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Difference between revisions of "2022 USAJMO Problems"

(→‎Problem 6)
(→‎Problem 6)
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Let <math>a_0,b_0,c_0</math> be complex numbers, and define
 
Let <math>a_0,b_0,c_0</math> be complex numbers, and define
−
\begin{align*}
+
 
−
a_{n+1}&=a_n^2+2b_nc_n\\
+
<cmath>a_{n+1}=a_n^2+2b_nc_n</cmath>
−
b_{n+1}&=b_n^2+2c_na_n\\
+
<cmath>b_{n+1}=b_n^2+2c_na_n</cmath>
−
c_{n+1}&=c_n^2+2a_nb_n
+
<cmath>c_{n+1}=c_n^2+2a_nb_n</cmath>
−
\end{align*}
 
 
for all nonnegative integers <math>n</math>.
 
for all nonnegative integers <math>n</math>.
  

Revision as of 20:14, 19 April 2022

Day 1

$\textbf{Note:}$ For any geometry problem whose statement begins with an asterisk $(*)$, the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.

Problem 1

For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1,a_2,\cdots$ and an infinite geometric sequence of integers $g_1,g_2,\cdots$ satisfying the following properties?

$\bullet$ $a_n-g_n$ is divisible by $m$ for all integers $n>1$;

$\bullet$ $a_2-a_1$ is not divisible by $m$.

Solution

Problem 2

Let $a$ and $b$ be positive integers. The cells of an $(a + b + 1)\times (a + b + 1)$ grid are colored amber and bronze such that there are at least $a^2+ab-b$ amber cells and at least $b^2+ab-a$ bronze cells. Prove that it is possible to choose $a$ amber cells and $b$ bronze cells such that no two of the $a+b$ chosen cells lie in the same row or column.

Solution

Problem 3

Solution Let $b\geq2$ and $w\geq2$ be fixed integers, and $n=b+w$. Given are $2b$ identical black rods and $2w$ identical white rods, each of side length 1.

We assemble a regular $2n$-gon using these rods so that parallel sides are the same color. Then, a convex $2b$-gon $B$ is formed by translating the black rods, and a convex $2w$-gon $W$ is formed by translating the white rods. An example of one way of doing the assembly when $b=3$ and $w=2$ is shown below, as well as the resulting polygons $B$ and $W$.

[image here]

Prove that the difference of the areas of $B$ and $W$ depends only on the numbers $b$ and $w$, and not on how the $2n$-gon was assembled.

Day 2

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

Let $a_0,b_0,c_0$ be complex numbers, and define

\[a_{n+1}=a_n^2+2b_nc_n\] \[b_{n+1}=b_n^2+2c_na_n\] \[c_{n+1}=c_n^2+2a_nb_n\] for all nonnegative integers $n$.

Suppose that $\max{|a_n|,|b_n|,|c_n|}\leq2022$ for all $n$. Prove that \[|a_0|^2+|b_0|^2+|c_0|^2\leq 1.\]

2021 USAJMO (Problems • Resources)
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2021 USAJMO 		Followed by
2023 USAJMO
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All USAJMO Problems and Solutions

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