Found problems: 876
2017 Korea USCM, 4
For a real coefficient cubic polynomial $f(x)=ax^3+bx^2+cx+d$, denote three roots of the equation $f(x)=0$ by $\alpha,\beta,\gamma$. Prove that the three roots $\alpha,\beta,\gamma$ are distinct real numbers iff the real symmetric matrix
$$\begin{pmatrix} 3 & p_1 & p_2 \\ p_1 & p_2 & p_3 \\ p_2 & p_3 & p_4 \end{pmatrix},\quad p_i = \alpha^i + \beta^i + \gamma^i$$
is positive definite.
2001 Putnam, 1
Let $n$ be an even positive integer. Write the numbers $1, 2, \cdots, n^2$ in the squares of an $n \times n$ grid so that the $k$th row, from left to right, is \[ (k-1)n + 1, \ (k-1)n + 2, \ \cdots, \ (k-1)n + n. \] Color the squares of the grid so that half of the squares in each row and in each column are red and the other half are black (a checkerboard coloring is one possibility). Prove that for each coloring, the sum of the numbers on the red squares is equal to the sum of the numbers on the black squares.
2006 Putnam, A4
Let $S=\{1,2\dots,n\}$ for some integer $n>1.$ Say a permutation $\pi$ of $S$ has a local maximum at $k\in S$ if
\[\begin{array}{ccc}\text{(i)}&\pi(k)>\pi(k+1)&\text{for }k=1\\ \text{(ii)}&\pi(k-1)<\pi(k)\text{ and }\pi(k)>\pi(k+1)&\text{for }1<k<n\\ \text{(iii)}&\pi(k-1)M\pi(k)&\text{for }k=n\end{array}\]
(For example, if $n=5$ and $\pi$ takes values at $1,2,3,4,5$ of $2,1,4,5,3,$ then $\pi$ has a local maximum of $2$ as $k=1,$ and a local maximum at $k-4.$)
What is the average number of local maxima of a permutation of $S,$ averaging over all permuatations of $S?$
2017 Miklós Schweitzer, 7
Characterize all increasing sequences $(s_n)$ of positive reals for which there exists a set $A\subset \mathbb{R}$ with positive measure such that $\lambda(A\cap I)<\frac{s_n}{n}$ holds for every interval $I$ with length $1/n$, where $\lambda$ denotes the Lebesgue measure.
2008 Putnam, A2
Alan and Barbara play a game in which they take turns filling entries of an initially empty $ 2008\times 2008$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?
1996 Putnam, 3
Let $S_n$ be the set of all permutations of $(1,2,\ldots,n)$. Then find :
\[ \max_{\sigma \in S_n} \left(\sum_{i=1}^{n} \sigma(i)\sigma(i+1)\right) \]
where $\sigma(n+1)=\sigma(1)$.
2013 Putnam, 2
Let $C=\bigcup_{N=1}^{\infty}C_N,$ where $C_N$ denotes the set of 'cosine polynomials' of the form \[f(x)=1+\sum_{n=1}^Na_n\cos(2\pi nx)\] for which:
(i) $f(x)\ge 0$ for all real $x,$ and
(ii) $a_n=0$ whenever $n$ is a multiple of $3.$
Determine the maximum value of $f(0)$ as $f$ ranges through $C,$ and prove that this maximum is attained.
1997 Putnam, 5
Let us define a sequence $\{a_n\}_{n\ge 1}$. Define as follows:
\[ a_1=2\text{ and }a_{n+1}=2^{a_n}\text{ for }n\ge 1 \]
Show this :
\[ a_{n}\equiv a_{n-1}\pmod n \]
1957 Miklós Schweitzer, 7
[b]7.[/b] Prove that any real number x satysfying the inequalities $0<x\leq 1$ can be represented in the form
$x= \sum_{k=1}^{\infty}\frac{1}{n_k}$
where $(n_k)_{k=1}^{\infty}$ is a sequence of positive integers such that $\frac{n_{k+1}}{n_k}$ assumes, for each $k$, one of the three values $2,3$ or $4$. [b](N. 14)[/b]
2011 Putnam, A5
Let $F:\mathbb{R}^2\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be twice continuously differentiable functions with the following properties:
• $F(u,u)=0$ for every $u\in\mathbb{R};$
• for every $x\in\mathbb{R},g(x)>0$ and $x^2g(x)\le 1;$
• for every $(u,v)\in\mathbb{R}^2,$ the vector $\nabla F(u,v)$ is either $\mathbf{0}$ or parallel to the vector $\langle g(u),-g(v)\rangle.$
Prove that there exists a constant $C$ such that for every $n\ge 2$ and any $x_1,\dots,x_{n+1}\in\mathbb{R},$ we have
\[\min_{i\ne j}|F(x_i,x_j)|\le\frac{C}{n}.\]
2003 Miklós Schweitzer, 7
Let $r$ be a nonnegative continuous function on the real line. Show that there exists a function $f\in C^1(\mathbb{R})$, not identically zero, such that $f'(x)=f(x-r(f(x)))$, $x\in\mathbb{R}$.
(translated by L. Erdős)
ICMC 2, 4
For \(u,v \in\mathbb{R}^4\), let \(<u,v>\) denote the usual dot product. Define a [i]vector field[/i] to be a map \(\omega:\mathbb{R}\to\mathbb{R}\) such that \(<\omega(z),z>=0,\ \forall z\in\mathbb{R}^4.\)
Find a maximal collection of vector fields \(\left\{\omega_1,...,\omega_k\right\}\) such that the map \(\Omega\) sending \(z\) to \(\lambda_1\omega_1(z)+\cdots+\lambda_k \omega_k(z)\), with \(\lambda_1,\ldots,\lambda_k\in\mathbb{R}\), is nonzero on \(\mathbb{R}^4\backslash\{0\}\) unless \(\lambda_1=\cdots=\lambda_k=0\)
1997 Putnam, 2
$f$ be a twice differentiable real valued function satisfying
\[ f(x)+f^{\prime\prime}(x)=-xg(x)f^{\prime}(x) \]
where $g(x)\ge 0$ for all real $x$. Show that $|f(x)|$ is bounded.
1966 Putnam, B2
Prove that among any ten consecutive integers at least one is relatively prime to each of the others.
2000 Putnam, 5
Let $S_0$ be a finite set of positive integers. We define finite sets $S_1, S_2, \cdots$ of positive integers as follows: the integer $a$ in $S_{n+1}$ if and only if exactly one of $a-1$ or $a$ is in $S_n$. Show that there exist infinitely many integers $N$ for which $S_N = S_0 \cup \{ N + a: a \in S_0 \}$.
2008 Miklós Schweitzer, 7
Let $f\colon \mathbb{R}^1\rightarrow \mathbb{R}^2$ be a continuous function such that $f(x)=f(x+1)$ for all $x$, and let $t\in [0,\frac14]$. Prove that there exists $x\in\mathbb{R}$ such that the vector from $f(x-t)$ to $f(x+t)$ is perpendicular to the vector from $f(x)$ to $f(x+\frac12)$.
(translated by Miklós Maróti)
2021 Alibaba Global Math Competition, 8
Let $f(z)$ be a holomorphic function in $\{\vert z\vert \le R\}$ ($0<R<\infty$). Define
\[M(r,f)=\max_{\vert z\vert=r} \vert f(z)\vert, \quad A(r,f)=\max_{\vert z\vert=r} \text{Re}\{f(z)\}.\]
Show that
\[M(r,f) \le \frac{2r}{R-r}A(R,f)+\frac{R+r}{R-r} \vert f(0)\vert, \quad \forall 0 \le r<R.\]
2009 IMC, 4
Let $p$ be a prime number and $\mathbf{W}\subseteq \mathbb{F}_p[x]$ be the smallest set satisfying the following :
[list]
(a) $x+1\in \mathbf{W}$ and $x^{p-2}+x^{p-3}+\cdots +x^2+2x+1\in \mathbf{W}$
(b) For $\gamma_1,\gamma_2$ in $\mathbf{W}$, we also have $\gamma(x)\in \mathbf{W}$, where $\gamma(x)$ is the remainder $(\gamma_1\circ \gamma_2)(x)\pmod {x^p-x}$.[/list]
How many polynomials are in $\mathbf{W}?$
2012 Putnam, 6
Let $f(x,y)$ be a continuous, real-valued function on $\mathbb{R}^2.$ Suppose that, for every rectangular region $R$ of area $1,$ the double integral of $f(x,y)$ over $R$ equals $0.$ Must $f(x,y)$ be identically $0?$
1957 Miklós Schweitzer, 5
[b]5.[/b] Find the continuous solutions of the functional equation $f(xyz)= f(x)+f(y)+f(z)$ in the following cases:
(a) $x,y,z$ are arbitrary non-zero real numbers;
(b) $a<x,y,z<b (1<a^{3}<b)$.
[b](R. 13)[/b]
2000 Miklós Schweitzer, 4
Let $a_1<a_2<a_3$ be positive integers. Prove that there are integers $x_1,x_2,x_3$ such that $\sum_{i=1}^3 |x_i | >0$, $\sum_{i=1}^3 a_ix_i= 0$ and
$$\max_{1\le i\le 3} | x_i|<\frac{2}{\sqrt{3}}\sqrt{a_3}+1$$.
2013 IMC, 5
Does there exist a sequence $\displaystyle{\left( {{a_n}} \right)}$ of complex numbers such that for every positive integer $\displaystyle{p}$ we have that $\displaystyle{\sum\limits_{n = 1}^{ + \infty } {a_n^p} }$ converges if and only if $\displaystyle{p}$ is not a prime?
[i]Proposed by Tomáš Bárta, Charles University, Prague.[/i]
2006 IMC, 1
Let $V$ be a convex polygon.
(a) Show that if $V$ has $3k$ vertices, then $V$ can be triangulated such that each vertex is in an odd number of triangles.
(b) Show that if the number of vertices is not divisible with 3, then $V$ can be triangulated such that exactly 2 vertices have an even number of triangles.
2018 Miklós Schweitzer, 8
Does there exist a piecewise linear, continuous, surjective mapping $f: [0,1]\to [0,1]$ such that $f(0)=f(1)=0$, and for all positive integer $n$,
$$2.0001^{(n-10)} <P_n(f)<2.9999^{(n+10)}$$holds, where $P_n(f)$ is the number of points $x$ such that $\underbrace{f(\dotsc f}_n(x)\dotsc )=x$?
2022 Miklós Schweitzer, 6
Let $\epsilon$ be a primitive seventh unit root. Which integers occur in $|\alpha|^2$ in form, where $\alpha$ is an element of the seventh circular field $\mathbb Q(\epsilon)$?