POTWs 4

Posted on May 22, 2018 by InfinityDots

Every week, we will select three notable or interesting problems, marked with E,M,H (“easy”, “medium”, “hard”) for the relative difficulty. Easy problems will be around TMO or easier than TMO; medium problems will be around Oct Camp / easy IMO, and hard problems will be around medium / hard IMO.

E4 [Classical]

Alice and Bob alternatively choose numbers from among $1,2,...,9$ , without replacement. The first to obtain $3$ numbers which sum to $15$ wins. Does Alice (the first to play) have a winning strategy?

M4 [The Guardian]

Given a $100\times 100$ grid with an arrow pointing to one of four direction (up/down or left/right) in each of $100^2$ cell. Initially you’re in a cell. The goal is to get out of the grid. In each turn, you must move in the neighbour cell, if not out of the grid, that the arrow in your current cell pointed to, and once you leave the old cell, you must also turn the arrow in that cell by $90$ degree clockwise. Is it true that no matter which square you chose to begin, and no matter what directions the arrows are initially pointing at, you will eventually get out of the grid?

H4 [@gausskarl on AoPS]

Let $\mathcal{P} =\{ P_1,P_2,...,P_{2018} \}$ be a set of $2018$ points in the interior of a circle of radius $1$ with $P_1$ be the center of the circle. For each $k=1,2,...,2018$ , let $d_k$ be the distance from $P_k$ to the other point in $\mathcal{P}$ that is closest to $P_k$ . Prove that

$d_1^2+d_2^2+...+d_{2018}^2\leq 9$ .

Solution will be available next week.

POTWs 3 Solutions

Posted on May 22, 2018 by InfinityDots

Here are the solutions to POTWs 3. Along with the solutions, we also give some comments and difficulty ratings $\beta$ in $[1,5]$ , where POSN Camp level is 1-2, TMO level is 1-3, October Camp is around 2-4, and IMO is around 3-5.

Continue reading →

POTWs 3

Posted on May 15, 2018 by InfinityDots

E3 [adapted from IMO 2006 P4]

Show that for all primes $p>3$ , $2^{p-2}+3^{p-2}+6^{p-2}-1$ is divisible by $p$ .

M3 [Bulgaria TST 2005]

Find the number of the subsets $B$ of the set $\{ 1,2,...,2005\}$ such that the sum of the elements of $B$ is congruent to $2006$ modulo $2048$ .

H3 [reddit]

From any pair of positive integers $(a,b)$ , in each turn, you can choose to move to either $(a+1,2b)$ or $(2a,b+1)$ . Show that, starting from any pair of positive integers $(m,n)$ , you can reach a pair of two equal positive integers.

Solution will be available next week.

POTWs 2 Solutions

Posted on May 15, 2018 by InfinityDots

Here are the solutions to POTWs 2. Along with the solutions, we also give some comments and difficulty ratings $\beta$ in $[1,5]$ , where POSN Camp level is 1-2, TMO level is 1-3, October Camp is around 2-4, and IMO is around 3-5.

Continue reading →

InfinityDots MO files now available!

Posted on May 12, 2018 by InfinityDots

The InfinityDots MOs are our Mock IMO contests on AoPS. We have held two InfinityDots MOs so far with lots of participants and great feedback.

Today we have added the InfinityDots MO and MO 2 problems and solutions to our archive. Here’s a direct link to the folder. Also, while this is the English site, we’d also like to note that we have translated the problems into Thai; the Thai versions are now available in our archives too.

We invite readers who are interested in IMO-like contests to try these problems. Read more about our contests on AoPS here: InfinityDots MO, InfinityDots MO 2.

TMO 15

Posted on May 11, 2018 by InfinityDots

Hi,

During the past several days, the $15^{th}$ Thailand Mathematics Olympiad was held in Nakhon Ratchasima. In this post, we are sharing our solutions, along with some comments, to the problems in the competition. But since it’s always good to try solving problems yourself before reading the solution, we’ve attached the solution file at the end of this post and give the non-spoiler version of comments to each problem first.

Note that we use the difficulty rates $\beta$ as in our POTWs, that is, POSN Camp level is 1-2, TMO level is 1-3, October Camp is around 2-4, and IMO is around 3-5.

Continue reading →

POTWs 2

Posted on May 8, 2018 by InfinityDots

E2 [@Konigsberg on AoPS]
In a convex pentagon, show that we can choose $3$ diagonals such that their lengths can form a triangle.

M2 [China TST 2007 Quiz]
Let $I$ be the incenter of triangle $ABC.$ Let $M,N$ be the midpoints of $AB,AC,$ respectively. Points $D,E$ lie on $AB,AC$ respectively such that $BD=CE=BC.$ The line perpendicular to $IM$ through $D$ intersects the line perpendicular to $IN$ through $E$ at $P.$ Prove that $AP\perp BC.$

H2 [Google CodeJam 2011]
Goro wants to sort a list of $n$ distinct numbers in an increasing order. In each round, Goro can fix some elements of the list. All non-fixed elements of the list will then be permuted randomly (with each permutation having equal probability.) Given $n$ and the initial list, determine the expected number of rounds Goro will need to sort the list, under Goro’s best strategy.

Solutions will be available next week.

POTWs 1 Solutions

Posted on May 8, 2018 by InfinityDots

Here are the solutions to POTWs 1. Along with the solutions, we also give some comments and difficulty ratings $\beta$ in $[1,5]$ , where POSN Camp level is 1-2, TMO level is 1-3, October Camp is around 2-4, and IMO is around 3-5.

Continue reading →

POTWs 1

Posted on May 1, 2018 by InfinityDots

E1 [Classical]
Starting from the origin $(0,0)$ . In each turn, you can move either up, down, or right (by one unit) but you cannot visit the same point twice and the points you visit must have coordinate $(x,y)$ such that $0 \leqslant x \leqslant m , 0 \leqslant y \leqslant n$ . How many paths are there to reach the point $(m,n)$ ?

M1 [Kvant Magazine]
Prove that for all positive integers $n>1,$

$\displaystyle \left\lfloor\sqrt{n}\right\rfloor + \left\lfloor\sqrt[3]{n}\right\rfloor + \dotsc + \left\lfloor\sqrt[n]{n} \right\rfloor = \left\lfloor \log_{2}{n} \right\rfloor + \left\lfloor \log_{3}{n}\right\rfloor + \dotsc + \left\lfloor\log_{n}{n}\right\rfloor.$

H1 [China MO 1993]
Let $f$ be a function from the positive real numbers to the positive real numbers such that $f(xy)\leqslant f(x)f(y)$ for any positive real numbers $x$ and $y$ . Prove that for any positive real number $x$ and all positive integers $n$ ,