USAJMO 2022

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

• $a_n - g_n$ is divisible by $m$ for all integers $n \ge 1$ ;
• $a_2 - a_1$ is not divisible by $m$ .

Holden Mui

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.

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 $W$ .

[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 + wdir(36); pair W3 = W2 + wdir(108); pair W4 = W3 + wdir(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--WtW1); draw(W2--WtW2); draw(W3--WtW3); draw(W4--WtW4); */

// Black shifted real Bk = 1.10; pair B1 = (1.5,-0.1); pair B2 = B1 + wdir(0); pair B3 = B2 + wdir(324); pair B4 = B3 + wdir(252); pair B5 = B4 + wdir(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 $B$ and $W$ depends only on the numbers $b$ and $w$ , and not on how the $2n$ -gon was assembled.

Proposed by Ankan Bhattacharya

Let $ABCD$ be a rhombus, and let $K$ and $L$ be points such that $K$ lies inside the rhombus, $L$ lies outside the rhombus, and $KA = KB = LC = LD$ . Prove that there exist points $X$ and $Y$ on lines $AC$ and $BD$ such that $KXLY$ is also a rhombus.

Proposed by Ankan Bhattacharya

Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.

Let $a_0, b_0, c_0$ be complex numbers, and define \begin{align*}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\end{align*}for all nonnegative integers $n.$

Suppose that $\max{\{|a_n|, |b_n|, |c_n|\}} \leq 2022$ for all $n.$ Prove that $$|a_0|^2 + |b_0|^2 + |c_0|^2 \leq 1.$$