Olympiad problems

MIPT student olimpiad spring 2024, 2

Let the matrix $S$ be orthogonal and the matrix $I-S$ be invertible, where I is the identity matrix of the same size as $S$. Find $x^T(I-S)^{-1}x$ Where $x$ is a real unit vector.

MIPT student olimpiad spring 2023, 3

Tags: geometry , 3D geometry , sphere , college contests , MIPT

Prove that if a set $X\subset S^n$ takes up more than half a Riemannian volume of a unit sphere $S^n$, then the set of all possible geodesic segments length less than $\pi$ with endpoints in the set $X$ covers the entire sphere. Geodetic on sphere $S^n$ is a curve lying on some circle of intersection of the sphere $S^n\subset R^{n+1}$ two-dimensional linear subspace $L \subset R^{n+1}$

MIPT student olimpiad spring 2024, 3

Tags: calculus , function , MIPT , college contests

Is it true that if a function $f: R \to R$ is continuous and takes rational values at rational points, then at least at one point it is differentiable?

MIPT student olimpiad spring 2022, 4

Tags: complex analysis , MIPT , college contests

Let us consider sequences of complex numbers that are infinite in both directions $c=(c_k) , k\in Z$ with finite norm $||c||= (\sum_{k \in Z} |c_k|^2)^{1/2}$ Let $T_m-$ this is a shift operation sequences on m ($(T_mc)_k=c_{k-m}$) Prove that: $\lim_{n \to \infty} \frac{\sum_{i=0}^{n-1} T_ic}{n} =0$ (Adding and multiplying a sequence by a number defined component by component)

MIPT student olimpiad spring 2024, 1

Tags: calculus , integration , college contests , MIPT

Find integral: $\int_{x^2+y^2\leq 1}e^xcos(y)dxdy$

MIPT student olimpiad autumn 2024, 3

Tags: function , real analysis , MIPT , college contests

$\exists ? f: R\to R$ continuos function that: $\forall x_0\in R \lim\limits_{x \to x_0} \frac{|f(x)-f(x_0)|}{|x-x_0|}=+\infty$

MIPT student olimpiad spring 2023, 2

Tags: linear algebra , matrix , MIPT , college contests

Let $A=a_{ij}$ is simetrical real matrix. Prove that : $\sum_i e^{a_{ii}} \leq tr (e^A)$

MIPT student olimpiad autumn 2022, 1

Tags: topology , college contests , MIPT

Prove that if a function $f:R \to R$ is bounded and its graph is closed as subset of the $R^2$ plane, then the function f is continuous.

MIPT student olimpiad autumn 2022, 2

Tags: geometry , polynomial , college contests , MIPT

Let $n \geq 3$ be an integer. Find the minimum degree of one algebraic (polynomial) equation that defines the set of vertices of the correct $n$-gon on plane $R^2$.

MIPT student olimpiad spring 2023, 1

Tags: Liniar algebra , vector , MIPT , college contests

In $R^n$ is given $n-1$ vectors, the coordinates of each are zero-sum integers. Prove that the $(n-1)$-dimensional volume of an $(n-1)$-dimensional parallelepiped $P$ stretched by these vectors, is the product of an integer and $\sqrt(n)$.

MIPT student olimpiad autumn 2024, 2

Tags: linear algebra , matrix , college contests , MIPT

$A,B \in M_{2\times 2}(C)$ Prove that: $Tr(AAABBABAABBB)=tr(BBBAABABBAAA)$

MIPT student olimpiad autumn 2024, 1

Tags: advanced fields , college contests , MIPT

$F$* is the multiplicative group of the field $F$. $F$* is of finitely generated. Is it true that $F$* is cyclic? Additional question: (wasn’t at the olympiad) $K$* is the multiplicative group of the field $K$. $L \subseteq $$K$* is a finitely generated subgroup. Is it true that $L$ is cyclic?

MIPT student olimpiad autumn 2024, 4

Tags: college contests , MIPT , analytic geometry , geometric inequality , combinatorial geometry

The ellipsoid $E$ is contained in the simplex $S$, which is located in the unit ball B space $R^n$. Prove that the sum of the principal semi-axes of the ellipsoid $E$ is no more than units.

MIPT student olimpiad spring 2022, 1

Tags: real analysis , function , MIPT , college contests

Sequence of uniformly continuous functions $f_n:R \to R$ uniformly converges to a function $f:R\to R$. Can we say that $f$ is uniformly continuous?

MIPT student olimpiad spring 2022, 2

Tags: geometry , 3D geometry , MIPT , college contests

Prove that every section of the cube $Q = {[-1,1]}^n \subset R^n$ linear k-dimensional subspace $L\subseteq R^n$ has a diameter of at least $2\sqrt k$.

MIPT student olimpiad spring 2024, 4

Tags: induction , High school math , college contests , MIPT

In some finite set of positive numbers, each number is expressed as a linear combination of the rest with rational non-negative coefficients. Prove that the ratio of some two numbers in the set is rational.

MIPT student olimpiad autumn 2022, 4

Tags: geometry , MIPT , college contests

In $R^n$ space is given a finite set of points $X$. It is known that for any subset $Y \subseteq X$ of at most $n+1$ points, there is a unit ball $B_Y$ containing $Y$ and not containing the origin. Prove that there is a unit a ball $B_X$ containing $X$ and not containing the origin.