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Difference between revisions of "2020 USAMO Problems"

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Let <math>ABC</math> be a fixed acute triangle inscribed in a circle <math>\omega</math> with center <math>O</math>. A variable point <math>X</math> is chosen on minor arc <math>AB</math> of <math>\omega</math>, and segments <math>CX</math> and <math>AB</math> meet at <math>D</math>. Denote by <math>O_1</math> and <math>O_2</math> the circumcenters of triangles <math>ADX</math> and <math>BDX</math>, respectively. Determine all points <math>X</math> for which the area of triangle <math>OO_1O_2</math> is minimized.
 
Let <math>ABC</math> be a fixed acute triangle inscribed in a circle <math>\omega</math> with center <math>O</math>. A variable point <math>X</math> is chosen on minor arc <math>AB</math> of <math>\omega</math>, and segments <math>CX</math> and <math>AB</math> meet at <math>D</math>. Denote by <math>O_1</math> and <math>O_2</math> the circumcenters of triangles <math>ADX</math> and <math>BDX</math>, respectively. Determine all points <math>X</math> for which the area of triangle <math>OO_1O_2</math> is minimized.
  
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[[2020 USOMO Problems/Problem 1|Solution]]
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[[2020 USAMO Problems/Problem 1|Solution]]
  
  
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An empty <math>2020 \times 2020 \times 2020</math> cube is given, and a <math>2020 \times 2020</math> grid of square unit cells is drawn on each of its six faces. A beam is a <math>1 \times 1 \times 2020</math> rectangular prism. Several beams are placed inside the cube subject to the following conditions:
 
An empty <math>2020 \times 2020 \times 2020</math> cube is given, and a <math>2020 \times 2020</math> grid of square unit cells is drawn on each of its six faces. A beam is a <math>1 \times 1 \times 2020</math> rectangular prism. Several beams are placed inside the cube subject to the following conditions:
  
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<math>\bullet</math> The two <math>1 \times 1</math> faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are <math>3 \cdot 2020^2</math> possible positions for a beam.)
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*The two <math>1 \times 1</math> faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are <math>3 \cdot 2020^2</math> possible positions for a beam.)
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*No two beams have intersecting interiors.
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<math>\bullet</math> No two beams have intersecting interiors.
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*The interiors of each of the four <math>1 \times 2020</math> faces of each beam touch either a face of the cube or the interior of the face of another beam.
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<math>\bullet</math> The interiors of each of the four <math>1 \times 2020</math> faces of each beam touch either a face
 
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of the cube or the interior of the face of another beam.
 
  
 
What is the smallest positive number of beams that can be placed to satisfy these conditions?
 
What is the smallest positive number of beams that can be placed to satisfy these conditions?
  
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[[2020 USOMO Problems/Problem 2|Solution]]
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[[2020 USAMO Problems/Problem 2|Solution]]
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===Problem 3===
 
===Problem 3===
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Denote by <math>A</math> the set of all integers <math>a</math> such that <math>1 \le a < p</math>, and both <math>a</math> and <math>4 - a</math> are quadratic non-residues. Calculate the remainder when the product of the elements of <math>A</math> is divided by <math>p</math>.
 
Denote by <math>A</math> the set of all integers <math>a</math> such that <math>1 \le a < p</math>, and both <math>a</math> and <math>4 - a</math> are quadratic non-residues. Calculate the remainder when the product of the elements of <math>A</math> is divided by <math>p</math>.
  
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[[2020 USOMO Problems/Problem 3|Solution]]
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[[2020 USAMO Problems/Problem 3|Solution]]
  
  
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Suppose that <math>(a_1, b_1), (a_2, b_2), \ldots , (a_{100}, b_{100})</math> are distinct ordered pairs of nonnegative integers. Let <math>N</math> denote the number of pairs of integers <math>(i, j)</math> satisfying <math>1 \le i < j \le 100</math> and <math>|a_ib_j - a_j b_i|=1</math>. Determine the largest possible value of <math>N</math> over all possible choices of the <math>100</math> ordered pairs.
 
Suppose that <math>(a_1, b_1), (a_2, b_2), \ldots , (a_{100}, b_{100})</math> are distinct ordered pairs of nonnegative integers. Let <math>N</math> denote the number of pairs of integers <math>(i, j)</math> satisfying <math>1 \le i < j \le 100</math> and <math>|a_ib_j - a_j b_i|=1</math>. Determine the largest possible value of <math>N</math> over all possible choices of the <math>100</math> ordered pairs.
  
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[[2020 USOMO Problems/Problem 4|Solution]]
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[[2020 USAMO Problems/Problem 4|Solution]]
  
  
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For each integer <math>n \ge 2</math>, find the largest integer <math>k</math> (in terms of <math>n</math>) such that there exists a set of <math>n</math> distinct points that is not overdetermined, but has <math>k</math> overdetermined subsets.
 
For each integer <math>n \ge 2</math>, find the largest integer <math>k</math> (in terms of <math>n</math>) such that there exists a set of <math>n</math> distinct points that is not overdetermined, but has <math>k</math> overdetermined subsets.
  
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[[2020 USOMO Problems/Problem 5|Solution]]
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[[2020 USAMO Problems/Problem 5|Solution]]
  
 
===Problem 6===
 
===Problem 6===
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<cmath>\sum_{i=1}^n(x_iy_i-x_iy_{n+1-i})\ge\frac{2}{\sqrt{n-1}}.</cmath>
 
<cmath>\sum_{i=1}^n(x_iy_i-x_iy_{n+1-i})\ge\frac{2}{\sqrt{n-1}}.</cmath>
  
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[[2020 USOMO Problems/Problem 6|Solution]]
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[[2020 USAMO Problems/Problem 6|Solution]]
  
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{{USAMO newbox|year= 2020 |before=[[2019 USAMO]]|after=[[2021 USAMO]]}}
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==See Also==
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{{USAMO newbox|year=2020|before=[[2019 USAMO Problems]]|after=[[2021 USAMO Problems]]}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 12:47, 22 November 2023

Day 1

Problem 1

Let $ABC$ be a fixed acute triangle inscribed in a circle $\omega$ with center $O$. A variable point $X$ is chosen on minor arc $AB$ of $\omega$, and segments $CX$ and $AB$ meet at $D$. Denote by $O_1$ and $O_2$ the circumcenters of triangles $ADX$ and $BDX$, respectively. Determine all points $X$ for which the area of triangle $OO_1O_2$ is minimized.

Solution


Problem 2

An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A beam is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:

  • The two $1 \times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \cdot 2020^2$ possible positions for a beam.)
  • No two beams have intersecting interiors.
  • The interiors of each of the four $1 \times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam.

What is the smallest positive number of beams that can be placed to satisfy these conditions?

Solution

Problem 3

Let $p$ be an odd prime. An integer $x$ is called a quadratic non-residue if $p$ does not divide $x - t^2$ for any integer $t$.

Denote by $A$ the set of all integers $a$ such that $1 \le a < p$, and both $a$ and $4 - a$ are quadratic non-residues. Calculate the remainder when the product of the elements of $A$ is divided by $p$.

Solution


Day 2

Problem 4

Suppose that $(a_1, b_1), (a_2, b_2), \ldots , (a_{100}, b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i, j)$ satisfying $1 \le i < j \le 100$ and $|a_ib_j - a_j b_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs.

Solution


Problem 5

A finite set $S$ of points in the coordinate plane is called overdetermined if $|S| \ge 2$ and there exists a nonzero polynomial $P(t)$, with real coefficients and of degree at most $|S| - 2$, satisfying $P(x) = y$ for every point $(x, y) \in S$.

For each integer $n \ge 2$, find the largest integer $k$ (in terms of $n$) such that there exists a set of $n$ distinct points that is not overdetermined, but has $k$ overdetermined subsets.

Solution

Problem 6

Let $n \ge 2$ be an integer. Let $x_1 \ge x_2 \ge \cdots \ge x_n$ and $y_1 \ge y_2 \ge \cdots \ge y_n$ be $2n$ real numbers such that \begin{align*} 0 &= x_1 + x_2 + \cdots + x_n = y_1 + y_2 + \cdots + y_n\\ \text{and }1 &= x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2. \end{align*} Prove that \[\sum_{i=1}^n(x_iy_i-x_iy_{n+1-i})\ge\frac{2}{\sqrt{n-1}}.\]

Solution

See Also

2020 USAMO (Problems • Resources)
Preceded by
2019 USAMO Problems 		Followed by
2021 USAMO Problems
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

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