truncated pyramids in a right prism with square base (2007 AMQ Concours p7)

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0 replies

jlacosta

Apr 2, 2025

0 replies

Eazy equation clap

giangtruong13 0

an hour ago

Find all $x,y,z$ satisfy that: $\frac{x}{y+z}=2x-1; \frac{y}{x+z}=3y-1;\frac{z}{x+y}=5x-1$

0 replies

giangtruong13

an hour ago

0 replies

inequality

revol_ufiaw 3

N 2 hours ago by MS_asdfgzxcvb

Prove that that for any real $x \ge 0$ and natural number $n$ ,
$x^n (n+1)^{n+1} \le n^n (x+1)^{n+1}.$

3 replies

revol_ufiaw

3 hours ago

MS_asdfgzxcvb

2 hours ago

How to judge a number is prime or not?

mingzhehu 0

3 hours ago

A=（10X1+1）(10X+1），X1，X∈N+
B=（10 X1+3）（10X+7），X∈N，X1∈N
C=(10 X1+9)(10X+9), X∈N，X1∈N
D=(10 X1+1)(10X+3), X1∈N+，X∈N
E=（10 X1+7）(10X+9），X∈N，X1∈N
F=（10 X1+1）（10X+7），X1∈N+，X∈N
G=（10 X1+3）（10X+9），X∈N，X1∈N
H=（10 X1+1）10X+9），X1∈N+，X∈N
I=（10 X1+3）（10X+3），X1∈N，X∈N
J=( 10X1+7)(10X+7),X∈N，X1∈N

For any natural number P∈{P=10N+1，n∈N}，make P=A or B or C
If P can make the roots of function group(ABC) without any root group completely made up of integer, P will be a prime
For any natural number P∈{P=10N+3，n∈N}，make P=D or E
If P can make the roots of function group(DE) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+7，n∈N}，make P=F or G
If P can make the roots of function group(FG) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+9，n∈N}，make P=H or I or J
If P can make the roots of function group(GIJ) without any root group completely made up
of integer, P will be a prime

0 replies

mingzhehu

3 hours ago

0 replies

What is an isogonal conjugate and why is it useful?

EaZ_Shadow 6

N 3 hours ago by maxamc

What is an isogonal conjugate and why is it useful? People use them in Olympiad geometry proofs but I don’t understand why and what is the purpose, as it complicates me because of me not understanding it.

6 replies

EaZ_Shadow

Dec 28, 2024

maxamc

3 hours ago

No more topics!

truncated pyramids in a right prism with square base (2007 AMQ Concours p7)

parmenides51 3

N Mar 5, 2021 by mop

Mr. A. Bajour makes lampshades in the shape of a truncated pyramid with a square base (see the first figure). The isosceles trapezoids forming the surface of its lampshades have the dimensions, in cm, given in the second figure. Each shade should be shipped in a box in the shape of a right prism with a square base that is as small as possible. What is the total area of such a box?

Note: the pyramid must be placed in the box so that its base touches the bottom of the box.

IMAGE

3 replies

parmenides51

Mar 5, 2021

mop

Mar 5, 2021

truncated pyramids in a right prism with square base (2007 AMQ Concours p7)

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parmenides51

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parmenides51

#1 h Mar 5, 2021, 8:55 AM

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parmenides51

#2 h Mar 5, 2021, 8:55 AM

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natmath

8219 posts

natmath

#3 h Mar 5, 2021, 4:18 PM

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The horizontal projection of the 19 edge has length $\frac{30\sqrt{2}-18\sqrt{2}}{2}=6\sqrt{2}$ .
So the altitude is $\sqrt{19^2-(6\sqrt{2})^2}=17$
The volume of the box is $30\cdot 30\cdot 17=15300$