Problem Statement Square $ABCD$ has sides of length 14. A circle is drawn through $A$ and $D$ so that it is tangent to $BC$, as shown. What is the radius of the circle? Problem Link Solutio...

Problem Statement In the diagram, the equation of the line $AD$ is $y = \sqrt{3}(x-1)$. $BD$ bisects $\angle ADC$. If the coordinates of $B$ are $(p, q)$, what is the value of $q$? Problem Lin...

Problem Statement (a) Determine all real numbers $x$ such that $(x + 3)(x − 6) = −14$ (b) Determine all real numbers $x$ such that $2^{2x} − 3(2^x) − 4 = 0$ (c) Determine all real numbers $x$ su...

Problem Statement The polynomial $f(x)$ satisfies the equation $f(x) - f(x - 2) = (2x - 1)^2 \;$ for all $x$. If $p$ and $q$ are the coefficients of $x^2$ and $x$, respectively, in $f(x)$, then $...

Problem Statement There are real numbers a and b for which the function f has the properties that \(f(x) = ax + b\) for all real numbers \(x\), and \(f(bx + a) = x\) for all real numbers \(x\). W...

Problem Statement The function $f$ has the properties that $f(1) = 6$ and $f(2x + 1) = 3f(x)$ for every integer $x$. What is the value of $f(63)$? Problem Link Solution $$f(1) = 6$$ $$f(2x...

Problem Statement The function $f(x)$ has the property that $f(x + y) = f(x) + f(y) + 2xy \;\;\; \forall \; x \in \mathbb{N}$. If $f(1) = 4$, then the numerical value of $f(8)$ is Problem Link ...

Problem Statement if $f(x) = px + q$ and $f(f(f(x))) = 8x + 21$, and if $p, q \in \mathbb{R}$, then find $p+q$ Problem Link Solution $$f(x) = px + q \; \; \; p, q \in \mathbb{R}$$ $$f(f(f(...