“An equilateral triangle ABC has side length 2. A square,PQRS, is such that P lies on AB, Q lies on BC, and R and S lie on AC as shown. The points P, Q, R, and S move so that P, Q and R always remain on the sides of the triangle and S moves […]

Notice that given a right triangle, the side length of the inscribed square as shown above can be expressed using two side lengths of the right triangle. Look at the following diagram. It’s easy to see $ML:EF=CL:CF$, then $w:EF=(CF-w):CF$. Therefore we have $w=\frac{EF*CF}{EF+CF}$. This result is general. So, similarly, we can have the following equation […]

The CEMC gave 4 solutions. Here I want to discuss a more general method using analytic geometry and trignometry. Let’s place the diagram on a coordinate plane as shown below. Let $k$ be the slope of the line DP. The problem is actually asking for the maximum possible value of $k$ such that the conditions […]

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 […]

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. […]

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 […]

The following is the 24th problem of the University of Waterloo’s Cayley Contest in 2022. I drew a cube using the Geogebra 3D Calculator. The cube can be rotated and panned-and-zoomed, which is helpful for understanding this problem. https://www.geogebra.org/m/ca3dfseb This post is also a test to embed LaTeX in WordPress. Thanks to QuickLaTeX.