Found problems: 85335
2015 Azerbaijan National Olympiad, 3
Find all polynomials $P(x)$ with real coefficents such that \[P(P(x))=(x^2+x+1)\cdot P(x)\] where $x \in \mathbb{R}$
2003 Gheorghe Vranceanu, 4
Let $ I $ be the incentre of $ ABC $ and $ D,E,F $ be the feet of the perpendiculars from $ I $ to $ BC,CA,AB, $ respectively. Show that
$$ \frac{AB}{DE} +\frac{BC}{EF} +\frac{CA}{FD}\ge 6. $$
2014 India IMO Training Camp, 3
For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$.
Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that $f^{k}((n,m))=(m,n)$,where
$f^{k}(x)=f(f(f(...f(x))))$,$f$ being composed with itself $k$ times.
2001 Moldova National Olympiad, Problem 1
Prove that $y\sqrt{3-2x}+x\sqrt{3-2y}\le x^2+y^2$ for any number $x,y\in\left[1,\frac32\right]$. When does equality occur?
2006 Romania Team Selection Test, 2
Let $m$ and $n$ be positive integers and $S$ be a subset with $(2^m-1)n+1$ elements of the set $\{1,2,3,\ldots, 2^mn\}$. Prove that $S$ contains $m+1$ distinct numbers $a_0,a_1,\ldots, a_m$ such that $a_{k-1} \mid a_{k}$ for all $k=1,2,\ldots, m$.
1999 Denmark MO - Mohr Contest, 1
In a coordinate system, a circle with radius $7$ and center is on the y-axis placed inside the parabola with equation $y = x^2$ , so that it just touches the parabola in two points. Determine the coordinate set for the center of the circle.
1993 Abels Math Contest (Norwegian MO), 1b
Given a triangle with sides of lengths $a,b,c$, prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}< 2$.
1988 IMO Longlists, 63
Let $ p$ be the product of two consecutive integers greater than 2. Show that there are no integers $ x_1, x_2, \ldots, x_p$ satisfying the equation
\[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1
\]
[b]OR[/b]
Show that there are only two values of $ p$ for which there are integers $ x_1, x_2, \ldots, x_p$ satisfying
\[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1
\]
1979 Romania Team Selection Tests, 6.
If $n>2$ is a positive integer, compute
\[\max_{1\leqslant k\leqslant n}\max_{n_1+...+n_k=n}
\binom{n_1}{2}\binom{n_2}{2}\ldots\binom{n_k}{2}.\]
[i]Ioan Tomescu[/i]
1998 Gauss, 6
In the multiplication question, the sum of the digits in the
four boxes is
[img]http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNy83L2NmMTU0MzczY2FhMGZhM2FjMjMwZDcwYzhmN2ViZjdmYjM4M2RmLnBuZw==&rn=U2NyZWVuc2hvdCAyMDE3LTAyLTI1IGF0IDUuMzguMjYgUE0ucG5n[/img]
$\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 22$
2020 Germany Team Selection Test, 3
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
2008 Puerto Rico Team Selection Test, 6
Let $n$ be a natural composite number. Prove that there are integers $a_1, a_2,. . . , a_k$ all greater than $ 1$, such that $$a_1 + a_2 +... + a_k = n \left(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}\right)$$
2014 MMATHS, 4
Determine, with proof, the maximum and minimum among the numbers
$$\sqrt5 - \lfloor \sqrt5 \rfloor, 2\sqrt5 - \lfloor 2\sqrt5 \rfloor, 3\sqrt5 - \lfloor 3
\sqrt5\rfloor, ..., 2013\sqrt5 - \lfloor 2013\sqrt5\rfloor, 2014\sqrt5 - \lfloor 2014\sqrt5\rfloor $$
1968 All Soviet Union Mathematical Olympiad, 108
Each of the $9$ referees on the figure skating championship estimates the program of $20$ sportsmen by assigning him a place (from $1$ to $20$). The winner is determined by adding those numbers. (The less is the sum - the higher is the final place). It was found, that for the each sportsman, the difference of the places, received from the different referees was not greater than $3$. What can be the maximal sum for the winner?
2006 District Olympiad, 2
A $9\times 9$ array is filled with integers from 1 to 81. Prove that there exists $k\in\{1,2,3,\ldots, 9\}$ such that the product of the elements in the row $k$ is different from the product of the elements in the column $k$ of the array.
2004 Iran MO (3rd Round), 27
$ \Delta_1,\ldots,\Delta_n$ are $ n$ concurrent segments (their lines concur) in the real plane. Prove that if for every three of them there is a line intersecting these three segments, then there is a line that intersects all of the segments.
2024 Singapore Senior Math Olympiad, Q3
Find the smallest positive integer $n$ for which there exist integers $x_{1} < x_{2} <...< x_{n}$ such that every integer from $1000$ to $2000$ can be written as a sum of some of the integers from $x_1,x_2,..,x_n$ without repetition.
2010 Contests, 1
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x, y\in\mathbb{R}$, we have
\[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]
2004 AIME Problems, 10
A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
1984 Spain Mathematical Olympiad, 5
Let $A$ and $A' $ be fixed points on two equal circles in the plane and let $AB$ and $A' B'$ be arcs of these circles of the same length $x$. Find the locus of the midpoint of segment $BB'$ when $x$ varies:
(a) if the arcs have the same direction,
(b) if the arcs have opposite directions.
2004 Harvard-MIT Mathematics Tournament, 5
A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a straight trajectory at $1$ meter per second. Suddenly someone turns on the turntable; it spins at $30$ rpm. Consider the set $S$ of points the mouse can reach in his car within $1$ second after the turntable is set in motion. What is the area of $S$, in square meters?
1986 AMC 8, 9
[asy]size(100);
draw((0,0)--(5,0),MidArrow);
draw((5,0)--(10,0),MidArrow);
draw((5,5sqrt(3))--(2.5,2.5sqrt(3)),MidArrow);
draw((2.5,2.5sqrt(3))--(0,0),MidArrow);
draw((5,5sqrt(3))--(7.5,2.5sqrt(3)),MidArrow);
draw((7.5,2.5sqrt(3))--(10,0),MidArrow);
draw((7.5,2.5sqrt(3))--(2.5,2.5sqrt(3)),MidArrow);
draw((7.5,2.5sqrt(3))--(5,0),MidArrow);
draw((2.5,2.5sqrt(3))--(5,0),MidArrow);
label("D",(0,0),SW);
label("C",(5,0),S);
label("N",(10,0),SE);
label("A",(2.5,2.5sqrt(3)),W);
label("B",(7.5,2.5sqrt(3)),E);
label("M",(5,5sqrt(3)),N);[/asy]
Using only the paths and the directions shown, how many different routes are there from $ M$ to $ N$?
\[ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ 6
\]
1983 IMO Longlists, 21
Prove that there are infinitely many positive integers $n$ for which it is possible for a knight, starting at one of the squares of an $n \times n$ chessboard, to go through each of the squares exactly once.
1958 Polish MO Finals, 2
Each side of a convex quadrilateral $ ABCD $ is divided into three equal parts; a straight line is drawn through the dividing points of sides $ AB $ and $ AD $ that lie closer to vertex $ A $, and similarly for vertices $ B $, $ C $, $ D $. Prove that the center of gravity of the quadrilateral formed by the drawn lines coincides with the center of gravity of quadrilateral $ ABCD $.
2016 AMC 12/AHSME, 23
Three numbers in the interval [0,1] are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area?
$\textbf{(A) }\frac16\qquad\textbf{(B) }\frac13\qquad\textbf{(C) }\frac12\qquad\textbf{(D) }\frac23\qquad\textbf{(E) }\frac56$