Found problems: 876
2002 Putnam, 5
Define a sequence by $a_0=1$, together with the rules $a_{2n+1}=a_n$ and $a_{2n+2}=a_n+a_{n+1}$ for each integer $n\ge0$. Prove that every positive rational number appears in the set $ \left\{ \tfrac {a_{n-1}}{a_n}: n \ge 1 \right\} = \left\{ \tfrac {1}{1}, \tfrac {1}{2}, \tfrac {2}{1}, \tfrac {1}{3}, \tfrac {3}{2}, \cdots \right\} $.
1953 Miklós Schweitzer, 10
[b]10.[/b] Consider a point performing a random walk on a planar triangular lattice and suppose that it moves away with equal probability from any lattice point along any one of the six lattice lines issuing from it. Prove that if the walk is continued indefinitely, then the point will return to its starting point with probability 1. [b](P. 5)[/b]
2001 Putnam, 6
Assume that $(a_n)_{n \ge 1}$ is an increasing sequence of positive real numbers such that $\lim \tfrac{a_n}{n}=0$. Must there exist infinitely many positive integers $n$ such that $a_{n-i}+a_{n+i}<2a_n$ for $i=1,2,\cdots,n-1$?
1994 Putnam, 1
Find all positive integers that are within $250$ of exactly $15$ perfect squares.
2013 Putnam, 4
A finite collection of digits $0$ and $1$ is written around a circle. An [i]arc[/i] of length $L\ge 0$ consists of $L$ consecutive digits around the circle. For each arc $w,$ let $Z(w)$ and $N(w)$ denote the number of $0$'s in $w$ and the number of $1$'s in $w,$ respectively. Assume that $|Z(w)-Z(w')|\le 1$ for any two arcs $w,w'$ of the same length. Suppose that some arcs $w_1,\dots,w_k$ have the property that \[Z=\frac1k\sum_{j=1}^kZ(w_j)\text{ and }N=\frac1k\sum_{j=1}^k N(w_j)\] are both integers. Prove that there exists an arc $w$ with $Z(w)=Z$ and $N(w)=N.$
2018 Brazil Undergrad MO, 13
A continuous function $ f: \mathbb {R} \to \mathbb {R} $ satisfies $ f (x) f (f (x)) = 1 $ for every real $ x $ and $ f (2020) = 2019 $ . What is the value of $ f (2018) $?
2012 Miklós Schweitzer, 4
Let $K$ be a convex shape in the $n$ dimensional space, having unit volume. Let $S \subset K$ be a Lebesgue measurable set with measure at least $1-\varepsilon$, where $0<\varepsilon<1/3$. Prove that dilating $K$ from its centroid by the ratio of $2\varepsilon \ln(1/\varepsilon)$, the shape obtained contains the centroid of $S$.
2004 IMC, 5
Let $S$ be a set of $\displaystyle { 2n \choose n } + 1$ real numbers, where $n$ is an positive integer. Prove that there exists a monotone sequence $\{a_i\}_{1\leq i \leq n+2} \subset S$ such that
\[ |x_{i+1} - x_1 | \geq 2 | x_i - x_1 | , \]
for all $i=2,3,\ldots, n+1$.
MIPT student olimpiad autumn 2024, 4
The ellipsoid $E$ is contained in the simplex $S$, which is located in the unit ball
B space $R^n$. Prove that the sum of the principal semi-axes of the ellipsoid $E$ is no more than
units.
2015 IMC, 4
Determine whether or not there exist 15 integers $m_1,\ldots,m_{15}$
such that~
$$\displaystyle \sum_{k=1}^{15}\,m_k\cdot\arctan(k) = \arctan(16). \eqno(1)$$
(Proposed by Gerhard Woeginger, Eindhoven University of Technology)
1977 Putnam, B3
An (ordered) triple $(x_1,x_2,x_3)$ of positive [i]irrational[/i] numbers with $x_1+x_2+x_3=1$ is called [b]balanced[/b] if each $x_i< 1/2.$ If a triple is not balanced, say if $x_j>1/2$, one performs the following [b] balancing act[/b] $$B(x_1,x_2,x_3)=(x'_1,x'_2,x'_3),$$ where $x'_i=2x_i$ if $i\neq j$ and $x'_j=2x_j-1.$ If the new triple is not balanced, one performs the balancing act on it. Does the continuation of this process always lead to a balanced triple after a finite number of performances of the balancing act?
2021 Alibaba Global Math Competition, 13
Let $M_n=\{(u,v) \in S^n \times S^n: u \cdot v=0\}$, where $n \ge 2$, and $u \cdot v$ is the Euclidean inner product of $u$ and $v$. Suppose that the topology of $M_n$ is induces from $S^n \times S^n$.
(1) Prove that $M_n$ is a connected regular submanifold of $S^n \times S^n$.
(2) $M_n$ is Lie Group if and only if $n=2$.
2014 Paenza, 6
(a) Show that if $f:[-1,1]\to \mathbb{R}$ is a convex and $C^2$ function such that $f(1),f(-1)\geq 0$, then:
\[\min_{x\in[-1,1]} \{f(x)\} \geq - \int_{-1}^1 f''\]
(b) Let $B\subset \mathbb{R}^2$ the closed ball with center $0$ and radius $1$. Show that if $f: B \to \mathbb{R}$ is a convex and $C^2$ function and $f\geq 0$ in $\partial B$, then:
\[f(0)\geq -\frac{1}{\sqrt{\pi}} \left( \int_{B} (f_{xx}f_{yy}-f_{xy}^2) \right)^{1/2}\]
1985 Miklós Schweitzer, 10
Show that any two intervals $A, B\subseteq \mathbb R$ of positive lengths can be countably disected into each other, that is, they can be written as countable unions $A=A_1\cup A_2\cup\ldots\,$ and $B=B_1\cup B_2\cup\ldots\,$ of pairwise disjoint sets, where $A_i$ and $B_i$ are congruent for every $i\in \mathbb N$ [Gy. Szabo]
2003 Miklós Schweitzer, 6
Show that the recursion $n=x_n(x_{n-1}+x_n+x_{n+1})$, $n=1,2,\ldots$, $x_0=0$ has exaclty one nonnegative solution.
(translated by L. Erdős)
2009 Putnam, A5
Is there a finite abelian group $ G$ such that the product of the orders of all its elements is $ 2^{2009}?$
1996 Putnam, 5
Let $p$ be a prime greater than $3$. Prove that
\[ p^2\Big| \sum_{i=1}^{\left\lfloor\frac{2p}{3}\right\rfloor}\dbinom{p}{i}. \]
2014 IMC, 5
Let $A_{1}A_{2} \dots A_{3n}$ be a closed broken line consisting of $3n$ lines segments in the Euclidean plane. Suppose that no three of its vertices are collinear, and for each index $i=1,2,\dots,3n$, the triangle $A_{i}A_{i+1}A_{i+2}$ has counterclockwise orientation and $\angle A_{i}A_{i+1}A_{i+2} = 60º$, using the notation $A_{3n+1} = A_{1}$ and $A_{3n+2} = A_{2}$. Prove that the number of self-intersections of the broken line is at most $\frac{3}{2}n^{2} - 2n + 1$
2017 Korea USCM, 5
Evaluate the following limit.
\[\lim_{n\to\infty} \sqrt{n} \int_0^\pi \sin^n x dx\]
ICMC 5, 6
Is it possible to cover a circle of area $1$ with finitely many equilateral triangles whose areas sum to $1.01$, all pointing in the same direction?
[i]Proposed by Ethan Tan[/i]
1958 Miklós Schweitzer, 4
[b]4.[/b] Let $P_1 P_2 P_3 P_4 P_5 P_6$ be a convex hexagon. Denote by $T$ its area and by $t$ the area of the triangle $Q_1 Q_2 Q_3$, where $Q_1,Q_2$ and $Q_3$ are the midpoints of $P_1P_4,P_2P_5,P_3P_6$ respectively. Prove that $t<\frac{1}{4}T$. [b](G. 3)[/b]
1999 Putnam, 4
Let $f$ be a real function with a continuous third derivative such that $f(x)$, $f^\prime(x)$, $f^{\prime\prime}(x)$, $f^{\prime\prime\prime}(x)$ are positive for all $x$. Suppose that $f^{\prime\prime\prime}(x)\leq f(x)$ for all $x$. Show that $f^\prime(x)<2f(x)$ for all $x$.
2018 VTRMC, 7
A continuous function $f : [a,b] \to [a,b]$ is called piecewise monotone if $[a, b]$ can be subdivided into finitely many subintervals
$$I_1 = [c_0,c_1], I_2 = [c_1,c_2], \dots , I_\ell = [ c_{\ell - 1},c_\ell ]$$
such that $f$ restricted to each interval $I_j$ is strictly monotone, either increasing or decreasing. Here we are assuming that $a = c_0 < c_1 < \cdots < c_{\ell - 1} < c_\ell = b$. We are also assuming that each $I_j$ is a maximal interval on which $f$ is strictly monotone. Such a maximal interval is called a lap of the function $f$, and the number $\ell = \ell (f)$ of distinct laps is called the lap number of $f$. If $f : [a,b] \to [a,b]$ is a continuous piecewise-monotone function, show that the sequence $( \sqrt[n]{\ell (f^n )})$ converges; here $f^n$ means $f$ composed with itself $n$-times, so $f^2 (x) = f(f(x))$ etc.
2015 VJIMC, 4
[b]Problem 4[/b]
Find all continuously differentiable functions $ f : \mathbb{R} \rightarrow \mathbb{R} $, such that for every $a \geq 0$ the following
relation holds:
$$\iiint \limits_{D(a)} xf \left( \frac{ay}{\sqrt{x^2+y^2}} \right) \ dx \ dy\ dz = \frac{\pi a^3}{8} (f(a) + \sin a -1)\ , $$
where $D(a) = \left\{ (x,y,z)\ :\ x^2+y^2+z^2 \leq a^2\ , \ |y|\leq \frac{x}{\sqrt{3}} \right\}\ .$
2018 Korea USCM, 1
Given vector $\mathbf{u}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)\in\mathbb{R}^3$ and recursively defined sequence of vectors $\{\mathbf{v}_n\}_{n\geq 0}$
$$\mathbf{v}_0 = (1,2,3),\quad \mathbf{v}_n = \mathbf{u}\times\mathbf{v}_{n-1}$$
Evaluate the value of infinite series $\sum_{n=1}^\infty (3,2,1)\cdot \mathbf{v}_{2n}$.