Difference between revisions of "2022 USAMO Problems"
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Let <math>b\geq2</math> and <math>w\geq2</math> be fixed integers, and <math>n=b+w</math>. Given are <math>2b</math> identical black rods and <math>2w</math> identical white rods, each of side length 1. | Let <math>b\geq2</math> and <math>w\geq2</math> be fixed integers, and <math>n=b+w</math>. Given are <math>2b</math> identical black rods and <math>2w</math> identical white rods, each of side length 1. | ||
− | We assemble a regular <math>2n | + | We assemble a regular <math>2n</math>-gon using these rods so that parallel sides are the same color. Then, a convex <math>2b</math>-gon <math>B</math> is formed by translating the black rods, and a convex <math>2w</math>-gon <math>W</math> is formed by translating the white rods. An example of one way of doing the assembly when <math>b=3</math> and <math>w=2</math> is shown below, as well as the resulting polygons <math>B</math> and <math>W</math>. |
+ | |||
+ | <asy> | ||
+ | size(10cm); | ||
+ | real w = 2*Sin(18); | ||
+ | real h = 0.10 * w; | ||
+ | real d = 0.33 * h; | ||
+ | picture wht; | ||
+ | picture blk; | ||
+ | |||
+ | draw(wht, (0,0)--(w,0)--(w+d,h)--(-d,h)--cycle); | ||
+ | fill(blk, (0,0)--(w,0)--(w+d,h)--(-d,h)--cycle, black); | ||
+ | |||
+ | // draw(unitcircle, blue+dotted); | ||
+ | |||
+ | // Original polygon | ||
+ | add(shift(dir(108))*blk); | ||
+ | add(shift(dir(72))*rotate(324)*blk); | ||
+ | add(shift(dir(36))*rotate(288)*wht); | ||
+ | add(shift(dir(0))*rotate(252)*blk); | ||
+ | add(shift(dir(324))*rotate(216)*wht); | ||
+ | |||
+ | add(shift(dir(288))*rotate(180)*blk); | ||
+ | add(shift(dir(252))*rotate(144)*blk); | ||
+ | add(shift(dir(216))*rotate(108)*wht); | ||
+ | add(shift(dir(180))*rotate(72)*blk); | ||
+ | add(shift(dir(144))*rotate(36)*wht); | ||
+ | |||
+ | // White shifted | ||
+ | real Wk = 1.2; | ||
+ | pair W1 = (1.8,0.1); | ||
+ | pair W2 = W1 + w*dir(36); | ||
+ | pair W3 = W2 + w*dir(108); | ||
+ | pair W4 = W3 + w*dir(216); | ||
+ | path Wgon = W1--W2--W3--W4--cycle; | ||
+ | draw(Wgon); | ||
+ | pair WO = (W1+W3)/2; | ||
+ | transform Wt = shift(WO)*scale(Wk)*shift(-WO); | ||
+ | draw(Wt * Wgon); | ||
+ | label("$W$", WO); | ||
+ | /* | ||
+ | draw(W1--Wt*W1); | ||
+ | draw(W2--Wt*W2); | ||
+ | draw(W3--Wt*W3); | ||
+ | draw(W4--Wt*W4); | ||
+ | */ | ||
+ | |||
+ | // Black shifted | ||
+ | real Bk = 1.10; | ||
+ | pair B1 = (1.5,-0.1); | ||
+ | pair B2 = B1 + w*dir(0); | ||
+ | pair B3 = B2 + w*dir(324); | ||
+ | pair B4 = B3 + w*dir(252); | ||
+ | pair B5 = B4 + w*dir(180); | ||
+ | pair B6 = B5 + w*dir(144); | ||
+ | path Bgon = B1--B2--B3--B4--B5--B6--cycle; | ||
+ | pair BO = (B1+B4)/2; | ||
+ | transform Bt = shift(BO)*scale(Bk)*shift(-BO); | ||
+ | fill(Bt * Bgon, black); | ||
+ | fill(Bgon, white); | ||
+ | label("$B$", BO); | ||
+ | </asy> | ||
Prove that the difference of the areas of <math>B</math> and <math>W</math> depends only on the numbers <math>b</math> and <math>w</math>, and not on how the <math>2n</math>-gon was assembled. | Prove that the difference of the areas of <math>B</math> and <math>W</math> depends only on the numbers <math>b</math> and <math>w</math>, and not on how the <math>2n</math>-gon was assembled. | ||
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[[2022 USAMO Problems/Problem 6|Solution]] | [[2022 USAMO Problems/Problem 6|Solution]] | ||
− | {{USAMO newbox|year= 2022 |before=[[2021 USAMO]]|after=[[2023 USAMO]]}} | + | ==See Also== |
+ | {{USAMO newbox|year=2022|before=[[2021 USAMO Problems]]|after=[[2023 USAMO Problems]]}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 13:45, 22 November 2023
Contents
Day 1
Problem 1
Let and
be positive integers. The cells of an
grid are colored amber and bronze such that there are at least
amber cells and at least
bronze cells. Prove that it is possible to choose
amber cells and
bronze cells such that no two of the
chosen cells lie in the same row or column.
Problem 2
Let and
be fixed integers, and
. Given are
identical black rods and
identical white rods, each of side length 1.
We assemble a regular -gon using these rods so that parallel sides are the same color. Then, a convex
-gon
is formed by translating the black rods, and a convex
-gon
is formed by translating the white rods. An example of one way of doing the assembly when
and
is shown below, as well as the resulting polygons
and
.
Prove that the difference of the areas of and
depends only on the numbers
and
, and not on how the
-gon was assembled.
Problem 3
Let be the set of all positive real numbers. Find all functions
such that for all
we have
Day 2
Problem 4
Find all pairs of primes for which
and
are both perfect squares.
Problem 5
A function is essentially increasing if
holds whenever
are real numbers such that
and
.
Find the smallest integer such that for any 2022 real numbers
there exist
essentially increasing functions
such that
Problem 6
There are 2022 users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.)
Starting now, Mathbook will only allow a new friendship to be formed between two users if they have at least two friends in common. What is the minimum number of friendships that must already exist so that every user could eventually become friends with every other user?
See Also
2022 USAMO (Problems • Resources) | ||
Preceded by 2021 USAMO Problems |
Followed by 2023 USAMO Problems | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.