MathB.in - Share Mathematics with LaTeX and Markdown

$\boxed{P1}$
We say a nondegenerate triangle whose angles have measures $\theta_1, \theta_2, \theta_3$ is quirky if there exist integers $r_1, r_2, r_3$, not all zero, such that,$$r_1\theta_1+r_2\theta_2+r_3\theta_3=0$$Find the sum of all integers $n\ge3$ for which a triangle with side lengths $n-1, n, n+1$ is quirky.

$\boxed{P2}$ 
In an acute scalene triangle ABC, points D,E,F lie on sides BC, CA, AB respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{BE}\perp\overline{CA}$, $\overline{CF}\perp\overline{AB}$. Altitudes $\overline{AD}$, $\overline{BE}$, $\overline{CF}$ meet at orthocenter $H$. Points $P$ and $Q$ lie on segment $\overline{EF}$ such that $\overline{AP}\perp\overline{EF}$ and $\overline{HQ}\perp\overline{EF}$. Line $DP$ and $QH$ intersect at point $R$. Compute $\frac{HQ}{HR}$.

$\boxed{P3}$
In   $\triangle ABC,CA=1960\sqrt{2},CB=6720$, and $\angle{C}=45^{\circ}$. Let $K, L, M$ lie on lines $BC$, $CA$ and $AB$ such that $\overline{AK}\perp\overline{BC}, \overline{BL}\perp\overline{CA}$ and $AM=BM$. Let $N, O, P$ lie on $\overline{KL}, \overline{BA}, \overline{BL}$ such that $AN=KN, BO=CO$ and $A$ lies on line $NP$. If $H$ is the orthocenter of $\triangle MOP$, compute $HK^{2}$

$\boxed{P4}$
The number $123454321$ is written on a blackboard. Anik walks by and erases some (but not all) of the digits, and notices that the resulting number (when spaces are removed) is divisible by 9. What is the fewest number of digits he could have erased?

$\boxed{P5}$ 
Two infinite sequences $\{s_i\}_{i\geq 1}$ and $\{t_i\}_{i\geq 1}$  are equivalent if, for all positive integers $i$ and $j$, $s_i = s_j$ if and only if $t_i = t_j$. A sequence $\{r_i\}_{i\geq 1}$ has equi-period $k$ if $r_1, r_2, \ldots $ and $r_{k+1}, r_{k+2}, \ldots$ are equivalent.

Suppose $M$ infinite sequences with equi-period $2022$ whose terms are in the set $\{1, \ldots, 2021\}$ can be chosen such that no two chosen sequences are equivalent to each other. Determine the largest possible value of $M$.

$\boxed{P6}$
Each cell of a $100×100$ grid is colored with one of $101$ colors. A cell is diverse if, among the $199$ cells in its row or column, every color appears at least once. Determine the maximum possible number of diverse cells.

$\boxed{P7}$
$2021$ people are sitting around a circular table. In one move, you may swap the positions of two people sitting next to each other. Determine the minimum number of moves necessary to make each person end up $1000$ positions to the left of their original position.

$\boxed{P8}$
The number $(2+2^{96})!$ has $2^{93}$ trailing zeroes when expressed in base $B$. Find the total number of possible B.

$\boxed{P9}$
The number $(2+2^{96})!$ has $2^{93}$ trailing zeroes when expressed in base $B$.Find the maximum possible B.

$\boxed{P10}$ 
The number $(2+2^{96})!$ has $2^{93}$ trailing zeroes when expressed in base $B$.Find the minimum possible B.

$\boxed{P11}$
Anik has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each.

Each morning, Anik performs exactly one of the following moves:

$(a)$ He clears every piece of rubbish from a single pile.

$(b)$ He clears one piece of rubbish from each pile.

However, every evening, a demon sneaks into the warehouse and performs exactly one of the following moves:

$(a)$ He adds one piece of rubbish to each non-empty pile.

$(b)$ He creates a new pile with one piece of rubbish.

What is the first morning when Anik can guarantee to have cleared all the rubbish from the warehouse?

$\boxed{P12}$
On a $3 \times 3$ board, the numbers from $1$ to $9$ are written in some order and without repeating. We say that the arrangement obtained is PhymTHian if the numbers in any two adjacent squares have different parity. Determine the number of different PhymTHian arrangements.

Note: Two arrangements are considered equal if one can be obtained from the other by rotating the board.

$\boxed{P13}$
Let $q$ be a positive rational number. Two ants are initially at the same point $X$ in the plane. In the $n$-th minute $(n = 1,2,...)$ each of them chooses whether to walk due north, east, south or west and then walks the distance of $q^n$ metres. After a whole number of minutes, they are at the same point in the plane (not necessarily $X$), but have not taken exactly the same route within that time. Determine all possible values of $q$.

$\boxed{P14}$
Compute $${\lceil}{\sum_{n=2021}^{\infty}\frac{2022!-2021!}{n!}}{\rceil}$$Note: The notation ${\lceil}x{\rceil}$ denotes the least integer $n$ such that $n\ge{x}$.

$\boxed{P15}$
The numbers $1,2,\cdots,2021$ are arranged in a circle. For any $1 \le i \le 2021$, if $i,i+1,i+2$ are three consecutive numbers in some order such that $i+1$ is not in the middle, then $i$ is said to be a good number. Indices are taken mod $2021$. What is the maximum possible number of good numbers?

$\boxed{P16}$
Consider an $2022×2023$ rectangular grid of points in the plane. Some $k$ of these points are coloured red in such a way that no three red points are the vertices of a rightangled triangle two of whose sides are parallel to the sides of the grid. Determine the greatest possible value of $k$.

$\boxed{P17}$
Let $2021$ be positive integer and fix $2×2021$ distinct points on a circle. Determine the number of ways to connect the points with $2021$ arrows (oriented line segments) such that all of the following conditions hold:
each of the $2×2021$ points is a startpoint or endpoint of an arrow;
no two arrows intersect; and
there are no two arrows $\overrightarrow{AB}$ and $\overrightarrow{CD}$ such that $A, B, C$ and $D$ appear in clockwise order around the circle (not necessarily consecutively).

$\boxed{P18}$
Let $2021$ be positive integer and fix $2×2021$ distinct points on a circle. Determine the number of ways to connect the points with $2021$ arrows (oriented line segments) such that all of the following conditions hold:
each of the $2×2021$ points is a startpoint or endpoint of an arrow;
no two arrows intersect; and
there are no two arrows $\overrightarrow{AB}$  and $\overrightarrow{CD}$ such that $A, B, C$ and $D$ appear in clockwise order around the circle (not necessarily consecutively).

$\boxed{P19}$
What is the greatest $5$-digit palindrome $n$ such that $7n$ is a $6$-digit palindrome?

$\boxed{P20}$
Find the remainder when $$\displaystyle{\sum_{k=1}^{2^{16}}\binom{2k}{k}(3\cdot2^{14}+1)^{k}(k-1)^{2^{16}-1}}$$ is divided by $2^{16}+1$.

Wed, 15 Sep 2021 15:28 GMT