Consider the net of the cube. Then it’s easy to see we just need to find the area of shaded region(light shaded plus dark shaded) in the following diagram. Notice that in $\Delta ONM, ON=OM=5, NM=4$. We can find the height $OV$. Then $BN=OV-OU=OV-2$. Therefore the area of BNMC=BN*4 can be calculated. Also notice that […]
Month: February 2023
Without loss of generality, we let $a, b, c, d$ be four side lengths of the ABCD as shown above and $d=7$. Then we have the following system of equations. $a+b+c+7=224$ …(1)(by perimeter) $ab+7c=4410$ …(2)(by area) $a^2+b^2=c^2+7^2$ …(3)(by pythagorean) Notice that $S=a^2+b^2+c^2+7^2=2(a^2+b^2)$. This suggests that if we can find the value of $a^2+b^2$, we’re done. […]
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Prerequisite: I assume you already know how to factor polynomials in the form of $ax^2+bx+c$ and $ax^2+bxy+cy^2$. Then it’s easy to factor the polynomials in the form of $Ax^2+Bxy+Cy^2+Dx+Ey+G$, if it is possible to do so. Let’s see an example from CEMC courseware: https://courseware.cemc.uwaterloo.ca/8/assignments/71/9. The link to the CEMC’s solution is :https://courseware.cemc.uwaterloo.ca/8/assignments/71/11. Here I want […]
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