Volume of right parallelepiped with minimal surface area

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0 replies

jlacosta

Apr 2, 2025

0 replies

Geometry Problem

Itoz 0

3 minutes ago

Source: Own

Given $\triangle ABC$ . Let the perpendicular line from $A$ to $BC$ meets $BC,\odot(ABC)$ at points $S,K$ , respectively, and the foot from $B$ to $AC$ is $L$ . $\odot (AKL)$ intersects line $AB$ at $T(\neq A)$ , $\odot(AST)$ intersects line $AC$ at $M(\neq A)$ , and lines $TM,CK$ intersect at $N$ .

Prove that $\odot(CNM)$ is tangent to $\odot (BST)$ .

0 replies

1 viewing

Itoz

3 minutes ago

0 replies

one cyclic formed by two cyclic

CrazyInMath 34

N 15 minutes ago by Assassino9931

Source: EGMO 2025/3

Let $ABC$ be an acute triangle. Points $B, D, E$ , and $C$ lie on a line in this order and satisfy $BD = DE = EC$ . Let $M$ and $N$ be the midpoints of $AD$ and $AE$ , respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$ , respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.

34 replies

CrazyInMath

Apr 13, 2025

Assassino9931

15 minutes ago

Board problem with complex numbers

egxa 1

N 18 minutes ago by hectorraul

Source: All Russian 2025 11.1

$777$ pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers $a$ and $b$ from the board such that:
$a^2 + b^2 + 1 = 2ab$ Ways that differ by the order of selection are considered the same. Prove that there exist two numbers $c$ and $d$ from the board such that:
$c^2 + d^2 + 2025 = 2cd$

1 reply

egxa

2 hours ago

hectorraul

18 minutes ago

parity of binomial coefficients

ThatApollo777 1

N 20 minutes ago by quasar_lord

Source: UMUMC 2011 P9

How many of the binomial coefficients $\binom{2011}{r}$ , $r = 0, 1, . . . , 2011$ are even?

1 reply

ThatApollo777

Oct 19, 2024

quasar_lord

20 minutes ago

No more topics!

Volume of right parallelepiped with minimal surface area

WakeUp 2

N Feb 2, 2011 by m.candales

Source: Romanian JBMO TST 2001

Determine a right parallelepiped with minimal area, if its volume is strictly greater than $1000$ , and the lengths of it sides are integer numbers.

2 replies

WakeUp

Jan 19, 2011

m.candales

Feb 2, 2011

Volume of right parallelepiped with minimal surface area

G H J

Reveal topic tags

geometry

number theory proposed

number theory

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Source: Romanian JBMO TST 2001

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WakeUp

1347 posts

WakeUp

#1 h Jan 19, 2011, 5:53 PM • 2 Y

Y by Adventure10, Mango247

Determine a right parallelepiped with minimal area, if its volume is strictly greater than $1000$ , and the lengths of it sides are integer numbers.

Z K Y

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mszew

1044 posts

mszew

#2 h Jan 28, 2011, 7:26 PM • 2 Y

Y by Adventure10, Mango247

Volume: $V=abc\ge 1001$
half Area $A_2=\frac{A}{2}=ab+bc+ca$

WLOG: $a \le b \le c$

$a=b=10$ $c=11$ , $V=1100$ , $A_2=320$ Then $a \le 10$

If $a\le 3$ then $334 \le bc \le A_2$
then $a \in \{4,5,6,7,8,9,10\}$

Notice that $\frac{1001}{a} \le bc$
and $A_2=a(b+c)+bc$
Checking some cases to find $(a,b,c)=(7,12,12)$

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m.candales

186 posts

m.candales

#3 h Feb 2, 2011, 4:12 PM • 2 Y

Y by Adventure10, Mango247

The right answer is $(8,9,14)$

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