Found problems: 85335
2013 HMNT, 8
Define the sequence $\{x_i\}_{i \ge 0}$ by $x_0 = x_1 = x_2 = 1$ and $x_k = \frac{x_{k-1}+x_{k-2}+1}{x_{k-3}}$ for $k > 2$. Find $x_{2013}$.
2015 India Regional MathematicaI Olympiad, 1
Let \(ABC\) be a triangle. Let \(B'\) denote the reflection of \(b\) in the internal angle bisector \(l\) of \(\angle A\).Show that the circumcentre of the triangle \(CB'I\) lies on the line \(l\) where \(I\) is the incentre of \(ABC\).
2003 Iran MO (3rd Round), 9
Does there exist an infinite set $ S$ such that for every $ a, b \in S$ we have $ a^2 \plus{} b^2 \minus{} ab \mid (ab)^2$.
2020 Malaysia IMONST 1, 14
A perfect square ends with the same two digits. How many possible values
of this digit are there?
2013 Putnam, 1
For positive integers $n,$ let the numbers $c(n)$ be determined by the rules $c(1)=1,c(2n)=c(n),$ and $c(2n+1)=(-1)^nc(n).$ Find the value of \[\sum_{n=1}^{2013}c(n)c(n+2).\]
2020 Durer Math Competition Finals, 16
Dora has $8$ rods with lengths $1, 2, 3, 4, 5, 6, 7$ and $8$ cm. Dora chooses $4$ of the rods and uses them to assemble a trapezoid (the $4$ chosen rods must be the $4$ sides). How many different trapezoids can she obtain in this way?
Two trapezoids are considered different if they are not congruent.
2013 National Chemistry Olympiad, 37
Three metals, $A, B $and $C$, with solutions of their respective cations are tested in a voltaic cell with the following results:
$A$ and $B$: $A$ is the cathode
$B$ and $C$: $C$ is the cathode
$A$ and $C$: $A$ is the anode
What is the order of the reduction potentials from highest to lowest for the cations of these metals?
$ \textbf{(A)}\ A>B>C \qquad\textbf{(B)}\ B>C>A\qquad$
${\textbf{(C)}\ C>A>B\qquad\textbf{(D}}\ B>A>C\qquad$
1951 Miklós Schweitzer, 8
Given a positive integer $ n>3$, prove that the least common multiple of the products $ x_1x_2\cdots x_k$ ($ k\geq 1$) whose factors $ x_i$ are positive integers with $ x_1\plus{}x_2\plus{}\cdots\plus{}x_k\le n$, is less than $ n!$.
2013 NIMO Problems, 4
Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \][i]Proposed by Evan Chen[/i]
2022 Sharygin Geometry Olympiad, 14
A triangle $ABC$ is given. Let $C'$ and $C'_{a}$ be the touching points of sideline $AB$ with the incircle and with the excircle touching the side $BC$. Points $C'_{b}$, $C'_{c}$, $A'$, $A'_{a}$, $A'_{b}$, $A'_{c}$, $B'$, $B'_{a}$, $B'_{b}$, $B'_{c}$ are defined similarly. Consider the lengths of $12$ altitudes of triangles $A'B'C'$, $A'_{a}B'_{a}C'_{a}$, $A'_{b}B'_{b}C'_{b}$, $A'_{c}B'_{c}C'_{c}$.
(a) (8-9) Find the maximal number of different lengths.
(b) (10-11) Find all possible numbers of different lengths.
2011 China Western Mathematical Olympiad, 4
In a circle $\Gamma_{1}$, centered at $O$, $AB$ and $CD$ are two unequal in length chords intersecting at $E$ inside $\Gamma_{1}$. A circle $\Gamma_{2}$, centered at $I$ is tangent to $\Gamma_{1}$ internally at $F$, and also tangent to $AB$ at $G$ and $CD$ at $H$. A line $l$ through $O$ intersects $AB$ and $CD$ at $P$ and $Q$ respectively such that $EP = EQ$. The line $EF$ intersects $l$ at $M$. Prove that the line through $M$ parallel to $AB$ is tangent to $\Gamma_{1}$
2016 EGMO TST Turkey, 5
A sequence $a_1, a_2, \ldots $ consisting of $1$'s and $0$'s satisfies for all $k>2016$ that
\[ a_k=0 \quad \Longleftrightarrow \quad a_{k-1}+a_{k-2}+\cdots+a_{k-2016}>23. \]
Prove that there exist positive integers $N$ and $T$ such that $a_k=a_{k+T}$ for all $k>N$.
2013 Sharygin Geometry Olympiad, 3
Let $X$ be a point inside triangle $ABC$ such that $XA.BC=XB.AC=XC.AC$. Let $I_1, I_2, I_3$ be the incenters of $XBC, XCA, XAB$. Prove that $AI_1, BI_2, CI_3$ are concurrent.
[hide]Of course, the most natural way to solve this is the Ceva sin theorem, but there is an another approach that may surprise you;), try not to use the Ceva theorem :))[/hide]
1973 All Soviet Union Mathematical Olympiad, 183
$N$ men are not acquainted each other. You need to introduce some of them to some of them in such a way, that no three men will have equal number of acquaintances. Prove that it is possible for all $N$.
1986 AMC 8, 18
A rectangular grazing area is to be fenced off on three sides using part of a $ 100$ meter rock wall as the fourth side. Fence posts are to be placed every $ 12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $ 36$ m by $ 60$ m?
\[ \textbf{(A)}\ 11 \qquad
\textbf{(B)}\ 12 \qquad
\textbf{(C)}\ 13 \qquad
\textbf{(D)}\ 14 \qquad
\textbf{(E)}\ 16
\]
2022 Federal Competition For Advanced Students, P2, 6
(a) Prove that a square with sides $1000$ divided into $31$ squares tiles, at least one of which has a side length less than $1$.
(b) Show that a corresponding decomposition into $30$ squares is also possible.
[i](Walther Janous)[/i]
2004 Hong kong National Olympiad, 2
In a school there $b$ teachers and $c$ students. Suppose that
a) each teacher teaches exactly $k$ students, and
b)for any two (distinct) students , exactly $h$ teachers teach both of them.
Prove that $\frac{b}{h}=\frac{c(c-1)}{k(k-1)}$.
2007 Hong Kong TST, 5
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url]
Problem 5
The sequence $\{a_{n}\}$ is defined by $a_{1}=0$ and $(n+1)^{3}a_{n+1}=2n^{2}(2n+1)a_{n}+2(3n+1)$ for all integers $\geq 1$. Show that infintely many members of the sequence are positive integers.
2003 Korea - Final Round, 1
Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively.
Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.
2020 May Olympiad, 5
On a table there are several cards, some face up and others face down. The allowed operation is to choose 4 cards and turn them over. The goal is to get all the cards in the same state (all face up or all face down). Determine if the objective can be achieved through a sequence of permitted operations if initially there are:
a) 101 cards face up and 102 face down;
b) 101 cards face up and 101 face down.
2017 IMC, 6
Let $f:[0;+\infty)\to \mathbb R$ be a continuous function such that $\lim\limits_{x\to +\infty} f(x)=L$ exists (it may be finite or infinite). Prove that $$ \lim\limits_{n\to\infty}\int\limits_0^{1}f(nx)\,\mathrm{d}x=L. $$
PEN A Problems, 2
Find infinitely many triples $(a, b, c)$ of positive integers such that $a$, $b$, $c$ are in arithmetic progression and such that $ab+1$, $bc+1$, and $ca+1$ are perfect squares.
1973 AMC 12/AHSME, 13
The fraction $ \frac{2(\sqrt2 \plus{} \sqrt6)}{3\sqrt{2\plus{}\sqrt3}}$ is equal to
$ \textbf{(A)}\ \frac{2\sqrt2}{3} \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ \frac{2\sqrt3}3 \qquad
\textbf{(D)}\ \frac43 \qquad
\textbf{(E)}\ \frac{16}{9}$
1999 Federal Competition For Advanced Students, Part 2, 1
Prove that for each positive integer $n$, the sum of the numbers of digits of $4^n$ and of $25^n$ (in the decimal system) is odd.
2015 Online Math Open Problems, 22
For a positive integer $n$ let $n\#$ denote the product of all primes less than or equal to $n$ (or $1$ if there are no such primes), and let $F(n)$ denote the largest integer $k$ for which $k\#$ divides $n$. Find the remainder when $F(1) + F(2) +F(3) + \dots + F(2015\#-1) + F(2015\#)$ is divided by $3999991$.
[i]Proposed by Evan Chen[/i]