IMC Problem Lab

International Mathematics Competition · 1994–present. Every problem with official solutions, difficulty from the published results, and a worksheet builder. How this works → About

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Hall of pain: least-solved problems

IMC 2012 · Day 2 · P10killeravg 0.0/10 · solved 0% · near-0 100% · disc 0.16

Let $c \ge 1$ be a real number. Let $G$ be an abelian group and let $A \subset G$ be a finite set satisfying $|A + A| \le c |A|$ , where $X + Y := \{ x + y \mid x \in X,\ y \in Y \}$ and $|Z|$ denotes the cardinality of $Z$ . Prove that …

combinatorics group theory

IMC 2002 · Day 2 · P12killeravg 0.1/10 · solved 0% · near-0 99% · disc 0.16

Let $f : \mathbb{R}^n \to \mathbb{R}$ be a convex function whose gradient $\nabla f = \left( \frac{\partial f}{\partial x_1}, \dots, \frac{\partial f}{\partial x_n} \right)$ exists at every point of $\mathbb{R}^n$ and satisfies the …

real analysis inequalities

IMC 2009 · Day 2 · P10killeravg 0.0/10 · solved 0% · near-0 99% · disc 0.19

Let $M$ be the vector space of $m \times p$ real matrices. For a vector subspace $S \subseteq M$ , denote by $\delta(S)$ the dimension of the vector space generated by all columns of all matrices in $S$ . Say that a vector subspace …

linear algebra

IMC 2007 · Day 2 · P12killeravg 0.1/10 · solved 0% · near-0 99% · disc 0.36

Let $f \ne 0$ be a polynomial with real coefficients. Define the sequence $f_0, f_1, f_2, \dots$ of polynomials by $f_0 = f$ and $f_{n+1} = f_n + f_n'$ for every $n \ge 0$ . Prove that there exists a number $N$ such that for every …

polynomials real analysis

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