Found problems: 85335
2012 Hanoi Open Mathematics Competitions, 3
[b]Q3.[/b] For any possitive integer $a$, let $\left[ a\right]$ denote the smallest prime factor of $a.$ Which of the following numbers is equal to $\left[ 35 \right]$ ?
\[(A) \; \left[10 \right]; \qquad (B) \; \left[ 15 \right]; \qquad (C ) \; \left[45 \right]; \qquad (D) \; \left[ 55 \right]; \qquad (E) \; \left[75 \right].\]
2000 Harvard-MIT Mathematics Tournament, 14
Define a sequence $<x_n>$ of real numbers by specifying an initial $x_0$ and by the recurrence $x_{n+1}=\frac{1+x_n}{1-x_n}$. Find $x_n$ as a function of $x_0$ and $n$, in closed form. There may be multiple cases.
2003 AIME Problems, 11
An angle $x$ is chosen at random from the interval $0^\circ < x < 90^\circ$. Let $p$ be the probability that the numbers $\sin^2 x$, $\cos^2 x$, and $\sin x \cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n$, where $d$ is the number of degrees in $\arctan m$ and $m$ and $n$ are positive integers with $m + n < 1000$, find $m + n$.
LMT Guts Rounds, 17
Determine the sum of the two largest prime factors of the integer $89! + 90!.$
2018 MIG, 2
The MIG is planning a lottery to give out prizes after the written tests, and the plan is very special. Contestants will be divided into prize groups in order to potentially receive a prize. However, based on the number of contestants, the ideal number of groups don't work. For example, when dividing into $4$ groups, there are $3$ left over. When dividing into $5$ groups, there's $2$ left over. When dividing into $6$ groups, theres $1$ left over. Finally, when dividing into $7$ groups, there are $2$ left over. With the knowledge that there are less than $300$ participants in the MIG, how many participants are there?
2000 Mongolian Mathematical Olympiad, Problem 5
Given a natural number $n$, find the number of quadruples $(x,y,u,v)$ of integers with $1\le x,y,y,v\le n$ satisfy the following inequalities:
\begin{align*}
&1\le v+x-y\le n,\\
&1\le x+y-u\le n,\\
&1\le u+v-y\le n,\\
&1\le v+x-u\le n.
\end{align*}
1988 IMO Longlists, 30
In the triangle $ABC$ let $D,E$ and $F$ be the mid-points of the three sides, $X,Y$ and $Z$ the feet of the three altitudes, $H$ the orthocenter, and $P,Q$ and $R$ the mid-points of the line segment joining $H$ to the three vertices. Show that the nine points $D,E,F,P,Q,R,X,Y,Z$ lie on a circle.
Novosibirsk Oral Geo Oly VIII, 2019.4
Given a triangle $ABC$, in which the angle $B$ is three times the angle $C$. On the side $AC$, point $D$ is chosen such that the angle $BDC$ is twice the angle $C$. Prove that $BD + BA = AC$.
2021 AMC 10 Spring, 22
Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$, the second sheet contains pages $3$ and $4$, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly $19$. How many sheets were borrowed?
$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 20$
2003 India IMO Training Camp, 1
Let $A',B',C'$ be the midpoints of the sides $BC, CA, AB$, respectively, of an acute non-isosceles triangle $ABC$, and let $D,E,F$ be the feet of the altitudes through the vertices $A,B,C$ on these sides respectively. Consider the arc $DA'$ of the nine point circle of triangle $ABC$ lying outside the triangle. Let the point of trisection of this arc closer to $A'$ be $A''$. Define analogously the points $B''$ (on arc $EB'$) and $C''$(on arc $FC'$). Show that triangle $A''B''C''$ is equilateral.
2008 Princeton University Math Competition, A6
Let $x$ be the largest root of $x^4 - 2009x + 1$. Find the nearest integer to $\frac{1}{x^3-2009}$ .
2017 Germany Team Selection Test, 1
The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?
1910 Eotvos Mathematical Competition, 2
Let $a, b, c, d$ and $u$ be integers such that each of the numbers
$$ac\ \ , \ \ bc + ad \ \ , \ \ bd$$
is a multiple of $u$. Show that $bc$ and $ad$ are multiples of $u$.
2019 Junior Balkan Team Selection Tests - Moldova, 5
Find all triplets of positive integers $(a, b, c)$ that verify $\left(\frac{1}{a}+1\right)\left(\frac{1}{b}+1\right)\left(\frac{1}{c}+1\right)=2$.
2004 All-Russian Olympiad, 2
Let $ABCD$ be a circumscribed quadrilateral (i. e. a quadrilateral which has an incircle). The exterior angle bisectors of the angles $DAB$ and $ABC$ intersect each other at $K$; the exterior angle bisectors of the angles $ABC$ and $BCD$ intersect each other at $L$; the exterior angle bisectors of the angles $BCD$ and $CDA$ intersect each other at $M$; the exterior angle bisectors of the angles $CDA$ and $DAB$ intersect each other at $N$. Let $K_{1}$, $L_{1}$, $M_{1}$ and $N_{1}$ be the orthocenters of the triangles $ABK$, $BCL$, $CDM$ and $DAN$, respectively. Show that the quadrilateral $K_{1}L_{1}M_{1}N_{1}$ is a parallelogram.
2020 Indonesia MO, 1
Since this is already 3 PM (GMT +7) in Jakarta, might as well post the problem here.
Problem 1. Given an acute triangle $ABC$ and the point $D$ on segment $BC$. Circle $c_1$ passes through $A, D$ and its centre lies on $AC$. Whereas circle $c_2$ passes through $A, D$ and its centre lies on $AB$. Let $P \neq A$ be the intersection of $c_1$ with $AB$ and $Q \neq A$ be the intersection of $c_2$ with $AC$. Prove that $AD$ bisects $\angle{PDQ}$.
1994 Tuymaada Olympiad, 1
World Cup in America introduced a new point system. For a victory $3$ points are given, for a draw $1$ point and for defeat $0$ points. In the preliminary games, the teams are divided into groups of $4$ teams. In groups, teams play with each other, once, then in accordance with the points scored $a,b,c$ and $d$ ($a>b>c>d$) teams take the first, second, third and fourth place in their groups. Give all possible options for the distribution points $a,b,c$ and $d$
2013 IFYM, Sozopol, 7
Let $a,b,c,$ and $d$ be real numbers and $k\geq l\geq m$ and $p\geq q\geq r$. Prove that
$f(x)=a(x+1)^k (x+2)^p+b(x+1)^l (x+2)^q+c(x+1)^m (x+2)^r-d=0$
has no more than 14 positive roots.
2015 Azerbaijan National Olympiad, 4
Natural number $M$ has $6$ divisors, such that sum of them are equal to $3500$.Find the all values of $M$.
2018 China Team Selection Test, 3
Prove that there exists a constant $C>0$ such that
$$H(a_1)+H(a_2)+\cdots+H(a_m)\leq C\sqrt{\sum_{i=1}^{m}i a_i}$$
holds for arbitrary positive integer $m$ and any $m$ positive integer $a_1,a_2,\cdots,a_m$, where $$H(n)=\sum_{k=1}^{n}\frac{1}{k}.$$
1988 Tournament Of Towns, (172) 5
Is it possible to cover a plane with circles in such a way that exactly $1988$ circles pass through each point?
( N . Vasiliev)
2002 Federal Math Competition of S&M, Problem 3
Find all pairs $(n,k)$ of positive integers such that $\binom nk=2002$.
2023 Israel Olympic Revenge, P1
Armadillo and Badger are playing a game. Armadillo chooses a nonempty tree (a simple acyclic graph) and places apples at some of its vertices (there may be several apples on a single vertex). First, Badger picks a vertex $v_0$ and eats all its apples. Next, Armadillo and Badger take turns alternatingly, with Armadillo starting. On the $i$-th turn the animal whose turn it is picks a vertex $v_i$ adjacent to $v_{i-1}$ that hasn't been picked before and eats all its apples. The game ends when there is no such vertex $v_i$.
Armadillo's goal is to have eaten more apples than Badger once the game ends. Can she fulfill her wish?
2005 Tournament of Towns, 2
The base-ten expressions of all the positive integers are written on an infinite ribbon without spacing: $1234567891011\ldots$. Then the ribbon is cut up into strips seven digits long. Prove that any seven digit integer will:
(a) appear on at least one of the strips; [i](3 points)[/i]
(b) appear on an infinite number of strips. [i](1 point)[/i]
2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2
For real numbers $b>a>0$, let $f : [0,\ \infty)\rightarrow \mathbb{R}$ be a continuous function.
Prove that :
(i) $\lim_{\epsilon\rightarrow +0} \int_{a\epsilon}^{b\epsilon} \frac{f(x)}{x}dx=f(0)\ln \frac{b}{a}.$
(ii) If $\int_1^{\infty} \frac{f(x)}{x}dx$ converges, then $\int_0^{\infty} \frac{f(bx)-f(ax)}{x}dx=f(0)\ln \frac{a}{b}.$