Found problems: 85335
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?
2004 Italy TST, 1
Two circles $\gamma_1$ and $\gamma_2$ intersect at $A$ and $B$. A line $r$ through $B$ meets $\gamma_1$ at $C$ and $\gamma_2$ at $D$ so that $B$ is between $C$ and $D$. Let $s$ be the line parallel to $AD$ which is tangent to $\gamma_1$ at $E$, at the smaller distance from $AD$. Line $EA$ meets $\gamma_2$ in $F$. Let $t$ be the tangent to $\gamma_2$ at $F$.
$(a)$ Prove that $t$ is parallel to $AC$.
$(b)$ Prove that the lines $r,s,t$ are concurrent.
2011 Saint Petersburg Mathematical Olympiad, 7
Sasha and Serg plays next game with $100$-angled regular polygon . In the beggining Sasha set natural numbers in every angle. Then they make turn by turn, first turn is made by Serg. Serg turn is to take two opposite angles and add $1$ to its numbers. Sasha turn is to take two neigbour angles and add $1$ to its numbers. Serg want to maximize amount of odd numbers. What maximal number of odd numbers can he get no matter how Sasha plays?
2015 Iran Team Selection Test, 6
$ABCD$ is a circumscribed and inscribed quadrilateral. $O$ is the circumcenter of the quadrilateral. $E,F$ and $S$ are the intersections of $AB,CD$ , $AD,BC$ and $AC,BD$ respectively. $E'$ and $F'$ are points on $AD$ and $AB$ such that $A\hat{E}E'=E'\hat{E}D$ and $A\hat{F}F'=F'\hat{F}B$. $X$ and $Y$ are points on $OE'$ and $OF'$ such that $\frac{XA}{XD}=\frac{EA}{ED}$ and $\frac{YA}{YB}=\frac{FA}{FB}$. $M$ is the midpoint of arc $BD$ of $(O)$ which contains $A$.
Prove that the circumcircles of triangles $OXY$ and $OAM$ are coaxal with the circle with diameter $OS$.