Problem Bank

Let $x, y, z > 0$. Find the maximum value of

As shown in the figure, let $C:\ y^2 = 2x$ be a parabola, and let $l$ be a line through the point $A(2,0)$ that int…

Let $M$ be a set with $6$ elements, and define

Find the nu…

In $\triangle ABC$, $M$ is the midpoint of $BC$, $BM = 2$, and $AM = AB - AC$. Find the maximum possible area of $\triangle ABC$.

An ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) passes through the point $P\left(\dfrac{\sqrt 6}2,\dfrac 12\right)$ and has eccentricity $\dfrac{\sqrt 2}2$. Let $M(2,t)$…

A sequence $\{x_n\}$ satisfies $x_1=1$ and $x_{n+1}=x_n+\lfloor\sqrt{x_n}\rfloor$. Find

Let $O$ be the circumcenter of $\triangle ABC$ with $A = \dfrac{\pi}{3}$, and suppose

Positive reals $a, b$ satisfy

Find the range of $3a + 4b$.

Let $P(x_0, y_0)$ be a point inside the ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$). A line $l$ through $P$ meets the ellipse at $A$ and $B$, and …

Suppose there exist $\alpha, \beta$ with $\alpha$, $\beta$, and $\alpha + \beta$ all acute, satisfying

Find the range of possible values of…

Let $f(x) = e^x - x$.

1. Find the equation of the tangent line to $y = f(x)$ at $(0, f(0))$. 2. Find the extreme values of $f(x)$. 3. Let $g(x) = f(x) + x^2 + x$. Prove that the m…

Let $P(a, b)$ be any point on the ellipse $\dfrac{x^2}{3} + \dfrac{y^2}{2} = 1$. From the origin $O$, two tangent lines to the circle