Found problems: 85335
2007 Moldova National Olympiad, 12.6
Show that the distance between a point on the hyperbola $xy=5$ and a point on the ellipse $x^{2}+6y^{2}=6$ is at least $\frac{9}{7}$.
2011 Indonesia TST, 3
Circle $\omega$ is inscribed in quadrilateral $ABCD$ such that $AB$ and $CD$ are not parallel and
intersect at point $O.$ Circle $\omega_1$ touches the side $BC$ at $K$ and touches line $AB$ and $CD$ at
points which are located outside quadrilateral $ABCD;$ circle $\omega_2$ touches side $AD$ at $L$ and
touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD.$ If $O,K,$
and $L$ are collinear$,$ then show that the midpoint of side $BC,AD,$ and the center of circle
$\omega$ are also collinear.
2016 Azerbaijan JBMO TST, 2
Let the angle bisectors of $\angle BAC,$ $\angle CBA,$ and $\angle ACB$ meets the circumcircle of $\triangle ABC$ at the points $M,N,$ and $K,$ respectively. Let the segments $AB$ and $MK$ intersects at the point $P$ and the segments $AC$ and $MN$ intersects at the point $Q.$ Prove that $PQ\parallel BC$
2004 Romania National Olympiad, 2
The sidelengths of a triangle are $a,b,c$.
(a) Prove that there is a triangle which has the sidelengths $\sqrt a,\sqrt b,\sqrt c$.
(b) Prove that $\displaystyle \sqrt{ab}+\sqrt{bc}+\sqrt{ca} \leq a+b+c < 2 \sqrt{ab} + 2 \sqrt{bc} + 2 \sqrt{ca}$.
2008 Tournament Of Towns, 6
Seated in a circle are $11$ wizards. A different positive integer not exceeding $1000$ is pasted onto the forehead of each. A wizard can see the numbers of the other $10$, but not his own. Simultaneously, each wizard puts up either his left hand or his right hand. Then each declares the number on his forehead at the same time. Is there a strategy on which the wizards can agree beforehand, which allows each of them to make the correct declaration?
2014 Iran MO (3rd Round), 5
Can an infinite set of natural numbers be found, such that for all triplets $(a,b,c)$ of it we have $abc + 1 $ perfect square?
(20 points )
2024/2025 TOURNAMENT OF TOWNS, P5
Given a polynomial with integer coefficients, which has at least one integer root. The greatest common divisor of all its integer roots equals $1$. Prove that if the leading coefficient of the polynomial equals $1$ then the greatest common divisor of the other coefficients also equals $1$.
2007 Princeton University Math Competition, 5
For how many integers $x \in [0, 2007]$ is $\frac{6x^3+53x^2+61x+7}{2x^2+17x+15}$ reducible?
2014 France Team Selection Test, 3
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
1979 IMO, 2
Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$
1979 IMO Longlists, 16
Let $Q$ be a square with side length $6$. Find the smallest integer $n$ such that in $Q$ there exists a set $S$ of $n$ points with the property that any square with side $1$ completely contained in $Q$ contains in its interior at least one point from $S$.
2018 District Olympiad, 4
Let $f:\mathbb{R} \to\mathbb{R}$ be a function. For every $a\in\mathbb{Z}$ consider the function $f_a : \mathbb{R} \to\mathbb{R}$, $f_a(x) = (x - a)f(x)$. Prove that if there exist infinitely many values $a\in\mathbb{Z}$ for which the functions $f_a$ are increasing, then the function $f$ is monotonic.
2016 China Northern MO, 3
Prove:
[b](a)[/b] There are infinitely many positive intengers $n$, satisfying:
$$\gcd(n,[\sqrt2n])=1.$$
[b](b)[/b] There are infinitely many positive intengers $n$, satisfying:
$$\gcd(n,[\sqrt2n])>1.$$
2009 Bosnia And Herzegovina - Regional Olympiad, 1
Prove that for every positive integer $m$ there exists positive integer $n$ such that $m+n+1$ is perfect square and $mn+1$ is perfect cube of some positive integers
2023 Durer Math Competition Finals, 5
We are given a triangle $ABC$ and two circles ($k_1$ and $k_2$) so the diameter of $k_1$ is $AB$ and the diameter of $k_2$ is $AC$. Let the intersection of $BC$ line segment and $k_1$ (that isn’t $B$) be $P,$ and the intersection of $BC$ line segment and $k_2$ (that isn’t $B$) be $Q$. We know, that $AB = 3003$ and $AC = 4004$ and $BC = 5005$. What is the distance between $P$ and $Q$?
2010 Gheorghe Vranceanu, 1
[b]a)[/b] Prove that any multiple of $ 6 $ is the sum of four cubes.
[b]b)[/b] Show that any integer is the sum of five cubes.
2010 239 Open Mathematical Olympiad, 6
We call natural numbers $n$ and $k$ are similar if they are multiples of square of a number greater than $1$. Let $f(n)$ denote the number of numbers from $1$ to $n$ similar to $n$ (for example, $f(16)=4$, since the number $16$ is similar to $4$, $8$, $12$ and $16$). What integer values can the quotient $\frac{n}{f(n)}$ take?
DMM Devil Rounds, 2010
[b]p1.[/b] Find all $x$ such that $(\ln (x^4))^2 = (\ln (x))^6$.
[b]p2.[/b] On a piece of paper, Alan has written a number $N$ between $0$ and $2010$, inclusive. Yiwen attempts to guess it in the following manner: she can send Alan a positive number $M$, which Alan will attempt to subtract from his own number, which we will call $N$. If $M$ is less than or equal $N$, then he will erase $N$ and replace it with $N -M$. Otherwise, Alan will tell Yiwen that $M > N$. What is the minimum number of attempts that Yiwen must make in order to determine uniquely what number Alan started with?
[b]p3.[/b] How many positive integers between $1$ and $50$ have at least $4$ distinct positive integer divisors? (Remember that both $1$ and $n$ are divisors of $n$.)
[b]p4.[/b] Let $F_n$ denote the $n^{th}$ Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. Find the last digit of $$\sum^{97!+4}_{i=0}F_i.$$
[b]p5.[/b] Find all prime numbers $p$ such that $2p + 1$ is a perfect cube.
[b]p6.[/b] What is the maximum number of knights that can be placed on a $9\times 9$ chessboard such that no two knights attack each other?
[b]p7.[/b] $S$ is a set of $9$ consecutive positive integers such that the sum of the squares of the $5$ smallest integers in the set is the sum of the squares of the remaining $4$. What is the sum of all $9$ integers?
[b]p8.[/b] In the following infinite array, each row is an arithmetic sequence, and each column is a geometric sequence. Find the sum of the infinite sequence of entries along the main diagonal.
[img]https://cdn.artofproblemsolving.com/attachments/5/1/481dd1e496fed6931ee2912775df630908c16e.png[/img]
[b]p9.[/b] Let $x > y > 0$ be real numbers. Find the minimum value of $\frac{x}{y} + \frac{4x}{x-y}$ .
[b]p10.[/b] A regular pentagon $P = A_1A_2A_3A_4A_5$ and a square $S = B_1B_2B_3B_4$ are both inscribed in the unit circle. For a given pentagon $P$ and square $S$, let $f(P, S)$ be the minimum length of the minor arcs $A_iB_j$ , for $1 \le i \le 5$ and $1 \le j \le4$. Find the maximum of $f(P, S)$ over all pairs of shapes.
[b]p11.[/b] Find the sum of the largest and smallest prime factors of $9^4 + 3^4 + 1$.
[b]p12.[/b] A transmitter is sending a message consisting of $4$ binary digits (either ones or zeros) to a receiver. Unfortunately, the transmitter makes errors: for each digit in the message, the probability that the transmitter sends the correct digit to the receiver is only $80\%$. (Errors are independent across all digits.) To avoid errors, the receiver only accepts a message if the sum of the first three digits equals the last digit modulo $2$. If the receiver accepts a message, what is the probability that the message was correct?
[b]p13.[/b] Find the integer $N$ such that $$\prod^{8}_{i=0}\sec \left( \frac{\pi}{9}2^i \right)= N.$$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1969 IMO Longlists, 35
$(HUN 2)$ Prove that $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}<\frac{5}{4}$
2011 ELMO Shortlist, 5
Prove there exists a constant $c$ (independent of $n$) such that for any graph $G$ with $n>2$ vertices, we can split $G$ into a forest and at most $cf(n)$ disjoint cycles, where
a) $f(n)=n\ln{n}$;
b) $f(n)=n$.
[i]David Yang.[/i]
2018 PUMaC Live Round, 4.1
The number $400000001$ can be written as $p\cdot q$, where $p$ and $q$ are prime numbers. Find the sum of the prime factors of $p+q-1$.
Indonesia MO Shortlist - geometry, g1.1
Given triangle $ ABC$. Points $ D,E,F$ outside triangle $ ABC$ are chosen such that triangles $ ABD$, $ BCE$, and $ CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.
2017 Denmark MO - Mohr Contest, 4
Let $A, B, C$ and $D$ denote the digits in a four-digit number $n = ABCD$. Determine the least $n$ greater than $2017$ satisfying that there exists an integer $x$ such that $$x =\sqrt{A +\sqrt{B +\sqrt{C +\sqrt{D + x}}}}.$$
1961 AMC 12/AHSME, 13
The symbol $|a|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number. For all real values of $t$ the expression $\sqrt{t^4+t^2}$ is equal to:
${{ \textbf{(A)}\ t^3 \qquad\textbf{(B)}\ t^2+t \qquad\textbf{(C)}\ |t^2+t| \qquad\textbf{(D)}\ t\sqrt{t^2+1} }\qquad\textbf{(E)}\ |t|\sqrt{1+t^2} } $
2016 Indonesia Juniors, day 1
p1. Find all real numbers that satisfy the equation $$(1 + x^2 + x^4 + .... + x^{2014})(x^{2016} + 1) = 2016x^{2015}$$
p2. Let $A$ be an integer and $A = 2 + 20 + 201 + 2016 + 20162 + ... + \underbrace{20162016...2016}_{40\,\, digits}$
Find the last seven digits of $A$, in order from millions to units.
p3. In triangle $ABC$, points $P$ and $Q$ are on sides of $BC$ so that the length of $BP$ is equal to $CQ$, $\angle BAP = \angle CAQ$ and $\angle APB$ is acute. Is triangle $ABC$ isosceles? Write down your reasons.
p4. Ayu is about to open the suitcase but she forgets the key. The suitcase code consists of nine digits, namely four $0$s (zero) and five $1$s. Ayu remembers that no four consecutive numbers are the same. How many codes might have to try to make sure the suitcase is open?
p5. Fulan keeps $100$ turkeys with the weight of the $i$-th turkey, being $x_i$ for $i\in\{1, 2, 3, ... , 100\}$. The weight of the $i$-th turkey in grams is assumed to follow the function $x_i(t) = S_it + 200 - i$ where $t$ represents the time in days and $S_i$ is the $i$-th term of an arithmetic sequence where the first term is a positive number $a$ with a difference of $b =\frac15$. It is known that the average data on the weight of the hundred turkeys at $t = a$ is $150.5$ grams. Calculate the median weight of the turkey at time $t = 20$ days.