This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 876

ICMC 6, 6
Tags: ICMC , college contests , real analysis , Convergence , Sequences , floor function
Consider the sequence defined by $a_1 = 2022$ and $a_{n+1} = a_n + e^{-a_n}$ for $n \geq 1$. Prove that there exists a positive real number $r$ for which the sequence $$\{ra_1\}, \{ra_{10}\}, \{ra_{100}\}, . . . $$converges. [i]Note[/i]: $\{x \} = x - \lfloor x \rfloor$ denotes the part of $x$ after the decimal point. [i]Proposed by Ethan Tan[/i]

1984 Miklós Schweitzer, 7
Tags: college contests , algebra , polynomial
[b]7.[/b] Let $V$ be a finite-dimensional subspace of $C[0,1]$ such that every nonzero $f\in V$ attains positive value at some point. Prove that there exists a polynomial $P$ that is strictly positive on $[0,1]$ and orthogonal to $V$, that is, for every $f \in V$, $\int_{0}^{1} f(x)P(x)dx =0$ ([b]F.39[/b]) [A. Pinkus, V. Totik]

2022 VTRMC, 5
Tags: VTRMC , college contests , linear algebra , matrix , eigenvalue
Let $A$ be an invertible $n \times n$ matrix with complex entries. Suppose that for each positive integer $m$, there exists a positive integer $k_m$ and an $n \times n$ invertible matrix $B_m$ such that $A^{k_m m} = B_m A B_m ^{-1}$. Show that all eigenvalues of $A$ are equal to $1$.

ICMC 5, 4
Tags: reflection , Groups , number theory , mod p , college contests , ICMC
Let $p$ be a prime number. Find all subsets $S\subseteq\mathbb Z/p\mathbb Z$ such that 1. if $a,b\in S$, then $ab\in S$, and 2. there exists an $r\in S$ such that for all $a\in S$, we have $r-a\in S\cup\{0\}$. [i]Proposed by Harun Khan[/i]

1985 Miklós Schweitzer, 2
Tags: college contests
[b]2.[/b] Let $S$ be a given finite set of hyperplanes in $\mathbb{R}^n$, and let $O$ be a point. Show that there exists a compact set $K \subseteq \mathbb{R}^n$ containing $O$ such that the orthogonal projection of any point of $K$ onto any hyperplane in $S$ is also in $K$. ([b]G.37[/b]) [Gy. Pap]

2000 Miklós Schweitzer, 7
Tags: college contests , Miklos Schweitzer , function , complex analysis , topology
Let $H(D)$ denote the space of functions holomorphic on the disc $D=\{ z\colon |z|<1 \}$, endowed with the topology of uniform convergence on each compact subset of $D$. If $f(z)=\sum_{n=0}^{\infty} a_nz^n$, then we shall denote $S_n(f,z)=\sum_{k=0}^n a_kz^k$. A function $f\in H(D)$ is called [i]universal[/i] if, for every continuous function $g\colon\partial D\rightarrow \mathbb{C}$ and for every $\varepsilon >0$, there are partial sums $S_{n(j)}(f,z)$ approximating $g$ uniformly on the arc $\{ e^{it} \colon 0\le t\le 2\pi - \varepsilon\}$. Prove that the set of universal functions contains a dense $G_{\delta}$ subset of $H(D)$.

1959 Miklós Schweitzer, 5
Tags: college contests
[b]5.[/b] Denote by $c_n$ the $n$th positive integer which can be represented in the form $c_n = k^{l} (k,l = 2,3, \dots )$. Prove that $\sum_{n=1}^{\infty}\frac{1}{c_n-1}=1$ [b](N. 18)[/b]

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?

2008 Miklós Schweitzer, 8
Tags: Miklos Schweitzer , college contests , topology , function
Let $S$ be the Sierpiński triangle. What can we say about the Hausdorff dimension of the elevation sets $f^{-1}(y)$ for typical continuous real functions defined on $S$? (A property is satisfied for typical continuous real functions on $S$ if the set of functions not having this property is of the first Baire category in the metric space of continuous $S\rightarrow\mathbb{R}$ functions with the supremum norm.) (translated by Miklós Maróti)

ICMC 6, 5
Tags: college contests , real analysis , continuous function , function , ICMC
Let $[0, 1]$ be the set $\{x \in \mathbb{R} : 0 \leq x \leq 1\}$. Does there exist a continuous function $g : [0, 1] \to [0, 1]$ such that no line intersects the graph of $g$ infinitely many times, but for any positive integer $n$ there is a line intersecting $g$ more than $n$ times? [i]Proposed by Ethan Tan[/i]

1962 Putnam, B3
Tags: Putnam , bounded , Convex Set , geometry , college contests , real analysis
Let $S$ be a convex region in the euclidean plane containing the origin. Assume that every ray from the origin has at least one point outside $S$. Prove that $S$ is bounded.

2002 Miklós Schweitzer, 3
Tags: college contests , Miklos Schweitzer , function , Boolean function
Put $\mathbb{A}=\{ \mathrm{yes}, \mathrm{no} \}$. A function $f\colon \mathbb{A}^n\rightarrow \mathbb{A}$ is called a [i]decision function[/i] if (a) the value of the function changes if we change all of its arguments; and (b) the values does not change if we replace any of the arguments by the function value. A function $d\colon \mathbb{A}^n \rightarrow \mathbb{A}$ is called a [i]dictatoric function[/i], if there is an index $i$ such that the value of the function equals its $i$th argument. The [i]democratic function[/i] is the function $m\colon \mathbb{A}^3 \rightarrow \mathbb{A}$ that outputs the majority of its arguments. Prove that any decision function is a composition of dictatoric and democratic functions.

1995 Miklós Schweitzer, 2
Tags: real analysis , calculus , integration , Miklos Schweitzer , college contests
Given $f,g\in L^1[0,1]$ and $\int_0^1 f = \int_0^1 g=1$, prove that there exists an interval I st $\int_I f = \int_I g=\frac12$.

2019 SEEMOUS, 4
Tags: real analysis , college contests
(a) Let $n$ is a positive integer. Calculate $\displaystyle \int_0^1 x^{n-1}\ln x\,dx$.\\ (b) Calculate $\displaystyle \sum_{n=0}^{\infty}(-1)^n\left(\frac{1}{(n+1)^2}-\frac{1}{(n+2)^2}+\frac{1}{(n+3)^2}-\dots \right).$

2018 Miklós Schweitzer, 4
Tags: college contests , Miklos Schweitzer , combinatorics
Let $P$ be a finite set of points in the plane. Assume that the distance between any two points is an integer. Prove that $P$ can be colored by three colors such that the distance between any two points of the same color is an even number.

2006 IMC, 5
Tags: algebra , polynomial , Vieta , IMC , college contests
Show that there are an infinity of integer numbers $m,n$, with $gcd(m,n)=1$ such that the equation $(x+m)^{3}=nx$ has 3 different integer sollutions.

2022 Miklós Schweitzer, 7
Tags: college contests
Point-like figures are placed in the vertices of a regular $k$-angle, and then we walk with them. In one step, a piece jumps over another piece, i.e. its new location will be a mirror image of its current location to the current location of another piece. In the case of $k \geq 3$ integers, it is possible to achieve with a series of such steps that the puppets form the vertices of a regular $k$-angle, different in size from the original?

2000 Putnam, 1
Tags: Putnam , vector , linear algebra , college contests , Putnam matrices , Parity
Let $a_j$, $b_j$, $c_j$ be integers for $1 \le j \le N$. Assume for each $j$, at least one of $a_j$, $b_j$, $c_j$ is odd. Show that there exists integers $r, s, t$ such that $ra_j+sb_j+tc_j$ is odd for at least $\tfrac{4N}{7}$ values of $j$, $1 \le j \le N$.

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 $2\sqrt k$.

2004 IMC, 1
Tags: IMC , college contests
Let $S$ be an infinite set of real numbers such that $|x_1+x_2+\cdots + x_n | \leq 1 $ for all finite subsets $\{x_1,x_2,\ldots,x_n\} \subset S$. Show that $S$ is countable.

2015 Miklos Schweitzer, 9
Tags: real analysis , college contests , complex analysis
For a function ${u}$ defined on ${G \subset \Bbb{C}}$ let us denote by ${Z(u)}$ the neignborhood of unit raduis of the set of roots of ${u}$. Prove that for any compact set ${K \subset G}$ there exists a constant ${C}$ such that if ${u}$ is an arbitrary real harmonic function on ${G}$ which vanishes in a point of ${K}$ then: \[\displaystyle \sup_{z \in K} |u(z)| \leq C \sup_{Z(u)\cap G}|u(z)|.\]

2019 SEEMOUS, 1
Tags: real analysis , college contests
A sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is called "Devin" if for any $f\in C[0,1]$ $$ \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n f(x_i)=\int_0^1 f(x)\,dx $$ Prove that a sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is "Devin" if and only if for any non-negative integer $k$ it holds $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n x_i^k=\frac{1}{k+1}.$$ [b]Remark[/b]. I left intact the text as it was proposed. Devin is a Bulgarian city and SPA resort, where this competition took place.

1954 Miklós Schweitzer, 2
Tags: series , Sequences , college contests
[b]2.[/b] Show that the series $\sum_{n=1}^{\infty}\frac{1}{n}sin(asin(\frac{2n\pi}{N}))e^{bcos(\frac{2n\pi}{N})}$ is convergent for every positive integer N and any real numbers a and b. [b](S. 25)[/b]

2005 Putnam, A1
Tags: Putnam , college contests
Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another. (For example, $23=9+8+6.)$

1977 Putnam, B1
Tags: college contests
Evaluate the infinite product $$\prod_{n=2}^{\infty} \frac{n^3-1}{n^3+1}.$$