
sides all whose vertices lie on these lines, and no line contains more than one vertex of the polygon.
(a) Show that are frameable.
(b) Show that any integer is not frameable.
(c) Determine whether is frameable.
Problem 6
A stromino is a rectangle. Show that a
board divided into twenty-five
squares cannot be covered by
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.)