Found problems: 85335
2022 South East Mathematical Olympiad, 7
Let $a,b$ be positive integers.Prove that there are no positive integers on the interval $\bigg[\frac{b^2}{a^2+ab},\frac{b^2}{a^2+ab-1}\bigg)$.
2025 Sharygin Geometry Olympiad, 8
The diagonals of a cyclic quadrilateral $ABCD$ meet at point $P$. Points $K$ and $L$ lie on $AC$, $BD$ respectively in such a way that $CK=AP$ and $DL=BP$. Prove that the line joining the common points of circles $ALC$ and $BKD$ passes through the mass-center of $ABCD$.
Proposed by:V.Konyshev
Oliforum Contest V 2017, 4
Let $p_n$ be the $n$-th prime, so that $p_1 = 2, p_2 = 3,...$ and de fine $$X_n = \{0\} \cup \{p_1,...,p_n\}$$ for each positive integer $n$. Find all $n$ for which there exist $A,B \subseteq N$ such that$ |A|,|B| \ge 2$ and
$$X_n = A + B$$, where $A + B :=\{a + b : a \in A; b \in B \}$ and $N := \{0,1, 2,...\}$.
(Salvatore Tringali)
2016 Tuymaada Olympiad, 8
The flights map of air company $K_{r,r}$ presents several cities. Some cities are connected by a direct (two way) flight, the total number of flights is m. One must choose two non-intersecting groups of r cities each so that every city of the first group is connected by a flight with every city of the second group. Prove that number of possible choices does not exceed $2*m^r$ .
2010 Purple Comet Problems, 1
If $125 + n + 135 + 2n + 145 = 900,$ find $n.$
2006 MOP Homework, 3
For positive integer $k$, let $p(k)$ denote the greatest odd divisor of $k$. Prove that for every positive integer $n$,
$$\frac{2n}{3} < \frac{p(1)}{1}+ \frac{p(2)}{2}+... +\frac{ p(n)}{n}<\frac{2(n + 1)}{3}$$
2011 Argentina Team Selection Test, 4
Determine all positive integers $n$ such that the number $n(n+2)(n+4)$ has at most $15$ positive divisors.
2005 Bulgaria National Olympiad, 5
For positive integers $t,a,b,$a $(t,a,b)$-[i]game[/i] is a two player game defined by the following rules. Initially, the number $t$ is written on a blackboard. At his first move, the 1st player replaces $t$ with either $t-a$ or $t-b$. Then, the 2nd player subtracts either $a$ or $b$ from this number, and writes the result on the blackboard, erasing the old number. After this, the first player once again erases either $a$ or $b$ from the number written on the blackboard, and so on. The player who first reaches a negative number loses the game. Prove that there exist infinitely many values of $t$ for which the first player has a winning strategy for all pairs $(a,b)$ with $a+b=2005$.
1998 India National Olympiad, 1
In a circle $C_1$ with centre $O$, let $AB$ be a chord that is not a diameter. Let $M$ be the midpoint of this chord $AB$. Take a point $T$ on the circle $C_2$ with $OM$ as diameter. Let the tangent to $C_2$ at $T$ meet $C_1$ at $P$. Show that $PA^2 + PB^2 = 4 \cdot PT^2$.
2009 Peru IMO TST, 1
Show that there are infinitely many triples $(x, y, z)$ of real numbers such that $$\displaystyle{x^2+y = y^2+z= z^2 + x}$$ and $x\ne y\ne z \ne x.$
2002 Estonia National Olympiad, 1
Peeter, Juri, Kati and Mari are standing at the entrance of a dark tunnel. They have one torch and none of them dares to be in the tunnel without it, but the tunnel is so narrow that at most two people can move together. It takes $1$ minute for Peeter, $2$ minutes for Juri, $5$ for Kati and $10$ for Mari to pass the tunnel. Find the minimum time in which they can all pass through the tunnel.
1967 Poland - Second Round, 4
Solve the equation in natural numbers $$
xy+yz+zx = xyz + 2.
$$
2023 239 Open Mathematical Olympiad, 7
The diagonals of convex quadrilateral $ABCD$ intersect at point $E$. Triangles $ABE$ and $CED$ have a common excircle $\Omega$, tangent to segments $AE$ and $DE$ at points $B_1$ and $C_1$, respectively. Denote by $I$ and $J$ the centers of the incircles of these triangles, respectively. Segments $IC_1$ and $JB_1$ intersect at point $S$. It is known that $S$ lies on $\Omega$. Prove that the circumcircle of triangle $AED$ is tangent to $\Omega$.
[i]Proposed by David Brodsky[/i]
2022 Malaysia IMONST 2, 3
Given an integer $n$. We rearrange the digits of $n$ to get another number $m$. Prove that it is impossible to get $m+n = 999999999$.
1971 Miklós Schweitzer, 1
Let $ G$ be an infinite compact topological group with a Hausdorff topology. Prove that $ G$ contains an element $ g \not\equal{} 1$ such that the set of all powers of $ g$ is either everywhere dense in $ G$ or nowhere dense in $ G$.
[i]J. Erdos[/i]
1996 USAMO, 2
For any nonempty set $S$ of real numbers, let $\sigma(S)$ denote the sum of the elements of $S$. Given a set $A$ of $n$ positive integers, consider the collection of all distinct sums $\sigma(S)$ as $S$ ranges over the nonempty subsets of $A$. Prove that this collection of sums can be partitioned into $n$ classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.
2020 LMT Fall, 5
For what digit $d$ is the base $9$ numeral $7d35_9$ divisible by $8?$
[i]Proposed by Alex Li[/i]
2002 Regional Competition For Advanced Students, 4
Let $a_0, a_1, ..., a_{2002}$ be real numbers.
a) Show that the smallest of the values $a_k (1-a_{2002-k})$ ($0 \le k \le 2002$) the following applies:
it is smaller or equal to $1/4$.
b) Does this statement always apply to the smallest of the values $a_k (1-a_{2003-k})$ ($1 \le k \le 2002$) ?
c) Show for positive real numbers $a_0, a_1, ..., a_{2002}$ :
the smallest of the values $a_k (1-a_{2003-k})$ ($1 \le k \le 2002$) is less than or equal to $1/4$.
2018 Puerto Rico Team Selection Test, 5
In the square shown in the figure, find the value of $x$.
[img]https://cdn.artofproblemsolving.com/attachments/0/1/4659d5afa5b409d9264924735297d1188b0be3.png[/img]
2007 Czech and Slovak Olympiad III A, 4
The set $M=\{1,2,\ldots,2007\}$ has the following property: If $n$ is an element of $M$, then all terms in the arithmetic progression with its first term $n$ and common difference $n+1$, are in $M$. Does there exist an integer $m$ such that all integers greater than $m$ are elements of $M$?
1994 India National Olympiad, 6
Find all real-valued functions $f$ on the reals such that $f(-x) = -f(x)$, $f(x+1) = f(x) + 1$ for all $x$, and $f\left(\dfrac{1}{x}\right) = \dfrac{f(x)}{x^2}$ for $x \not = 0$.
1971 AMC 12/AHSME, 21
If $\log_2(\log_3(\log_4 x))=\log_3(\log_4(\log_2 y))=\log_4(\log_2(\log_3 z))=0$, then the sum $x+y+z$ is equal to
$\textbf{(A) }50\qquad\textbf{(B) }58\qquad\textbf{(C) }89\qquad\textbf{(D) }111\qquad \textbf{(E) }1296$
2022-IMOC, A2
Given positive integer $n>2,$ consider real numbers $a_1,a_2,\dots, a_n$ satisfying $a^{2}_1+a^2_2+\dots a^2_n=1.$ Find the maximal value of $$|a_1-a_2|+|a_2-a_3| +\dots +|a_n-a_1|.$$
[i]Proposed by ltf0501[/i]
LMT Speed Rounds, 2011.12
In a round robin tournament of $7$ people, each person plays every other person exactly once in a game of table tennis. For each game played, the winner is given $2$ points, the loser is given $0$ points, and in the event of a tie, each player gets $1$ point. At the end of the tournament, what is the average score of the $7$ people?
2019 CMI B.Sc. Entrance Exam, 6
$(a)$ Compute -
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg]
\end{align*}
$(b)$ For $x > 0 $ define $F ( x ) = \int_{1}^{x} t \log ( t ) \mathrm{d}t . $\\
\\$1.$ Determine the open interval(s) (if any) where $F ( x )$ is decreasing and all the open interval(s) (if any) where $F ( x )$ is increasing.\\
\\$2.$ Determine all the local minima of $F ( x )$ (if any) and all the local maxima of $F ( x )$ (if any) $.$