p1. Find the shortest path from point $A$ to point $B$ that does not pass through the interior of the circular region. p2. A giant, circular and perfectly flat pizza must be shared by $22$ people. Which is the least number of cuts to be made on the pizza so that each person can have a piece of pizza? (the pieces are not necessarily the same shape or with the same area). p3.Are there integers $n, m$ such that the equation $n\sqrt2 + m\sqrt3 = 2011$ will hold? p4. An convex quadrilateral is drawn and the $4$ triangles obtained by $3$ of the $4$ vertices of the quadrilateral . If the area of the largest of these triangles is $1888$ and the area of the the smallest of these triangles is $123$, determine what is the greatest value that the area of the quadrilateral can have. p5. Determine whether or not there are two digits other than $a, b$ such that the number $\overline{ab}$ is a multiple of the number $\overline{ba}$ (both written in decimal notation). p6. On an infinite lagoon there are arranged lotus flowers numbered $f_1, f_2, f_3, ...$ .Above each of the first $25$ lotus flowers there is a little frog. The frog jump out of a lotus flower according to the following rule: if a frog is on the $f_n$ , he can jump to the $f_{n + 1}$ or at $f_{n + 30}$ (as it likes) . Show that frogs can jump so that each lotus flower is visited by a frog exactly once. PS. Juniors P1, P2, P3, P5, P6 were also proposed as Seniors harder P1, P2, P3, P5, harder P6.
Problem
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Tags: algebra, geometry, number theory, combinatorics, chilean NMO
biscuit02
10.10.2021 10:37
Draw tangent from A and tangent from B.
Go along the circumference of the circle between the tangent points.
@post #5. Yes you're right. Thanks for pointing it out.
Initially there's 1 slice.
Strategy : Each cut should pass through the maximum number of current slices to create more slices.
After 1st cut, there's 2 slices.
After 2nd cut, there's 4 slices.
After 3rd cut, there's 7 slices. This cut passes through 3 previous slices.
After 4th cut, there's 11 slices. This cut passes through 4 previous slices.
After $n$ th cut, there's $1+\frac{n(n+1)}{2}$ slices. (Lazy Caterer's sequence)
When n=6, 22 slices are formed
So $\boxed{6}$
$n\sqrt2 = 2011 - m\sqrt3$
$(n\sqrt2)^2 = (2011 - m\sqrt3)^2$
$2n^2 -3m^2 -2011^2 = 4022m\sqrt{3}$
Since $n,m$ are integers, left hand side will be an integer but right hand side will not be an integer.
$\therefore$ no solutions
pilion
10.10.2021 22:39
The answer is 1888+123=2011.
Look at the part of the quadrilateral outside the triangle with area 123, it is another triangle whose area is at most 1888, so the area of the whole quadrilateral is at most 1888+123.
take a=b
or take a=any digit and b=0
If you can't do the above two things then the answer is no (try all the numbers $\overline{ba}$ less than 50 and with $a>b$)
If the frog at the front keeps jumping one space forward then every flower past 25 will be visited by that frog
mop
11.10.2021 00:36
biscuit02 wrote:
Draw tangent from A and tangent from B. Call point of intersection Q.
Path $AQ + QB$
Initially there's 1 slice.
Strategy : Each cut should pass through the maximum number of current slices to create more slices.
After 1st cut, there's 2 slices.
After 2nd cut, there's 4 slices.
After 3rd cut, there's 7 slices. This cut passes through 3 previous slices.
After 4th cut, there's 11 slices. This cut passes through 4 previous slices.
After $n$ th cut, there's $1+\frac{n(n+1)}{2}$ slices.
When n=6, 22 slices are formed
So $\boxed{6}$
$n\sqrt2 = 2011 - m\sqrt3$
$(n\sqrt2)^2 = (2011 - m\sqrt3)^2$
$2n^2 -3m^2 -2011^2 = 4022m\sqrt{3}$
Since $n,m$ are integers, left hand side will be an integer but right hand side will not be an integer.
$\therefore$ no solutions
shouldn't it go along the circumference of the circle between the tangent points?