Problem Bank

In trapezoid $ABCD$ (with $AD \parallel BC$), points $E$ and $F$ lie on leg $AB$ with $AE = BF$. Segments $CE$ and $DF…

In a cube $ABCD\text{-}A_1B_1C_1D_1$ of edge length $2$, let $E$ be the midpoint of edge $AA_1$. A point $P$ moves over the surface of the cube subject to the condition $\overright…

In a tetrahedron $D\text{-}ABC$, $AB = 2$ and $\overrightarrow{AC}\cdot\overrightarrow{BD} = -3$. With $AD = a$, $BC = b$, $CD = c$, find the minimum value of

Let $E, F$ be the trisection points of the hypotenuse $AB$ of an isosceles right triangle $ABC$. Find $\tan\angle ECF$.

Consider the line $l \colon kx - y - k + 1 = 0$ and

The line $l$ meets the grap…

The hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a>0$, $b>0$) has left and right foci $F_1$, $F_2$. The parabola $y^2=2px$ ($p>0$) has focus $F_2$, and meets the hyperbola at …

Let $a_1, a_2, \ldots, a_n > 0$ be real numbers. Prove that

where indices…

The Riemann function is defined on $[0, 1]$ by

The function

has a local minimum, and for every value of $b$ for which this happens, that local minimum is positive. Find the minimum poss…

As shown in the figure, the right prism $ABC\text{-}A_1B_1C_1$ (its lateral edges perpendicular to the base)…

An ellipse centered at the origin $O$, with foci on the $x$-axis and eccentricity $\dfrac{\sqrt 3}2$, passes through the point $\left(\sqrt 2,\dfrac{\sqrt 2}2\right)$. A line $l$ n…

Evaluate