Found problems: 876
2000 Putnam, 2
Prove that the expression \[ \dfrac {\text {gcd}(m, n)}{n} \dbinom {n}{m} \] is an integer for all pairs of integers $ n \ge m \ge 1 $.
2018 Putnam, A5
Let $f: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function satisfying $f(0) = 0$, $f(1) = 1$, and $f(x) \ge 0$ for all $x \in \mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x) < 0$.
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.
1996 Putnam, 6
Let $c\ge 0$ be a real number. Give a complete description with proof of the set of all continuous functions $f: \mathbb{R}\to \mathbb{R}$ such that $f(x)=f(x^2+c)$ for all $x\in \mathbb{R}$.
1976 Putnam, 6
Suppose $f(x)$ is a twice continuously differentiable real valued function defined for all real numbers $x$ and satisfying $$|f(x)| \leq 1$$ for all x and $$(f(0))^2+(f'(0))^2=4.$$ Prove that there exists a real number $x_0$ such that $$f(x_0)+f''(x_0)=0.$$
2016 Korea USCM, 4
Suppose a continuous function $f:[-\frac{\pi}{4},\frac{\pi}{4}]\to[-1,1]$ and differentiable on $(-\frac{\pi}{4},\frac{\pi}{4})$. Then, there exists a point $x_0\in (-\frac{\pi}{4},\frac{\pi}{4})$ such that
$$|f'(x_0)|\leq 1+f(x_0)^2$$
2017 IMC, 9
Define the sequence $f_1,f_2,\ldots :[0,1)\to \mathbb{R}$ of continuously differentiable functions by the following recurrence: $$ f_1=1; \qquad \quad f_{n+1}'=f_nf_{n+1} \quad\text{on $(0,1)$}, \quad \text{and}\quad f_{n+1}(0)=1. $$
Show that $\lim\limits_{n\to \infty}f_n(x)$ exists for every $x\in [0,1)$ and determine the limit function.
2000 Putnam, 4
Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$. Show that $f(x)=0$ for $-1\le x \le 1$.
2020 Brazil Undergrad MO, Problem 6
Let $f(x) = 2x^2 + x - 1, f^{0}(x) = x$, and $f^{n+1}(x) = f(f^{n}(x))$ for all real $x>0$ and $n \ge 0$ integer (that is, $f^{n}$ is $f$ iterated $n$ times).
a) Find the number of distinct real roots of the equation $f^{3}(x) = x$
b) Find, for each $n \ge 0$ integer, the number of distinct real solutions of the equation $f^{n}(x) = 0$
1986 Miklós Schweitzer, 8
Let $a_0=0$, $a_1, \ldots, a_k$ and $b_1, \ldots, b_k$ be arbitrary real numbers.
(i) Show that for all sufficiently large $n$ there exist polynomials $p_n$ of degree at most $n$ for which
$$p_n^{(i)} (-1)=a_i,\,\,\,\,\, p_n^{(i)} (1)=b_i,\,\,\,\,\, i=0, 1, \ldots, k$$
and
$$\max_{|x|\leq 1} |p_n (x)|\leq \frac{c}{n^2}\,\,\,\,\,\,\,\,\,\, (*)$$
where the constant $c$ depends only on the numbers $a_i, b_i$.
(ii) Prove that, in general, (*) cannot be replaced by the relation
$$\lim_{n\to\infty} n^2\cdot \max_{|x|\leq 1} |p_n (x)| = 0$$
[J. Szabados]
2018 Miklós Schweitzer, 5
For every positive integer $n$, define
$$f(n)=\sum_{p\mid n}{p^{k_p}},$$where the sum is taken over all positive prime divisors $p$ of $n$, and $k_p$ is the unique integer satisfying $$p^{k_p}\leqslant n<p^{k_p+1}.$$Find$$\limsup_{n\to \infty} \frac{f(n)\log \log n}{n\log n} .$$
1953 Miklós Schweitzer, 4
[b]4.[/b] Show that every closed curve c of length less than $ 2\pi $ on the surface of the unit sphere lies entirely on the surface of some hemisphere of the unit sphere. [b](G. 8)[/b]
2014 IMC, 3
Let $n$ be a positive integer. Show that there are positive real numbers $a_0, a_1, \dots, a_n$ such that for each choice of signs the polynomial
$$\pm a_nx^n\pm a_{n-1}x^{n-1} \pm \dots \pm a_1x \pm a_0$$
has $n$ distinct real roots.
(Proposed by Stephan Neupert, TUM, München)
2002 IMC, 8
200 students participated in a math contest. They had 6 problems to solve. Each problem was correctly solved by at least 120 participants. Prove that there must be 2 participants such that every problem was solved by at least one of these two students.
2018 IMC, 9
Determine all pairs $P(x),Q(x)$ of complex polynomials with leading coefficient $1$ such that $P(x)$ divides $Q(x)^2+1$ and $Q(x)$ divides $P(x)^2+1$.
[i]Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro[/i]
2004 IMC, 2
Let $f,g:[a,b]\to [0,\infty)$ be two continuous and non-decreasing functions such that each $x\in [a,b]$ we have
\[ \int^x_a \sqrt { f(t) }\ dt \leq \int^x_a \sqrt { g(t) }\ dt \ \ \textrm{and}\ \int^b_a \sqrt {f(t)}\ dt = \int^b_a \sqrt { g(t)}\ dt. \]
Prove that
\[ \int^b_a \sqrt { 1+ f(t) }\ dt \geq \int^b_a \sqrt { 1 + g(t) }\ dt. \]
1986 Miklós Schweitzer, 4
Determine all real numbers $x$ for which the following statement is true: the field $\mathbb C$ of complex numbers contains a proper subfield $F$ such that adjoining $x$ to $F$ we get $\mathbb C$. [M. Laczkovich]
1995 Putnam, 4
Suppose we have a necklace of $n$ beads. Each bead is labelled with an integer and the sum of all these labels is $n-1$. Prove that we can cut the necklace to form a string whose consecutive labels $x_1, x_2,\cdots , x_n$ satisfy
\[ \sum_{i=1}^{k}x_i\le k-1\quad \forall \;\;1\le k\le n \]
1959 Miklós Schweitzer, 9
[b]9.[/b] Let $f(z)= z^n +a_1 z^{n-1}+\dots + a_n$ be a polynomial over the field of the complex numbers and let $E_f$ denote the closed (not necessarily connected) domain of complex numbers $z$ for which $\mid f(z) \mid \leq 1$. Show that there exists a point $z_0 \in E_f$ such that $\mid f'(z_0) \mid \geq n$. [b](F. 5)[/b]
2019 IMC, 7
Let $C=\{4,6,8,9,10,\ldots\}$ be the set of composite positive integers. For each $n\in C$ let $a_n$ be the smallest positive integer $k$ such that $k!$ is divisible by $n$. Determine whether the following series converges:
$$\sum_{n\in C}\left(\frac{a_n}{n}\right)^n.$$
[i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan[/i]
2015 Kyoto University Entry Examination, 1
1. The Line $y=px+q$ intersects $y=x^2-x$, but not intersect $y=|x|+|x-1|+1$, then illustlate range of $(p,q)$ and find the area.
2003 Miklós Schweitzer, 3
Let $Z=\{ z_1,\ldots, z_{n-1}\}$, $n\ge 2$, be a set of different complex numbers such that $Z$ contains the conjugate of any its element.
a) Show that there exists a constant $C$, depending on $Z$, such that for any $\varepsilon\in (0,1)$ there exists an algebraic integer $x_0$ of degree $n$, whose algebraic conjugates $x_1, x_2, \ldots, x_{n-1}$ satisfy $|x_1-z_1|\le \varepsilon, \ldots, |x_{n-1}-z_{n-1}|\le \varepsilon$ and $|x_0|\le \frac{C}{\varepsilon}$.
b) Show that there exists a set $Z=\{ z_1, \ldots, z_{n-1}\}$ and a positive number $c_n$ such that for any algebraic integer $x_0$ of degree $n$, whose algebraic conjugates satisfy $|x_1-z_1|\le \varepsilon,\ldots, |x_{n-1}-z_{n-1}|\le \varepsilon$, it also holds that $|x_0|>\frac{c_n}{\varepsilon}$.
(translated by L. Erdős)
2008 Putnam, A6
Prove that there exists a constant $ c>0$ such that in every nontrivial finite group $ G$ there exists a sequence of length at most $ c\ln |G|$ with the property that each element of $ G$ equals the product of some subsequence. (The elements of $ G$ in the sequence are not required to be distinct. A [i]subsequence[/i] of a sequence is obtained by selecting some of the terms, not necessarily consecutive, without reordering them; for example, $ 4,4,2$ is a subesequence of $ 2,4,6,4,2,$ but $ 2,2,4$ is not.)
2014 Contests, 1
Determine all pairs $(a, b)$ of real numbers for which there exists a unique symmetric $2\times 2$ matrix $M$ with real entries satisfying $\mathrm{trace}(M)=a$ and $\mathrm{det}(M)=b$.
(Proposed by Stephan Wagner, Stellenbosch University)
1976 Putnam, 5
In the $(x,y)-$plane, if $R$ is the set of points inside and on a convex polygon, let $D(x,y)$ be the distance from $(x,y)$ to the nearest point of $R.$
(a) Show that there exists constants $a,b,c,$ independent of $R$, such that $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-D(x,y)} dxdy =a+bL+cA,$$ where $L$ is the perimeter of $R$ and $A$ is the area of $R.$
(b) Find the values of $a,b$ and $c.$