Problem Bank

Plane vectors $\vec a, \vec b, \vec c$ satisfy

Find the range of $\left|\vec a + \vec…

A family of curves $\Gamma_m$ is called triangle-preserving if for every triangle $ABC$, some curve in the family contains three points $P,Q,R$ with $\triangle PQR$ congruent to …

Let $(1+x+x^2)^{1000}=a_0+a_1x+a_2x^2+\cdots+a_{2000}x^{2000}$. Find the value of

Let $f(x) = ax^2 + 2x + 1$. If

find the range of possible values of the real number $a$.

Let $M$ be a point of the midsphere (the sphere tangent to all edges) of a cube $ABCD\text{-}A_1B_1C_1D_1$ of edge length $2$, and let $N$ be a point of the circumcircle of the tri…

Let $A,B,C,D$ be four points on the hyperbola $xy=k$. Let $AD$ meet $BC$ at $P$, let $BD$ meet $AC$ at $Q$, and let $CD$ meet $AB$ at $R$. Let $O$ be the origin. Prove that $O,P,Q,…

Vectors $\vec{a}, \vec{b}$ make an angle of $\dfrac{\pi}{3}$ and $\left|\vec{a} - \vec{b}\right| = 5$; the vectors $\vec{c} - \vec{a}$ and $\vec{c} - \vec{b}$ make an angle of $\df…

Through the left focus $F_1$ of the ellipse $W:\dfrac{x^2}{2}+y^2=1$, a line $l_1$ meets the ellipse at points $A$ and $B$, where $A(0,1)$. A second line $l_2$ through $F_1$ meets …

Let $f(x)=\sqrt{2x^4-18x^2+12x+68}+x^2-x+1$. Determine, with proof, the minimum value of $f(x)$ and the number of points at which it is attained.

Let $f(x)$ be defined on an interval $I$, and suppose that for all $x_1, x_2 \in I$ and all $t$ with $0 \leqslant t \leqslant 1$,

In a tetrahedron $S\text{-}ABC$, the base $\triangle ABC$ is equilateral. Let $\alpha,\beta,\gamma$ be the plane angles of the dihedral angles along $AB$, $BC$, $CA$ (i.e., $S\text…

Let $f(x) = x\ln x - ax^2 + a$ with $a > 0$.

1. If $f(x) < 0$ for all $x > 1$, find the range of possible values of $a$. 2. For integers $n \geqslant 2$, prove that