Found problems: 85335
2021 Romanian Master of Mathematics Shortlist, A2
Let $n$ be a positive integer and let $x_1,\ldots,x_n,y_1,\ldots,y_n$ be integers satisfying the following
condition: the numbers $x_1,\ldots,x_n$ are pairwise distinct and for every positive integer $m$ there
exists a polynomial $P_m$ with integer coefficients such that $P_m(x_i) - y_i$, $i=1,\ldots,n$, are all divisible by $m$. Prove that there exists a polynomial $P$ with integer coefficients such that $P(x_i) = y_i$ for all $i=1,\ldots,n$.
2010 Slovenia National Olympiad, 3
Find all functions $f: [0, +\infty) \to [0, +\infty)$ satisfying the equation
\[(y+1)f(x+y) = f\left(xf(y)\right)\]
For all non-negative real numbers $x$ and $y.$
1998 Estonia National Olympiad, 3
A function $f$ satisfies the conditions $f (x) \ne 0$ and $f (x+2) = f (x-1) f (x+5)$ for all real x. Show that $f (x+18) = f (x)$ for any real $x$.
2016 Stars of Mathematics, 3
Let $ n $ be a natural number, and $ 2n $ nonnegative real numbers $ a_1,a_2,\ldots ,a_{2n} $ such that $ a_1a_2\cdots a_{2n}=1. $ Show that
$$ 2^{n+1} +\left( a_1^2+a_2^2 \right)\left( a_3^2+a_4^2 \right)\cdots\left( a_{2n-1}^2+a_{2n}^2 \right) \ge 3\left( a_1+a_2 \right)\left( a_3+a_4 \right)\cdots\left( a_{2n-1}+a_{2n} \right) , $$
and specify in which circumstances equality happens.
2018 HMNT, 1
Four standard six-sided dice are rolled. Find the probability that, for each pair of dice, the product of the two numbers rolled on those dice is a multiple of 4.
2012 Balkan MO Shortlist, A3
Determine the maximum possible number of distinct real roots of a polynomial $P(x)$ of degree $2012$ with real coefficients satisfying the condition
\begin{align*} P(a)^3 + P(b)^3 + P(c)^3 \geq 3 P(a) P(b) P(c) \end{align*}
for all real numbers $a,b,c \in \mathbb{R}$ with $a+b+c=0$
2022 Canadian Mathematical Olympiad Qualification, 2
Determine all pairs of integers $(m, n)$ such that $m^2 + n$ and $n^2 + m$ are both perfect squares.
2020 LIMIT Category 1, 15
In a $4\times 4$ chessboard, in how many ways can you place $3$ rooks and one bishop such that none of these pieces threaten another piece?
2020 Online Math Open Problems, 24
In graph theory, a [i]triangle[/i] is a set of three vertices, every two of which are connected by an edge. For an integer $n \geq 3$, if a graph on $n$ vertices does not contain two triangles that do not share any vertices, let $f(n)$ be the maximum number of triangles it can contain. Compute $f(3) + f(4) + \cdots + f(100).$
[i]Proposed by Edward Wan[/i]
2000 Switzerland Team Selection Test, 13
The incircle of a triangle $ABC$ touches the sides $AB,BC,CA$ at points $D,E,F$ respectively. Let $P$ be an internal point of triangle $ABC$ such that the incircle of triangle $ABP$ touches $AB$ at $D$ and the sides $AP$ and $BP$ at $Q$ and $R$. Prove that the points $E,F,R,Q$ lie on a circle.
2012 Middle European Mathematical Olympiad, 1
Find all triplets $ (x,y,z) $ of real numbers such that
\[ 2x^3 + 1 = 3zx \]\[ 2y^3 + 1 = 3xy \]\[ 2z^3 + 1 = 3yz \]
1997 Canadian Open Math Challenge, 5
Two cubes have their faces painted either red or blue. The 1st cube has five red faces and one blue face. When the two cubes are rolled simultaneously, the probability that the two top faces show the same color is $\frac{1}{2}$. How many red faces are there on the second cube?
1995 Turkey Team Selection Test, 2
Let $n\in\mathbb{N}$ be given. Prove that the following two conditions are equivalent:
$\quad(\text{i})\: n|a^n-a$ for any positive integer $a$;
$\quad(\text{ii})\:$ For any prime divisor $p$ of $n$, $p^2 \nmid n$ and $p-1|n-1$.
2012 Olympic Revenge, 2
We define $(x_1, x_2, \ldots , x_n) \Delta (y_1, y_2, \ldots , y_n) = \left( \sum_{i=1}^{n}x_iy_{2-i}, \sum_{i=1}^{n}x_iy_{3-i}, \ldots , \sum_{i=1}^{n}x_iy_{n+1-i} \right)$, where the indices are taken modulo $n$.
Besides this, if $v$ is a vector, we define $v^k = v$, if $k=1$, or $v^k = v \Delta v^{k-1}$, otherwise.
Prove that, if $(x_1, x_2, \ldots , x_n)^k = (0, 0, \ldots , 0)$, for some natural number $k$, then $x_1 = x_2 = \ldots = x_n = 0$.
2021 USAJMO, 2
Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that \[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\] Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.
2002 District Olympiad, 2
a) Let $x$ be a real number such that $x^2+x$ and $x^3+2x$ are rational numbers. Show that $x$ is a rational number.
b) Show that there exist irrational numbers $x$ such that $x^2+x$and $x^3-2x$ are rational.
2013 Purple Comet Problems, 12
How many four-digit positive integers have no adjacent equal even digits? For example, count numbers such as $1164$ and $2035$ but not $6447$ or $5866$.
2018 Yasinsky Geometry Olympiad, 5
In the trapezium $ABCD$ ($AD // BC$), the point $M$ lies on the side of $CD$, with $CM:MD=2:3$, $AB=AD$, $BC:AD=1:3$. Prove that $BD \perp AM$.
2013 AMC 12/AHSME, 25
Let $f : \mathbb{C} \to \mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \text{Im}(z) > 0 $ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $ 10 $?
${ \textbf{(A)} \ 399 \qquad \textbf{(B)} \ 401 \qquad \textbf{(C)} \ 413 \qquad \textbf{(D}} \ 431 \qquad \textbf{(E)} \ 441 $
2011 IMO, 4
Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
Determine the number of ways in which this can be done.
[i]Proposed by Morteza Saghafian, Iran[/i]
2008 Gheorghe Vranceanu, 3
If the circumradius of any three consecutive vertices of a convex polygon is at most $ 1, $ show that the discs of radius $ 1 $ centered at each vertex cover the polygon and its interior.
2006 Belarusian National Olympiad, 3
A finite set $V \in Z^2$ of vectors with integer coordinates is chosen on the plane. Each of them is painted one of the $n$ colors. The color is [i]suitable[/i] for the vector if this vector may be presented as' a linear combination (with integer coefficients) of the vectors from $V$ of this color. It is known,that for any vector from $Z^2$ there exist a suitable color. Find all $n$ such that there must exist a color which is suitable for any vector from $Z^2$ .
(V. Lebed)
2009 Korea Junior Math Olympiad, 5
Acute triangle $\triangle ABC$ satis es $AB < AC$. Let the circumcircle of this triangle be $O$, and the midpoint of $BC,CA,AB$ be $D,E,F$. Let $P$ be the intersection of the circle with $AB$ as its diameter and line $DF$, which is in the same side of $C$ with respect to $AB$. Let $Q$ be the intersection of the circle with $AC$ as its diameter and the line $DE$, which is in the same side of $B$ with respect to $AC$. Let $PQ \cap BC = R$, and let the line passing through $R$ and perpendicular to $BC$ meet $AO$ at $X$. Prove that $AX = XR$.
2012 Kosovo National Mathematical Olympiad, 3
Prove that for any integer $n\geq 2$ it holds that
$\dbinom {2n}{n}>\frac {4^n}{2n}$.
2016 Romania Team Selection Tests, 3
Given a positive integer $n$, show that for no set of integers modulo $n$, whose size exceeds $1+\sqrt{n+4}$, is it possible that the pairwise sums of unordered pairs be all distinct.