Found problems: 876
1961 Putnam, A7
Let $S$ be a nonempty closed set in the euclidean plane for which there is a closed disk $D$ containing $S$ such that $D$ is a subset of every closed disk that contains $S$. Prove that every point inside $D$ is the midpoint of a segment joining two points of $S.$
2019 Miklós Schweitzer, 1
Prove that if every subspace of a Hausdorff space $X$ is $\sigma$-compact, then $X$ is countable.
ICMC 2, 4
Let \(f:\{0, 1\}^n \to \{0, 1\} \subseteq \mathbb{R}\) be a function. Call such a function a Boolean function.
Let \(\wedge\) denote the component-wise multiplication in \(\{0,1\}^n\). For example, for \(n = 4\), \[(0,0,1,1) \wedge (0,1,0,1) = (0,0,0,1).\]
Let \(S = \left\{i_1,i_2,\ldots ,i_k\right\} \subseteq \left\{1,2,\ldots ,n\right\}\). \(f\) is called the oligarchy function over \(S\) if \[f (x) = x_{i_1},x_{i_2},\ldots,x_{i_k}\ \text{ (with the usual multiplication),}\]
where \(x_i\) denotes the \(i\)-th component of \(x\). By convention, \(f\) is called the oligarchy function over \(\emptyset\) if \(f\) is constantly 1.
(i) Suppose \(f\) is not constantly zero. Show that \(f\) is an oligarchy function [u]if and only if[/u] \(f\) satisfies \[f(x\wedge y)=f(x)f(y),\ \forall x,y\in\left\{0,1\right\}^n.\]
Let \(Y\) be a uniformly distributed random variable over \(\left\{0, 1\right\}^n\). Let \(T\) be an operator that maps Boolean functions to functions \(\left\{0, 1\right\}^n\to\mathbb{R}\), such that
\[(Tf)(x)=E_Y(f(x\wedge Y)),\ \forall x\in\left\{0,1\right\}^n\]
where \(E_Y()\) denotes the expectation over \(Y\). \(f\) is called an eigenfunction of \(T\) if \(\exists\lambda\in\mathbb{R}\backslash\left\{0\right\}\) such that
\[(Tf)(x)=\lambda f(x),\ \forall x\in\left\{0,1\right\}^n\]
(ii) Prove that \(f\) is an eigenfunction of \(T\) [u]if and only if[/u] \(f\) is an oligarchy function.
2003 Putnam, 2
Let $n$ be a positive integer. Starting with the sequence $1,\frac{1}{2}, \frac{1}{3} , \cdots , \frac{1}{n}$, form a new sequence of $n -1$ entries $\frac{3}{4}, \frac{5}{12},\cdots ,\frac{2n -1}{2n(n -1)}$, by taking the averages of two consecutive entries in the first sequence. Repeat the averaging of neighbors on the second sequence to obtain a third sequence of $n -2$ entries and continue until the final sequence consists of a single number $x_n$. Show that $x_n < \frac{2}{n}$.
1990 Putnam, B4
Let $G$ be a finite group of order $n$ generated by $a$ and $b$. Prove or disprove: there is a sequence \[ g_1, g_2, g_3, \cdots, g_{2n} \] such that:
$(1)$ every element of $G$ occurs exactly twice, and
$(2)$ $g_{i+1}$ equals $g_{i}a$ or $g_ib$ for $ i = 1, 2, \cdots, 2n $. (interpret $g_{2n+1}$ as $g_1$.)
1971 Putnam, B1
Let $S$ be a set and let $\circ$ be a binary operation on $S$ satisfying two laws
$$x\circ x=x \text{ for all } x \text{ in } S, \text{ and}$$
$$(x \circ y) \circ z= (y\circ z) \circ x \text{ for all } x,y,z \text{ in } S.$$
Show that $\circ$ is associative and commutative.
2014 IMC, 1
For a positive integer $x$, denote its $n^{\mathrm{th}}$ decimal digit by $d_n(x)$, i.e. $d_n(x)\in \{ 0,1, \dots, 9\}$ and $x=\sum_{n=1}^{\infty} d_n(x)10^{n-1}$. Suppose that for some sequence $(a_n)_{n=1}^{\infty}$, there are only finitely many zeros in the sequence $(d_n(a_n))_{n=1}^{\infty}$. Prove that there are infinitely many positive integers that do not occur in the sequence $(a_n)_{n=1}^{\infty}$.
(Proposed by Alexander Bolbot, State University, Novosibirsk)
1998 Putnam, 3
Let $f$ be a real function on the real line with continuous third derivative. Prove that there exists a point $a$ such that \[f(a)\cdot f^\prime(a)\cdot f^{\prime\prime}(a)\cdot f^{\prime\prime\prime}(a)\geq 0.\]
1971 Putnam, B2
Let $F(x)$ be a real valued function defined for all real $x$ except for $x=0$ and $x=1$ and satisfying the functional equation $F(x)+F\{(x-1)/x\}=1+x.$ Find all functions $F(x)$ satisfying these conditions.
2014 Contests, 1
Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation:
\[a_{n+1}=10^n a_n^2\]
(a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$.
(b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.
2004 Miklós Schweitzer, 3
Prove that there is a constant $c>0$ such that for any $n>3$ there exists a planar graph $G$ with $n$ vertices such that every straight-edged plane embedding of $G$ has a pair of edges with ratio of lengths at least $cn$.
2002 Putnam, 4
In Determinant Tic-Tac-Toe, Player $1$ enters a $1$ in an empty $3 \times 3$ matrix. Player $0$ counters with a $0$ in a vacant position and play continues in turn intil the $ 3 \times 3 $ matrix is completed with five $1$’s and four $0$’s. Player $0$ wins if the determinant is $0$ and player $1$ wins otherwise. Assuming both players pursue optimal strategies, who will win and how?
1971 Putnam, A1
Let there be given nine lattice points (points with integral coordinates) in three dimensional Euclidean space. Show that there is a lattice point on the interior of one of the line segments joining two of these points.
2012 Putnam, 4
Let $q$ and $r$ be integers with $q>0,$ and let $A$ and $B$ be intervals on the real line. Let $T$ be the set of all $b+mq$ where $b$ and $m$ are integers with $b$ in $B,$ and let $S$ be the set of all integers $a$ in $A$ such that $ra$ is in $T.$ Show that if the product of the lengths of $A$ and $B$ is less than $q,$ then $S$ is the intersection of $A$ with some arithmetic progression.
2018 ISI Entrance Examination, 8
Let $n\geqslant 3$. Let $A=((a_{ij}))_{1\leqslant i,j\leqslant n}$ be an $n\times n$ matrix such that $a_{ij}\in\{-1,1\}$ for all $1\leqslant i,j\leqslant n$. Suppose that $$a_{k1}=1~~\text{for all}~1\leqslant k\leqslant n$$ and $~~\sum_{k=1}^n a_{ki}a_{kj}=0~~\text{for all}~i\neq j$.
Show that $n$ is a multiple of $4$.
2016 IMC, 1
Let $f : \left[ a, b\right]\rightarrow\mathbb{R}$ be continuous on $\left[ a, b\right]$ and differentiable on $\left( a, b\right)$. Suppose that $f$ has infinitely many zeros, but there is no $x\in \left( a, b\right)$ with $f(x)=f'(x)=0$.
(a) Prove that $f(a)f(b)=0$.
(b) Give an example of such a function on $\left[ 0, 1\right]$.
(Proposed by Alexandr Bolbot, Novosibirsk State University)
2022 District Olympiad, P1
Let $e$ be the identity of monoid $(M,\cdot)$ and $a\in M$ an invertible element. Prove that
[list=a]
[*]The set $M_a:=\{x\in M:ax^2a=e\}$ is nonempty;
[*]If $b\in M_a$ is invertible, then $b^{-1}\in M_a$ if and only if $a^4=e$;
[*]If $(M_a,\cdot)$ is a monoid, then $x^2=e$ for all $x\in M_a.$
[/list]
[i]Mathematical Gazette[/i]
2012 Putnam, 5
Prove that, for any two bounded functions $g_1,g_2 : \mathbb{R}\to[1,\infty),$ there exist functions $h_1,h_2 : \mathbb{R}\to\mathbb{R}$ such that for every $x\in\mathbb{R},$\[\sup_{s\in\mathbb{R}}\left(g_1(s)^xg_2(s)\right)=\max_{t\in\mathbb{R}}\left(xh_1(t)+h_2(t)\right).\]
ICMC 4, 3
Let $\displaystyle s_n=\int_0^1 \text{sin}^n(nx) \,dx$.
(a) Prove that $s_n \leq \dfrac 2n$ for all odd $n$.
(b) Find all the limit points of the sequence $s_1, s_2, s_3, \dots$.
[i]Proposed by Cristi Calin[/i]
1986 Miklós Schweitzer, 9
Consider a latticelike packing of translates of a convex region $K$. Let $t$ be the area of the fundamental parallelogram of the lattice defining the packing, and let $t_{\min} (K)$ denote the minimal value of $t$ taken for all latticelike packings. Is there a natural number $N$ such that for any $n>N$ and for any $K$ different from a parallelogram, $nt_{\min} (K)$ is smaller that the area of any convex domain in which $n$ translates to $K$ can be placed without overlapping? (By a [i]latticelike packing[/i] of $K$ we mean a set of nonoverlapping translates of $K$ obtained from $K$ by translations with all vectors of a lattice.) [G. and L. Fejes-Toth]
2015 Kyoto University Entry Examination, 3
3. Six points A, B, C, D, E, F are connected with segments length of $1$. Each segment is painted red or black probability of $\frac{1}{2}$ independence. When point A to Point E exist through segments painted red, let $X$ be. Let $X=0$ be non-exist it. Then, for $n=0,2,4$, find the probability of $X=n$.
2012 Miklós Schweitzer, 5
Let $V_1,V_2,V_3,V_4$ be four dimensional linear subspaces in $\mathbb{R}^8$ such that the intersection of any two contains only the zero vector. Prove that there exists a linear four dimensional subspace $W$ in $\mathbb{R}^8$ such that all four vector spaces $W\cap V_i$ are two dimensional.
1996 Putnam, 2
Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be circles whose centers are $10$ units apart, and whose radii are $1$ and $3$. Find, with proof, the locus of all points $M$ for which there exists points $X\in \mathcal{C}_1,Y\in \mathcal{C}_2$ such that $M$ is the midpoint of $XY$.
2017 VJIMC, 2
Prove or disprove the following statement. If $g:(0,1) \to (0,1)$ is an increasing function and satisfies
$g(x) > x$ for all $x \in (0,1)$, then there exists a continuous function $f:(0,1) \to \mathbb{R}$ satisfying $f(x) < f(g(x)) $ for all $x \in (0,1)$, but $f$ is not an increasing function.
2018 Korea USCM, 6
Suppose a continuous function $f:[0,1]\to\mathbb{R}$ is differentiable on $(0,1)$ and $f(0)=1$, $f(1)=0$. Then, there exists $0<x_0<1$ such that
$$|f'(x_0)| \geq 2018 f(x_0)^{2018}$$