Problem Bank

Real numbers $x_1, x_2, \dots, x_{2018}$ satisfy

Prove that $x_{1009}\…

Let $A$ and $B$ be the common vertices of the ellipse $\dfrac{x^2}4+\dfrac{y^2}3=1$ and the hyperbola $\dfrac{x^2}4-\dfrac{y^2}3=1$. Let $M$ be a point on the hyperbola; the lines …

In $\triangle ABC$, the sides opposite $A, B, C$ are $a, b, c$. A point $M$ lies on side $AB$, and a point $P$ on line $CM$ satisfies

Let $f(x) = (a - x)\mathrm{e}^x \ln x$, defined for $x > 0$.

1. If $f(x) \leqslant 0$ for all $x > 0$, find $a$. 2. For this value of $a$, prove that $f(x) < x^a$ for all $x > 0$.

A point $P$ moves along the line $l \colon 2x + y - 2 = 0$. Two mutually perpendicular lines $PA, PB$ through $P$ meet the $x$-axis at $A$ and the $y$-axis at $B$; $M$ is the midpo…

You have $20$ grams of sugar water at $15\%$ sugar concentration and $15$ grams of sugar water at $40\%$ concentration, plus an unlimited supply of pure sugar and pure water. You w…

The line $\ell$ meets the circle $O: x^2 + y^2 = 4$ at $P, Q$. Let $A(2, 2)$. If

find the maximum length of the chord $PQ$.

In a right triangle $ABC$ with the right angle at $B$, $AB=1$ and $BC=2$. A point $D$ moves along the hypotenuse $AC$. The triangle $ABD$ is folded along $BD$ to a triangle $A_1BD$…

Let $P$ be a point of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) in the first quadrant, and $F_1, F_2$ the left and right foci. The bisector of $\angl…

Let $f(x) = ax^3 + bx^2 + cx + d$ with $a, b, c, d \in \mathbb{R}$ and $a \neq 0$. If

find the range of possible values of $f(1) + f(5)$.

In a triangular frustum $ABC\text{-}DEF$, $AB\perp AC$, $AB=2DE=2$, $AC=2\sqrt 2$, $CF=2$, and $CF$ is perpendicular to the plane $ABC$. Let $P$, $Q$, $R$ be the midpoints of $AC$,…

In a pyramid $P\text{-}ABCD$, $AB=AD=\sqrt{10}$, $CB=CD=5$, $\angle BAD=90^\circ$, $PB=4$, $PC=3$. A point $Q$ inside $\triangle PBC$ is such that the pyramid $Q\text{-}ABCD$ and t…