Found problems: 876
2015 IMC, 7
Compute
$$ \lim_{A\to+\infty}\frac1A\int_1^A
A^{\frac1x}\, dx . $$
Proposed by Jan Šustek, University of Ostrava
2007 Putnam, 6
For each positive integer $ n,$ let $ f(n)$ be the number of ways to make $ n!$ cents using an unordered collection of coins, each worth $ k!$ cents for some $ k,\ 1\le k\le n.$ Prove that for some constant $ C,$ independent of $ n,$
\[ n^{n^2/2\minus{}Cn}e^{\minus{}n^2/4}\le f(n)\le n^{n^2/2\plus{}Cn}e^{\minus{}n^2/4}.\]
1984 Miklós Schweitzer, 1
[b]1.[/b] Let $k$ be an arbitrary cardinality. Show that there exists a tournament $T_k = (V_n , E_n)$ such that for any coloring $f: E_n \to k$ of the edge set $E_n$, there are three different vertices $x_0 , x_1 , x_2 \in V_n$ such that
$x_0 x_1 , x_1 x_2 , x_2 x_0 \in E_n$
and
$\left | \{ f(x_0 x_1), f(x_1 x_2), f(x_2 x_0)\} \right |\leq 2$
(A [i]tounament[/i] is a directed graph such that for any vertices $x, y \in V_n, x \neq y$ exactly one of the relations $xy \in E_n$ holds.) ([b]C.19[/b]) [A. Hajnal]
1953 Miklós Schweitzer, 3
[b]3.[/b] Denoting by $E$ the class of trigonometric polynomials of the form $f(x)=c_{0}+c_{1}cos(x)+\dots +c_{n} cos(nx)$, where $c_{0} \geq c_{1} \geq \dots \geq c_{n}>0$, prove that
$(1-\frac{2}{\pi})\frac{1}{n+1}\leq min_{{f\epsilon E}}( \frac{max_{\frac{\pi}{2}\leq x\leq \pi} \left | f(x) \right |}{max_{0\leq x\leq 2\pi} \left | f(x) \right |})\leq (\frac{1}{2}+\frac{1}{\sqrt{2}})\frac{1}{n+1}$.
[b](S. 24)[/b]
2023 SG Originals, Q1
Two straight lines divide a square of side length $1$ into four regions. Show that at least one of the regions has a perimeter greater than or equal to $2$.
[i]Proposed by Dylan Toh[/i]
2023 Olimphíada, 4
We all know the Fibonacci sequence. However, a slightly less known sequence is the $k$-bonacci sequence. In it, we have $F_1^{(k)} = F_2^{(k)} = \cdots = F_{k-1}^{(k)} = 0, F_k^{(k)} = 1$ and $$F^{(k)}_{n+k} = F^{(k)}_{n+k-1} + F^{(k)}_{n+k-2} + \cdots + F^{(k)}_n,$$for all $n \geq 1$. Find all positive integers $k$ for which there exists a constant $N$ such that $$F^{(k)}_{n-1}F^{(k)}_{n+1} - (F ^{(k)}_n)^2 = (-1)^n$$ for every positive integer $n \geq N$.
ICMC 5, 1
Let $S$ be a set of $2022$ lines in the plane, no two parallel, no three concurrent. $S$ divides the plane into finite regions and infinite regions. Is it possible for all the finite regions to have integer area?
[i]Proposed by Tony Wang[/i]
2022 District Olympiad, P2
Let $A,B\in\mathcal{M}_3(\mathbb{R})$ de matrices such that $A^2+B^2=O_3.$ Prove that $\det(aA+bB)=0$ for any real numbers $a$ and $b.$
1997 Putnam, 6
The dissection of the $3-4-5$ triangle shown below has diameter $5/2$.
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(23cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = 1.42, xmax = 24.42, ymin = 3.8, ymax = 15.54; /* image dimensions */
Label laxis; laxis.p = fontsize(10);
xaxis(xmin, xmax,defaultpen+black, Ticks(laxis, Step = 1, Size = 2, NoZero), Arrows(6), above = true);
yaxis(ymin, ymax,defaultpen+black, Ticks(laxis, Step = 1, Size = 2, NoZero), Arrows(6), above = true); /* draws axes; NoZero hides '0' label */
/* draw figures */
draw((9.44,8.52)--(12.44,8.52));
draw((9.44,12.52)--(9.44,8.52));
draw((9.44,12.52)--(12.44,8.52));
draw((9.44,10.52)--(10.94,10.52));
draw((10.94,10.52)--(10.94,8.52));
draw((9.44,8.52)--(10.94,10.52));
/* dots and labels */
dot((9.44,8.52),dotstyle);
label("$A$", (9.52,8.64), NE * labelscalefactor);
dot((12.44,8.52),dotstyle);
label("$B$", (12.52,8.64), NE * labelscalefactor);
dot((9.44,12.52),dotstyle);
label("$C$", (9.52,12.64), NE * labelscalefactor);
dot((10.94,8.52),dotstyle);
label("$D$", (11.02,8.64), NE * labelscalefactor);
dot((9.44,10.52),dotstyle);
label("$E$", (9.52,10.64), NE * labelscalefactor);
dot((10.94,10.52),dotstyle);
label("$F$", (11.02,10.64), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy]
Find the least diameter of this triangle into four parts. (The diameter of a dissection is the least upper bound of the distances between pairs of points belonging to the same part.)
2001 Putnam, 2
For each $k$, $\mathcal{C}_k$ is biased so that, when tossed, it has probability $\tfrac{1}{(2k+1)}$ of falling heads. If the $n$ coins are tossed, what is the probability that the number of heads is odd? Express the answer as a rational function $n$.
1994 Putnam, 3
Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f^{\prime}(x) > f(x)$ for all $x,$ there is some number $N$ such that $f(x) > e^{kx}$ for all $x > N.$
2019 Simon Marais Mathematical Competition, B3
Let $G$ be a finite simple graph and let $k$ be the largest number of vertices of any clique in $G$. Suppose that we label each vertex of $G$ with a non-negative real number, so that the sum of all such labels is $1$. Define the [i]value of an edge[/i] to be the product of the labels of the two vertices at its ends. Define the [i]value of a labelling[/i] to be the sum of values of the edges.
Prove that the maximum possible value of a labelling of $G$ is $\frac{k-1}{2k}$.
(A [i]finite simple graph[/i] is a graph with finitely many vertices, in which each edge connects two distinct vertices and no two edges connect the same two vertices. A [i]clique[/i] in a graph is a set of vertices in which any two are connected by an edge.)
1985 Miklós Schweitzer, 8
Let $\frac{2}{\sqrt5+1}\leq p < 1$, and let the real sequence $\{ a_n \}$ have the following property: for every sequence $\{ e_n \}$ of $0$'s and $\pm 1$'s for which $\sum_{n=1}^\infty e_np^n=0$, we also have $\sum_{n=1}^\infty e_na_n=0$. Prove that there is a number $c$ such that $a_n=cp^n$ for all $n$. [Z. Daroczy, I. Katai]
ICMC 6, 2
Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f'(x) > f(x)>0$ for all real numbers $x$. Show that $f(8) > 2022f(0)$.
[i]Proposed by Ethan Tan[/i]
ICMC 6, 4
Do there exist infinitely many positive integers $m$ such that the sum of the positive divisors of $m$ (including $m$ itself) is a perfect square?
[i]Proposed by Dylan Toh[/i]
2023 SG Originals, Q5
A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$?
[i]Proposed by Dylan Toh[/i]
2009 Putnam, B3
Call a subset $ S$ of $ \{1,2,\dots,n\}$ [i]mediocre[/i] if it has the following property: Whenever $ a$ and $ b$ are elements of $ S$ whose average is an integer, that average is also an element of $ S.$ Let $ A(n)$ be the number of mediocre subsets of $ \{1,2,\dots,n\}.$ [For instance, every subset of $ \{1,2,3\}$ except $ \{1,3\}$ is mediocre, so $ A(3)\equal{}7.$] Find all positive integers $ n$ such that $ A(n\plus{}2)\minus{}2A(n\plus{}1)\plus{}A(n)\equal{}1.$
2018 Brazil Undergrad MO, 8
A student will take an exam in which they have to solve three chosen problems by chance of a list of $10$ possible problems. It will be approved if it correctly resolves two problems. Considering that the student can solve five of the problems on the list and not know how to solve others, how likely is he to pass the exam?
2019 SEEMOUS, 3
Let $A,B$ be $n\times n$ matrices, $n\geq 2$, and $B^2=B$. Prove that:
$$\text{rank}\,(AB-BA)\leq \text{rank}\,(AB+BA)$$
2021 Alibaba Global Math Competition, 18
Let $p$ be an odd prime number, and let $m \ge 0$ and $N \ge 1$ be integers. Let $\Lambda$ be a free $\mathbb{Z}/p^N\mathbb{Z}$-module of rank $2m+1$, equipped with a perfect symmetric $\mathbb{Z}/p^N\mathbb{Z}$-bilinear form
\[(\, ,\,): \Lambda \times \Lambda \to \mathbb{Z}/p^N\mathbb{Z}.\]
Here ``perfect'' means that the induced map
\[\Lambda \to \text{Hom}_{\mathbb{Z}/p^N\mathbb{Z}}(\Lambda, \mathbb{Z}/p^N\mathbb{Z}), \quad x \mapsto (x,\cdot)\]
is an isomorphism. Find the cardinality of the set
\[\{x \in \Lambda: (x,x)=0\},\]
expressed in terms of $p,m,N$.
2017 IMC, 7
Let $p(x)$ be a nonconstant polynomial with real coefficients. For every positive integer~$n$, let $$q_n(x) = (x+1)^np(x)+x^n p(x+1) .$$
Prove that there are only finitely many numbers $n$ such that all roots of $q_n(x)$ are real.
2014 IMC, 4
Let $n>6$ be a perfect number, and let $n=p_1^{e_1}\cdot\cdot\cdot p_k^{e_k}$ be its prime factorisation with $1<p_1<\dots <p_k$. Prove that $e_1$ is an even number.
A number $n$ is [i]perfect[/i] if $s(n)=2n$, where $s(n)$ is the sum of the divisors of $n$.
(Proposed by Javier Rodrigo, Universidad Pontificia Comillas)
2016 Korea USCM, 2
Suppose $\{a_n\}$ is a decreasing sequence of reals and $\lim\limits_{n\to\infty} a_n = 0$. If $S_{2^k} - 2^k a_{2^k} \leq 1$ for any positive integer $k$, show that
$$\sum_{n=1}^{\infty} a_n \leq 1$$
(At here, $S_m = \sum_{n=1}^m a_n$ is a partial sum of $\{a_n\}$.)
2024 CIIM, 4
Given the points $O = (0, 0)$ and $A = (2024, -2024)$ in the plane. For any positive integer $n$, Damian draws all the points with integer coordinates $B_{i,j} = (i, j)$ with $0 \leq i, j \leq n$ and calculates the area of each triangle $OAB_{i,j}$. Let $S(n)$ denote the sum of the $(n+1)^2$ areas calculated above. Find the following limit:
\[
\lim_{n \to \infty} \frac{S(n)}{n^3}.
\]
MIPT student olimpiad spring 2024, 4
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.