Found problems: 85335
1971 IMO Longlists, 38
Let $A,B,C$ be three points with integer coordinates in the plane and $K$ a circle with radius $R$ passing through $A,B,C$. Show that $AB\cdot BC\cdot CA\ge 2R$, and if the centre of $K$ is in the origin of the coordinates, show that $AB\cdot BC\cdot CA\ge 4R$.
1963 AMC 12/AHSME, 9
In the expansion of $\left(a-\dfrac{1}{\sqrt{a}}\right)^7$ the coefficient of $a^{-\dfrac{1}{2}}$ is:
$\textbf{(A)}\ -7 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ -21 \qquad
\textbf{(D)}\ 21 \qquad
\textbf{(E)}\ 35$
2023 CMIMC Team, 8
NASA is launching a spaceship at the south pole, but a sudden earthquake shock caused the spaceship to be launched at an angle of $\theta$ from vertical ($0 < \theta < 90^\circ$). The spaceship crashed back to Earth, and NASA found the debris floating in the ocean in the northern hemisphere. NASA engineers concluded that $\tan \theta > M$, where $M$ is maximal. Find $M$.
Assume that the Earth is a sphere, and the trajectory of the spaceship (in the reference frame of Earth) is an ellipse with the center of the Earth one of the foci.
[i]Proposed by Kevin You[/i]
2020 ITAMO, 4
Let $ABC$ be an acute-angled triangle with $AB=AC$, let $D$ be the foot of perpendicular, of the point $C$, to the line $AB$ and the point $M$ is the midpoint of $AC$. Finally, the point $E$ is the second intersection of the line $BC$ and the circumcircle of $\triangle CDM$. Prove that the lines $AE, BM$ and $CD$ are concurrents if and only if $CE=CM$.
2017 F = ma, 3
A ball of radius R and mass m is magically put inside a thin shell of the same mass and radius 2R. The system is at rest on a horizontal frictionless surface initially. When the ball is, again magically, released inside the shell, it sloshes around in the shell and eventually stops at the bottom of the shell. How far does the shell move from its initial contact point with the surface?
$\textbf{(A)}R\qquad
\textbf{(B)}\frac{R}{2}\qquad
\textbf{(C)}\frac{R}{4}\qquad
\textbf{(D)}\frac{3R}{8}\qquad
\textbf{(E)}\frac{R}{8}$
2018 May Olympiad, 4
Anna must write $7$ positive integers, not necessarily distinct, around a circle such that the following conditions are met:
$\bullet$ The sum of the seven numbers equals $36$.
$\bullet$ If two numbers are neighbours, the difference between the largest and the smallest is equal to $2$ or $3$.
Find the maximum value of the largest of the numbers that Anna can write.
2011 IFYM, Sozopol, 5
The vertices of $\Delta ABC$ lie on the graphics of the function $f(x)=x^2$ and its centroid is $M(1,7)$. Determine the greatest possible value of the area of $\Delta ABC$.
2021 Novosibirsk Oral Olympiad in Geometry, 1
Cut the $9 \times 10$ grid rectangle along the grid lines into several squares so that there are exactly two of them with odd sidelengths.
2018 Thailand TST, 2
Find all pairs $(p,q)$ of prime numbers which $p>q$ and
$$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$
is an integer.
2015 Portugal MO, 3
The numbers from $1$ to $2015$ are written on sheets so that if if $n-m$ is a prime, then $n$ and $m$ are on different sheets. What is the minimum number of sheets required?
2022 Romania National Olympiad, P1
Let $a\neq 1$ be a positive real number. Find all real solutions to the equation $a^x=x^x+\log_a(\log_a(x)).$
[i]Mihai Opincariu[/i]
2009 Croatia Team Selection Test, 2
Every natural number is coloured in one of the $ k$ colors. Prove that there exist four distinct natural numbers $ a, b, c, d$, all coloured in the same colour, such that $ ad \equal{} bc$, $ \displaystyle \frac b a$ is power of 2 and $ \displaystyle \frac c a$ is power of 3.
2002 All-Russian Olympiad, 3
On a plane are given finitely many red and blue lines, no two parallel, such that any intersection point of two lines of the same color also lies on another line of the other color. Prove that all the lines pass through a single point.
2022 Girls in Math at Yale, 11
Georgina calls a $992$-element subset $A$ of the set $S = \{1, 2, 3, \ldots , 1984\}$ a [b]halfthink set[/b] if
[list]
[*] the sum of the elements in $A$ is equal to half of the sum of the elements in $S$, and
[*] exactly one pair of elements in $A$ differs by $1$.
[/list]
She notices that for some values of $n$, with $n$ a positive integer between $1$ and $1983$, inclusive, there are no halfthink sets containing both $n$ and $n+1$. Find the last three digits of the product of all possible values of $n$.
[i]Proposed by Andrew Wu and Jason Wang[/i]
(Note: wording changed from original to specify what $n$ can be.)
2005 Tournament of Towns, 5
A cube lies on the plane. After being rolled a few times (over its edges), it is brought back to its initial location with the same face up. Could the top face have been rotated by 90 degrees?
[i](5 points)[/i]
1994 Bundeswettbewerb Mathematik, 2
Two students $ A$ and $ B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $ A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $ B.$ If $ B$ answers “no,” the referee puts the question back to $ A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”
2021 Thailand Online MO, P2
Determine all integers $n>1$ that satisfy the following condition: for any positive integer $x$, if gcd$(x,n)=1$, then gcd$(x+101,n)=1$.
2013 Today's Calculation Of Integral, 878
A cubic function $f(x)$ satisfies the equation $\sin 3t=f(\sin t)$ for all real numbers $t$.
Evaluate $\int_0^1 f(x)^2\sqrt{1-x^2}\ dx$.
2020 Princeton University Math Competition, A5/B7
We say that a positive integer $n$ is [i]divable [/i] if there exist positive integers $1 < a < b < n$ such that, if the base-$a$ representation of $n$ is $\sum_{i=0}^{k_1} a_ia^i$ , and the base-$b$ representation of $n$ is $\sum_{i=0}^{k_2} b_ib^i$ , then for all positive integers $c > b$, we have that $\sum_{i=0}^{k_2} b_ic^i$ divides $\sum_{i=0}^{k_1} a_ic^i$. Find the number of non-divable $n$ such that $1 \le n \le 100$.
2020 May Olympiad, 5
We say that a positive integer $n$ is circular if it is possible to place the numbers $1, 2, \cdots , n$ in a
circumference so that there are no three adjacent numbers whose sum is a multiple of 3.
a) Show that 9 is not circular
b) Show that any integer greater than 9 is circular.
2015 VTRMC, Problem 1
Find all n such that $n^{4}+6n^{3}+11n^{2}+3n+31$ is a perfect square.
1987 Traian Lălescu, 2.1
Any polynom, with coefficients in a given division ring, that is irreducible over it, is also irreducible over a given extension skew ring of it that's finite. Prove that the ring and its extension coincide.
2016 IFYM, Sozopol, 3
Find the least natural number $n\geq 5$, for which $x^n\equiv 16\, (mod\, p)$ has a solution for any prime number $p$.
2019 Saudi Arabia JBMO TST, 4
A positive integer $n$ is called $nice$, if the sum of the squares of all its positive divisors is equal to $(n+3)^2$. Prove that if $n=pq$ is nice, where $p, q$ are not necessarily distinct primes, then $n+2$ and $2(n+1)$ are simultaneously perfect squares.
2012 USAMO, 6
For integer $n\geq2$, let $x_1, x_2, \ldots, x_n$ be real numbers satisfying \[x_1+x_2+\ldots+x_n=0, \qquad \text{and}\qquad x_1^2+x_2^2+\ldots+x_n^2=1.\]For each subset $A\subseteq\{1, 2, \ldots, n\}$, define\[S_A=\sum_{i\in A}x_i.\](If $A$ is the empty set, then $S_A=0$.)
Prove that for any positive number $\lambda$, the number of sets $A$ satisfying $S_A\geq\lambda$ is at most $2^{n-3}/\lambda^2$. For which choices of $x_1, x_2, \ldots, x_n, \lambda$ does equality hold?