angle wanted, folding a square (Oral Moscow 2022 VII p8)

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0 replies

jlacosta

Wednesday at 3:18 PM

0 replies

Inequalities

sqing 23

N 18 minutes ago by sqing

Let $a,b,c\geq 0$ and $a+b+c=1$ . Prove that
$a^2+b^2+ ab +21abc\leq\frac{512}{441}$ Equality holds when $a=b=\frac{38}{21},c=\frac{5}{214}.$
$a^2+b^2+ ab +19abc\leq\frac{10648}{9747}$ Equality holds when $a=b=\frac{22}{57},c=\frac{13}{57}.$
$a^2+b^2+ ab +22abc\leq\frac{15625}{13068}$ Equality holds when $a=b=\frac{25}{66},c=\frac{8}{33}.$

23 replies

1 viewing

sqing

Mar 26, 2025

sqing

18 minutes ago

Inequalities

sqing 3

N 34 minutes ago by sqing

Let $a, b,c\geq 0$ and $2a+3b+ 4c=11.$ Prove that
$a+ab+abc\leq\frac{49}{6}$ Let $a, b,c\geq 0$ and $2a+3b+ 4c=10.$ Prove that
$a+ab+abc\leq\frac{169}{24}$ Let $a, b,c\geq 0$ and $2a+3b+ 4c=14.$ Prove that
$a+ab+abc\leq\frac{63+5\sqrt 5}{6}$ Let $a, b,c\geq 0$ and $2a+3b+ 4c=32.$ Prove that
$a+ab+abc\leq48+\frac{64\sqrt{2}}{3}$

3 replies

sqing

Apr 1, 2025

sqing

34 minutes ago

Puzzling p&c question

Hunter87 0

3 hours ago

There are 9 numbered tickets (distinct) to be distributed (1-9), and 7 contestants. Each contestant must get atleast one ticket. Two PARTICULAR contestants (say, A and B) can never get tickets with adjacent numbers and any contestant can get more than 1 ticket. All tickets are to be distributed.
In how many ways can this be done?

0 replies

Hunter87

3 hours ago

0 replies

.problem.

Cobedangiu 1

N 3 hours ago by Cobedangiu

Find the integer coefficients after expanding Newton's binomial:
$(\frac{3}{2}-\frac{2}{3}x^2)^n (n \in Z)$

1 reply

Cobedangiu

6 hours ago

Cobedangiu

3 hours ago

No more topics!

angle wanted, folding a square (Oral Moscow 2022 VII p8)

parmenides51 3

N Apr 17, 2022 by samrocksnature

The paper square $ABCD$ is folded along a straight line so that that the vertex $A$ coincides with the interior point $M$ of the side $CD$ , and side $AB$ (in the new position) crosses side $BC$ at point $N$ (see figure). Find the angle $\angle MAN$ .
IMAGE

3 replies