Found problems: 85335
1996 Cono Sur Olympiad, 2
Consider a sequence of real numbers defined by:
$a_{n + 1} = a_n + \frac{1}{a_n}$ for $n = 0, 1, 2, ...$
Prove that, for any positive real number $a_0$, is true that $a_{1996}$ is greater than $63$.
2023 Romania National Olympiad, 2
We say that a natural number is called special if all of its digits are non-zero and any two adjacent digits in its decimal representation are consecutive (not necessarily in ascending order).
a) Determine the largest special number $m$ whose sum of digits is equal to $2023$.
b) Determine the smallest special number $n$ whose sum of digits is equal to $2022$.
2015 FYROM JBMO Team Selection Test, 3
Let $a, b$ and $c$ be positive real numbers. Prove that $\prod_{cyc}(16a^2+8b+17)\geq2^{12}\prod_{cyc}(a+1)$.
2013 Dutch Mathematical Olympiad, 5
The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 +...+ 10^{2013}$
If one writes down the number $S$, how often does the digit `$5$' occur in the result?
1986 Bulgaria National Olympiad, Problem 6
Let $0<k<1$ be a given real number and let $(a_n)_{n\ge1}$ be an infinite sequence of real numbers which satisfies $a_{n+1}\le\left(1+\frac kn\right)a_n-1$. Prove that there is an index $t$ such that $a_t<0$.
KoMaL A Problems 2023/2024, A. 882
Let $H_1, H_2,\ldots, H_m$ be non-empty subsets of the positive integers, and let $S$ denote their union. Prove that
\[\sum_{i=1}^m \sum_{(a,b)\in H_i^2}\gcd(a,b)\ge\frac1m \sum_{(a,b)\in S^2}\gcd(a,b).\]
[i]Proposed by Dávid Matolcsi, Berkeley[/i]
2010 Baltic Way, 12
Let $ABCD$ be a convex quadrilateral with precisely one pair of parallel sides.
$(a)$ Show that the lengths of its sides $AB,BC,CD, DA$ (in this order) do not form an arithmetic progression.
$(b)$ Show that there is such a quadrilateral for which the lengths of its sides $AB ,BC,CD,DA$ form an arithmetic progression after the order of the lengths is changed.
2020 Taiwan TST Round 1, 3
Let $N>2^{5000}$ be a positive integer. Prove that if $1\leq a_1<\cdots<a_k<100$ are distinct positive integers then the number
\[\prod_{i=1}^{k}\left(N^{a_i}+a_i\right)\]
has at least $k$ distinct prime factors.
Note. Results with $2^{5000}$ replaced by some other constant $N_0$ will be awarded points depending on the value of $N_0$.
[i]Proposed by Evan Chen[/i]
1996 Putnam, 1
Define a $\emph{selfish}$ set to be a set which has its own cardinality as its element. And a set is a $\emph{minimal }\text{ selfish}$ set if none of its proper subsets are $\emph{selfish}$. Find with proof the number of $\text{minimal selfish}$ subsets of $\{1,2,\cdots ,n\}$.
2014 Romania National Olympiad, 4
Let $ ABCD $ be a quadrilateral inscribed in a circle of diameter $ AC. $ Fix points $ E,F $ of segments $ CD, $ respectively, $ BC $ such that $ AE $ is perpendicular to $ DF $ and $ AF $ is perpendicular to $ BE. $
Show that $ AB=AD. $
1994 AIME Problems, 4
Find the positive integer $n$ for which \[ \lfloor \log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994. \] (For real $x$, $\lfloor x\rfloor$ is the greatest integer $\le x.$)
II Soros Olympiad 1995 - 96 (Russia), 11.6
For what natural number $x$ will the value of the polynomial $x^3+7x^2+6x+1$ be the cube of a natural number?
1972 Bundeswettbewerb Mathematik, 2
Prove: out of $ 79$ consecutive positive integers, one can find at least one whose sum of digits is divisible by $ 13$. Show that this isn't true for $ 78$ consecutive integers.
MathLinks Contest 3rd, 2
Prove that for all positive reals $a, b, c$ the following double inequality holds:
$$\frac{a+b+c}{3}\ge \sqrt[3]{\frac{(a+b)(b+c)(c+a)}{8}}\ge \frac{\sqrt{ab}+\sqrt{bc}\sqrt{ca}}{3}$$
1985 IMO Longlists, 78
The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by
\[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\]
Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$
2014 ELMO Shortlist, 8
Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that
\[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]
2012 Online Math Open Problems, 37
In triangle $ABC$, $AB = 1$ and $AC = 2$. Suppose there exists a point $P$ in the interior of triangle $ABC$ such that $\angle PBC = 70^{\circ}$, and that there are points $E$ and $D$ on segments $AB$ and $AC$, such that $\angle BPE = \angle EPA = 75^{\circ}$ and $\angle APD = \angle DPC = 60^{\circ}$. Let $BD$ meet $CE$ at $Q,$ and let $AQ$ meet $BC$ at $F.$ If $M$ is the midpoint of $BC$, compute the degree measure of $\angle MPF.$
[i]Authors: Alex Zhu and Ray Li[/i]
2007 Pre-Preparation Course Examination, 13
Let $\{a_i\}_{i=1}^{\infty}$ be a sequence of positive integers such that $a_1<a_2<a_3\cdots$ and all of primes are members of this sequence. Prove that for every $n<m$
\[\dfrac{1}{a_n} + \dfrac{1}{a_{n+1}} + \cdots + \dfrac{1}{a_m} \not \in \mathbb N\]
2024 Harvard-MIT Mathematics Tournament, 7
Let $ABCDEF$ be a regular hexagon with $P$ as a point in its interior. Prove that of the three values $\tan \angle APD$, $\tan \angle BPE$ and $\tan \angle CPF$, two of them sum to the third one.
2013 Kurschak Competition, 2
Consider the closed polygonal discs $P_1$, $P_2$, $P_3$ with the property that for any three points $A\in P_1$, $B\in P_2$, $C\in P_3$, we have $[\triangle ABC]\le 1$. (Here $[X]$ denotes the area of polygon $X$.)
(a) Prove that $\min\{[P_1],[P_2],[P_3]\}<4$.
(b) Give an example of polygons $P_1,P_2,P_3$ with the above property such that $[P_1]>4$ and $[P_2]>4$.
2009 Sharygin Geometry Olympiad, 21
The opposite sidelines of quadrilateral $ ABCD$ intersect at points $ P$ and $ Q$. Two lines passing through these points meet the side of $ ABCD$ in four points which are the vertices of a parallelogram. Prove that the center of this parallelogram lies on the line passing through the midpoints of diagonals of $ ABCD$.
Kvant 2024, M2817
We are given fixed circles $\Omega$ and $\omega$ such that there exists a hexagon $ABCDEF$ inscribed in $\Omega$ and circumscribed around $\omega$. (Note that then, by virtue of Poncelet's theorem, there is an infinite family of such hexagons.) Prove that the value of $\dfrac{S_{ABCDEF}}{AD+BE+CF}$ it does not depend on the choice of the hexagon $ABCDEF$.
[i]A. Zaslavsky and Tran Quang Hung[/i]
IMSC 2023, 2
There are $n!$ empty baskets in a row, labelled $1, 2, . . . , n!$. Caesar
first puts a stone in every basket. Caesar then puts 2 stones in every second basket.
Caesar continues similarly until he has put $n$ stones into every nth basket. In
other words, for each $i = 1, 2, . . . , n,$ Caesar puts $i$ stones into the baskets labelled
$i, 2i, 3i, . . . , n!.$
Let $x_i$ be the number of stones in basket $i$ after all these steps. Show that
$n! \cdot n^2 \leq \sum_{i=1}^{n!} x_i^2 \leq n! \cdot n^2 \cdot \sum_{i=1}^{n} \frac{1}{i} $
2005 Grigore Moisil Urziceni, 2
Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R}_{\ge 0} $ that admits primitives and such that $ \lim_{x\to 0 } \frac{f(x)}{x} =0. $ Prove that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} , $ defined as
$$ g(x)=\left\{ \begin{matrix} f(x)/x ,&\quad x\neq 0\\ 0,& \quad x=0 \end{matrix} \right. , $$
is primitivable.
1998 Harvard-MIT Mathematics Tournament, 10
Lukas is playing pool on a table shaped like an equilateral triangle.
The pockets are at the corners of the triangle and are labeled $C$, $H$, and $T$. Each side of the table is $16$ feet long. Lukas shoots a ball from corner $C$ of the table in such a way that on the second bounce, the ball hits $2$ feet away from him along side $CH$.
a. How many times will the ball bounce before hitting a pocket?
b. Which pocket will the ball hit?
c. How far will the ball travel before hitting the pocket?