Found problems: 876
2019 IMC, 3
Let $f:(-1,1)\to \mathbb{R}$ be a twice differentiable function such that
$$2f’(x)+xf''(x)\geqslant 1 \quad \text{ for } x\in (-1,1).$$
Prove that
$$\int_{-1}^{1}xf(x)dx\geqslant \frac{1}{3}.$$
[i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karim Rakhimov, Scuola Normale Superiore and National University of Uzbekistan[/i]
2019 Korea USCM, 3
Two vector fields $\mathbf{F},\mathbf{G}$ are defined on a three dimensional region $W=\{(x,y,z)\in\mathbb{R}^3 : x^2+y^2\leq 1, |z|\leq 1\}$.
$$\mathbf{F}(x,y,z) = (\sin xy, \sin yz, 0),\quad \mathbf{G} (x,y,z) = (e^{x^2+y^2+z^2}, \cos xz, 0)$$
Evaluate the following integral.
\[\iiint_{W} (\mathbf{G}\cdot \text{curl}(\mathbf{F}) - \mathbf{F}\cdot \text{curl}(\mathbf{G})) dV\]
2009 Putnam, B6
Prove that for every positive integer $ n,$ there is a sequence of integers $ a_0,a_1,\dots,a_{2009}$ with $ a_0\equal{}0$ and $ a_{2009}\equal{}n$ such that each term after $ a_0$ is either an earlier term plus $ 2^k$ for some nonnnegative integer $ k,$ or of the form $ b\mod{c}$ for some earlier positive terms $ b$ and $ c.$ [Here $ b\mod{c}$ denotes the remainder when $ b$ is divided by $ c,$ so $ 0\le(b\mod{c})<c.$]
2000 Miklós Schweitzer, 1
Prove that there exists a function $f\colon [\omega_1]^2 \rightarrow \omega _1$ such that
(i) $f(\alpha, \beta)< \mathrm{min}(\alpha, \beta)$ whenever $\mathrm{min}(\alpha,\beta)>0$; and
(ii) if $\alpha_0<\alpha_1<\ldots<\alpha_i<\ldots<\omega_1$ then $\sup\left\{ a_i \colon i<\omega \right\} =\sup \left\{ f(\alpha_i, \alpha_j)\colon i,j<\omega\right\}$.
MIPT student olimpiad spring 2023, 2
Let $A=a_{ij}$ is simetrical real matrix. Prove that :
$\sum_i e^{a_{ii}} \leq tr (e^A)$
2018 Brazil Undergrad MO, 1
An equilateral triangle is cut as shown in figure 1 and the parts are used to form figure 2. What is the shape of figure 2?
2008 Miklós Schweitzer, 1
Let $H \subset P(X)$ be a system of subsets of $X$ and $\kappa > 0$ be a cardinal number such that every $x \in X$ is contained in less than $\kappa$ members of $H$. Prove that there exists an $f \colon X \rightarrow \kappa$ coloring, such that every nonempty $A \in H$ has a “unique” point, that is, an element $x \in A$ such that $f(x) \neq f(y)$ for all $x \neq y \in A$.
(translated by Miklós Maróti)
1959 Miklós Schweitzer, 3
[b]3.[/b]Let $G$ be an arbitrary group, $H_1,\dots ,H_n$ some (not necessarily distinet) subgroup of $G$ and $g_1, \dots , g_n$ elements of $G$ such that each element of $G$ belongs at least to one of the right cosets $H_1 g_1, \dots , H_n g_n$. Show that if, for any $k$, the set-union of the cosets $H_i g_i (i=1, \dots , k-1, k+1, \dots , n)$ differs from $G$, then every $H_k (k=1, \dots , n)$ is of finite index in $G$. [b](A. 15)[/b]
2019 Brazil Undergrad MO, 3
Let $a,b,c$ be constants and $a,b,c$ are positive real numbers. Prove that the equations
$2x+y+z=\sqrt{c^2+z^2}+\sqrt{c^2+y^2}$
$x+2y+z=\sqrt{b^2+x^2}+\sqrt{b^2+z^2}$
$x+y+2z=\sqrt{a^2+x^2}+\sqrt{a^2+y^2}$
have exactly one real solution $(x,y,z)$ with $x,y,z \geq 0$.
2012 Miklós Schweitzer, 1
Is there any real number $\alpha$ for which there exist two functions $f,g: \mathbb{N} \to \mathbb{N}$ such that
$$\alpha=\lim_{n \to \infty} \frac{f(n)}{g(n)},$$
but the function which associates to $n$ the $n$-th decimal digit of $\alpha$ is not recursive?
2015 IMC, 1
For any integer $n\ge 2$ and two $n\times n$ matrices with real
entries $A,\; B$ that satisfy the equation
$$A^{-1}+B^{-1}=(A+B)^{-1}\;$$
prove that $\det (A)=\det(B)$.
Does the same conclusion follow for matrices with complex entries?
(Proposed by Zbigniew Skoczylas, Wroclaw University of Technology)
2018 VTRMC, 4
Let $m, n$ be integers such that $n \geq m \geq 1$. Prove that $\frac{\text{gcd} (m,n)}{n} \binom{n}{m}$ is an integer. Here $\text{gcd}$ denotes greatest common divisor and $\binom{n}{m} = \frac{n!}{m!(n-m)!}$ denotes the binomial coefficient.
2000 Miklós Schweitzer, 10
Joe generates 4 independent random numbers in $(0,1)$ according to the uniform distribution. He shows one the numbers to Bill, who has to guess whether the number shown is one of the extremal numbers (that is, the smallest or the greatest) of the four numbers or not. Can Joe have a deterministic strategy such that no matter what Bill's method is, the probability of the right guess of Bill is at most $\frac12$?
1986 Miklós Schweitzer, 5
Prove that existence of a constant $c$ with the following property: for every composite integer $n$, there exists a group whose order is divisible by $n$ and is less than $n^c$, and that contains no element of order $n$. [P. P. Palfy]
1961 Miklós Schweitzer, 7
[b]7.[/b] For the differential equation
$ \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}= 2\frac{\partial^2 u}{\partial x \partial y} $
find all solutions of the form $u(x,y)=f(x)g(y)$. [b](R. 14)[/b]
2021 Alibaba Global Math Competition, 14
Let $f$ be a smooth function on $\mathbb{R}^n$, denote by $G_f=\{(x,f(x)) \in \mathbb{R}^{n+1}: x \in \mathbb{R}^n\}$. Let $g$ be the restriction of the Euclidean metric on $G_f$.
(1) Prove that $g$ is a complete metric.
(2) If there exists $\Lambda>0$, such that $-\Lambda I_n \le \text{Hess}(f) \le \Lambda I_n$, where $I_n$ is the unit matrix of order $n$, and $\text{Hess}8f)$ is the Hessian matrix of $f$, then the injectivity radius of $(G_f,g)$ is at least $\frac{\pi}{2\Lambda}$.
2018 Brazil Undergrad MO, 15
A real number $ to $ is randomly and uniformly chosen from the $ [- 3,4] $ interval. What is the probability that all roots of the polynomial $ x ^ 3 + ax ^ 2 + ax + 1 $ are real?
1971 Putnam, A3
The three vertices of a triangle of sides $a,b,$ and $c$ are lattice points and lie on a circle of radius $R$. Show that $abc \geq 2R.$ (Lattice points are points in Euclidean plane with integral coordinates.)
2002 Putnam, 6
Fix an integer $ b \geq 2$. Let $ f(1) \equal{} 1$, $ f(2) \equal{} 2$, and for each $ n \geq 3$, define $ f(n) \equal{} n f(d)$, where $ d$ is the number of base-$ b$ digits of $ n$. For which values of $ b$ does
\[ \sum_{n\equal{}1}^\infty \frac{1}{f(n)}
\]
converge?
1999 Putnam, 5
For an integer $n\geq 3$, let $\theta=2\pi/n$. Evaluate the determinant of the $n\times n$ matrix $I+A$, where $I$ is the $n\times n$ identity matrix and $A=(a_{jk})$ has entries $a_{jk}=\cos(j\theta+k\theta)$ for all $j,k$.
1999 Putnam, 4
Sum the series \[\sum_{m=1}^\infty\sum_{n=1}^\infty\dfrac{m^2n}{3^m(n3^m+m3^n)}.\]
1953 Miklós Schweitzer, 6
[b]6.[/b] Let $H_{n}(x)$ be the [i]n[/i]th Hermite polynomial. Find
$ \lim_{n \to \infty } (\frac{y}{2n})^{n} H_{n}(\frac{n}{y})$
For an arbitrary real y. [b](S.5)[/b]
$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{{-x^2}}\right)$
2022 IMC, 3
Let $p$ be a prime number. A flea is staying at point $0$ of the real line. At each minute,
the flea has three possibilities: to stay at its position, or to move by $1$ to the left or to the right.
After $p-1$ minutes, it wants to be at $0$ again. Denote by $f(p)$ the number of its strategies to do this
(for example, $f(3) = 3$: it may either stay at $0$ for the entire time, or go to the left and then to the
right, or go to the right and then to the left). Find $f(p)$ modulo $p$.
2022 Miklós Schweitzer, 4
Consider the integral $$\int_{-1}^1 x^nf(x) \; dx$$ for every $n$-th degree polynomial $f$ with integer coefficients. Let $\alpha_n$ denote the smallest positive real number that such an integral can give. Determine the limit value $$\lim_{n\to \infty} \frac{\log \alpha_n}n.$$
2014 IMC, 4
We say that a subset of $\mathbb{R}^n$ is $k$-[i]almost contained[/i] by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points $k$-[i]generic[/i] if there is no hyperplane that $k$-almost contains the set. For each pair of positive integers $(k, n)$, find the minimal number of $d(k, n)$ such that every finite $k$-generic set in $\mathbb{R}^n$ contains a $k$-generic subset with at most $d(k, n)$ elements.
(Proposed by Shachar Carmeli, Weizmann Inst. and Lev Radzivilovsky, Tel Aviv Univ.)