Complex number

by ronitdeb, May 29, 2025, 6:13 PM

Let $z_1, ... ,z_5$ be vertices of regular pentagon inscribed in a circle whose radius is $2$ and center is at $6+i8$ . Find all possible values of $z_1^2+z_2^2+...+z_5^2$

This post has been edited 1 time. Last edited by ronitdeb, 3 hours ago

complex numbers

geometry

Cup of Combinatorics

by M11100111001Y1R, May 27, 2025, 7:24 AM

There are $n$ cups labeled $1, 2, \dots, n$ , where the $i$ -th cup has capacity $i$ liters. In total, there are $n$ liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.

$a)$ Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most $\frac{4n}{3}$ steps.

$b)$ Prove that in at most $\frac{5n}{3}$ steps, one can go from any configuration with integer water amounts to any other configuration with the same property.

This post has been edited 1 time. Last edited by M11100111001Y1R, May 27, 2025, 7:26 AM

combinatorics

Elementary Problems Compilation

by Saucepan_man02, May 26, 2025, 1:44 PM

Could anyone send some elementary problems, which have tricky and short elegant methods to solve?

For example like this one:
Solve over reals: $a^2 + b^2 + c^2 + d^2 -ab-bc-cd-d +2/5=0$

Shortest number theory you might've seen in your life

by AlperenINAN, May 11, 2025, 7:51 PM

Let $p$ and $q$ be prime numbers. Prove that if $pq(p+1)(q+1)+1$ is a perfect square, then $pq + 1$ is also a perfect square.

This post has been edited 3 times. Last edited by AlperenINAN, May 12, 2025, 10:09 AM
Reason: Typo

number theory

TST

Arc Midpoints Form Cyclic Quadrilateral

by ike.chen, Jul 9, 2023, 4:36 AM

In the acute-angled triangle $ABC$ , the point $F$ is the foot of the altitude from $A$ , and $P$ is a point on the segment $AF$ . The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$ , respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$ , respectively, such that $DA = DX$ and $EA = EY$ .
Prove that $B, C, X,$ and $Y$ are concyclic.

This post has been edited 4 times. Last edited by ike.chen, Jul 9, 2023, 5:15 PM

Odd digit multiplication

by JuanDelPan, Oct 6, 2021, 10:21 PM

Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$ .

$\textit{Proposed by Pablo Serrano, Ecuador}$

This post has been edited 2 times. Last edited by JuanDelPan, Oct 7, 2021, 12:36 AM

Digits

number theory

PAGMO

Drawing Triangles Against Your Clone

by pieater314159, Jun 27, 2019, 9:38 PM

Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$ .)

Proposed by Milan Haiman

Inequality

by knm2608, Jun 25, 2017, 1:04 PM

If the non-negative reals $x,y,z$ satisfy $x^2+y^2+z^2=x+y+z$ . Prove that
$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$ When does the equality occur?

Proposed by Dorlir Ahmeti, Albania

This post has been edited 4 times. Last edited by v_Enhance, Jun 25, 2017, 2:33 PM

JBMO Shortlist

Inequality

Circumcircle of XYZ is tangent to circumcircle of ABC

by mathuz, May 22, 2013, 3:39 PM

Let $\omega$ be the incircle of the triangle $ABC$ and with centre $I$ . Let $\Gamma$ be the circumcircle of the triangle $AIB$ . Circles $\omega$ and $\Gamma$ intersect at the point $X$ and $Y$ . Let $Z$ be the intersection of the common tangents of the circles $\omega$ and $\Gamma$ . Show that the circumcircle of the triangle $XYZ$ is tangent to the circumcircle of the triangle $ABC$ .

geometry

circumcircle

ratio

geometric transformation

geometry proposed

n-term Sequence

by MithsApprentice, Oct 22, 2005, 11:53 PM

An $n$ -term sequence $(x_1, x_2, \ldots, x_n)$ in which each term is either 0 or 1 is called a binary sequence of length $n$ . Let $a_n$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$ .

combinatorics unsolved

combinatorics