Problem Bank

An arithmetic sequence $\{a_n\}$ has its first term equal to its common difference, and its partial sums $S_n$ satisfy $S_{10} = 55$. A sequence $\{b_n\}$ satisfies $b_1 = 1$ and, …

Real numbers $a, b, c, d$ satisfy $a \geqslant b \geqslant d > 0$ and $a + b - 3c - 3d \leqslant 0$. Find the minimum value of

Points $T, M, N$ are three distinct points on the circle $C: (x-1)^2 + y^2 = 4$, and

for positive reals …

As shown in the figure, $\triangle ABC$ has a right angle at $C$, and two squares are inscribed in it. Square $CPQR$ h…

Prove that infinitely many positive integers $n$ admit an arrangement of $1, 2, \dots, 3n$ into an $3 \times n$ table

A regular tetrahedron $ABCD$ has edge length $1$. The plane $\alpha$ is parallel to edge $AB$, and $E$, $F$ are the midpoints of edges $AD$ and $BC$. As the tetrahedron makes a ful…

A ring-shaped flower bed is divided into $5$ regions, as shown in the figure. Each region is to be planted with one kind of …

In triangle $QRS$, the bisector of angle $QRS$ meets $QS$ at $T$. Let $QT = m$ and $TS = n$, where $m, n$ are positive integers with $n > m$. Suppose the perimeter $p$ of triangle …

Let $[x]$ and $\{x\}$ denote the integer and fractional parts of $x$.

1. Find the number of zeros of $f(x)=\sin^2[x]+\sin^2\{x\}-1$ on $[0,100]$. 2. Find the number of zeros of $g…

Let $A_1,A_2$ be the left and right vertices of the ellipse $C_1:\dfrac{x^2}3+\dfrac{y^2}{b^2}=1$ ($0<b<\sqrt 3$). Let $M$ be a point on $C_1$ and $N$ a point on the hyperbola $C_2…

Real numbers $a, b, c$ satisfy

Arrange $a, b, c$ in increasing order, with proof.

For all reals $m, n$ with $0 < n < m < \dfrac{1}{a}$, the inequality

always holds. Find the minimum value of the real number…