Problem Bank

The parabola $y=ax^2$ passes through the point $P(-1,1)$. Through the point $Q\left(-\dfrac 12,0\right)$, a line $l$ of positive slope meets the parabola at two points $M$ and $N$,…

Let $f_1(x)=f(x)=x(x-1)$ and $f_n(x)=f(f_{n-1}(x))$ for $n\geqslant 2$. Prove:

1. For $x>2$, $f_n(x)$ has no zeros. 2. For $1\leqslant x\leqslant 2$, $f_n(x)$ has at least $n$ zer…

In a round-robin tournament, $2023$ teams each play every other team once. A win is worth $3$ points, a loss $0$, and a draw gives each team $1$ point. After the tournament the tea…

Consider the sequence $1,1,2,1,2,4,1,2,4,8,1,2,4,8,16,\dots$, where the first term is $2^0$, the next two terms are $2^0,2^1$, the next three are $2^0,2^1,2^2$, and so on. Find the…

In triangle $ABC$ shown, $AD$ is the bisector of angle $A$, with $D$ on $BC$, and $AB<AC$. Point $E$ lies on segment $…

In the coordinate plane, let $A(3, 0)$ and $B(0, t)$ with $t > 0$. If there is no point $P$ in the plane simultaneously satisfying

Let $x_1, x_2, x_3$ be the roots of $x^3 - 17x - 18 = 0$, with $-4 < x_1 < -3$ and $4 < x_3 < 5$.

1. Find the integer part of $x_2$. 2. Evaluate $\arctan x_1 + \arctan x_2 + \arct…

Let $f(x) = \ln x - \dfrac{1}{x}$ and $g(x) = ax + b$.

1. If $h(x) = f(x) - g(x)$ is increasing on $(0, +\infty)$, find the range of possible values of $a$. 2. If the line $y = ax…

A line $\ell$ meets the ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{2} = 1$ at $A$ and $B$, the negative $x$-axis at $C$, and the positive $y$-axis at $D$, with $A, C, D, B$ in order alo…

Real numbers $a, b, c$ satisfy

Find the minimum value of $a^3 + b^3 + c^3$.

A sequence has $a_1 = \dfrac13$ and

1. Prove $a_n < a_{n+1} < 1$ for all $n$. 2. Prove $a_n > \dfrac12 - \dfrac{1}{4n}$ for all $n$.

For positive reals $a, b, c$, prove that