Found problems: 85335
2009 Germany Team Selection Test, 3
Let $ A,B,C,M$ points in the plane and no three of them are on a line. And let $ A',B',C'$ points such that $ MAC'B, MBA'C$ and $ MCB'A$ are parallelograms:
(a) Show that
\[ \overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} < \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}.\]
(b) Assume segments $ AA', BB'$ and $ CC'$ have the same length. Show that $ 2 \left(\overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} \right) \leq \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}.$ When do we have equality?
2019 Tournament Of Towns, 5
Consider a sequence of positive integers with total sum $2019$ such that no number and no sum of a set of consecutive num bers is equal to $40$. What is the greatest possible length of such a sequence?
(Alexandr Shapovalov)
2002 IMO Shortlist, 3
Let $P$ be a cubic polynomial given by $P(x)=ax^3+bx^2+cx+d$, where $a,b,c,d$ are integers and $a\ne0$. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root.
2023 CCA Math Bonanza, L2.1
A rectangle has been divided into 8 smaller rectangles as shown below. Given the area of seven of these rectangles, find the area of the shaded rectangle.
[i]Lightning 2.1[/i]
2017 Taiwan TST Round 1, 5
Let $n$ be an odd number larger than 1, and $f(x)$ is a polynomial with degree $n$ such that $f(k)=2^k$ for $k=0,1,\cdots,n$. Prove that there is only finite integer $x$ such that $f(x)$ is the power of two.
2015 Costa Rica - Final Round, N4
Show that there are no triples $(a, b, c)$ of positive integers such that
a) $a + c, b + c, a + b$ do not have common multiples in pairs.
b)$\frac{c^2}{a + b},\frac{b^2}{a + c},\frac{a^2}{c + b}$ are integer numbers.
2019 Junior Balkan Team Selection Tests - Romania, 4
Consider two disjoint finite sets of positive integers, $A$ and $B$, have $n$ and $m$ elements, respectively. It is knows that all $k$ belonging to $A \cup B$ satisfies at least one of the conditions $k + 17 \in A$ and $k - 31 \in B$.
Prove that $17n = 31m$.
2014 Singapore Senior Math Olympiad, 25
Find the number of ordered pairs of integers (p,q) satisfying the equation $p^2-q^2+p+q=2014$.
2014 Contests, 4
Let $x_1,x_2,\dots,x_{2014}$ be integers among which no two are congurent modulo $2014$. Let $y_1,y_2,\dots,y_{2014}$ be integers among which no two are congurent modulo $2014$. Prove that one can rearrange $y_1,y_2,\dots,y_{2014}$ to $z_1,z_2,\dots,z_{2014}$, so that among \[x_1+z_1,x_2+z_2,\dots,x_{2014}+z_{2014}\] no two are congurent modulo $4028$.
2005 Belarusian National Olympiad, 3
Solve in positive integers $a>b$:
$$(a-b)^{ab}=a^bb^a$$
2007 District Olympiad, 3
Let $b>a\geq 2$ be positive integers. Prove that if number $a+k$ is coprime to number $b+k$, for all $k=1,2,...,b-a$, then $a,b$ are consecutive numbers
1990 India Regional Mathematical Olympiad, 2
For all positive real numbers $ a,b,c$, prove that
\[ \frac {a}{b \plus{} c} \plus{} \frac {b}{c \plus{} a} \plus{} \frac {c}{a \plus{} b} \geq \frac {3}{2}.\]
2021 Bangladeshi National Mathematical Olympiad, 10
A positive integer $n$ is called [i]nice[/i] if it has at least $3$ proper divisors and it is equal to the sum of its three largest proper divisors. For example, $6$ is [i]nice[/i] because its largest three proper divisors are $3,2,1$ and $6=3+2+1$. Find the number of [i]nice[/i] integers not greater than $3000$.
2007 Stanford Mathematics Tournament, 2
Given that $x_1>0$ and $x_2=4x_1$ are solutions to $ax^2+bx+c$ and that $3a=2(c-b)$, what is $x_1$?
2001 District Olympiad, 3
Consider four points $A,B,C,D$ not in the same plane such that
\[AB=BD=CD=AC=\sqrt{2} AD=\frac{\sqrt{2}}{2}BC=a\]
Prove that:
a)There is a point $M\in [BC]$ such that $MA=MB=MC=MD$.
b)$2m(\sphericalangle(AD,BC))=3m(\sphericalangle((ABC),(BCD)))$
c)$6(d(A,CD))^2=7(d(A,(BCD)))^2$
[i]Ion Trandafir[/i]
1991 Denmark MO - Mohr Contest, 4
Let $a, b, c$ and $d$ be arbitrary real numbers. Prove that if $$a^2+b^2+c^2+d^2=ab+bc+cd+da,$$ then $a=b=c=d$.
1993 Chile National Olympiad, 5
Let $a,b,c$ three positive numbers less than $ 1$. Prove that cannot occur simultaneously these three inequalities:
$$a (1- b)>\frac14$$
$$b (1-c)>\frac14 $$
$$c (1-a)>\frac14$$
2006 CentroAmerican, 1
For $0 \leq d \leq 9$, we define the numbers \[S_{d}=1+d+d^{2}+\cdots+d^{2006}\]Find the last digit of the number \[S_{0}+S_{1}+\cdots+S_{9}.\]
2004 AMC 10, 6
Bertha has $ 6$ daughters and no sons. Some of her daughters have $ 6$ daughters and the rest have none. Bertha has a total of $ 30$ daughters and granddaughters, and no great-grand daughters. How many of Bertha's daughters and granddaughters have no daughters?
$ \textbf{(A)}\ 22\qquad
\textbf{(B)}\ 23\qquad
\textbf{(C)}\ 24\qquad
\textbf{(D)}\ 25\qquad
\textbf{(E)}\ 26$
2022 Israel TST, 1
A triangle $ABC$ with orthocenter $H$ is given. $P$ is a variable point on line $BC$. The perpendicular to $BC$ through $P$ meets $BH$, $CH$ at $X$, $Y$ respectively. The line through $H$ parallel to $BC$ meets $AP$ at $Q$. Lines $QX$ and $QY$ meet $BC$ at $U$, $V$ respectively. Find the shape of the locus of the incenters of the triangles $QUV$.
2022 Azerbaijan Junior National Olympiad, A3
Let $x,y,z \in \mathbb{R}^{+}$ and $x^2+y^2+z^2=x+y+z$. Prove that
$$x+y+z+3 \ge 6 \sqrt[3]{\frac{xy+yz+zx}{3}}$$
2021 Bangladesh Mathematical Olympiad, Problem 5
How many ways can you roll three 20-sided dice such that the sum of the three rolls is exactly $42$? Here the order of the rolls matter. [i](Note that a 20-sided die is is very much like a regular 6-sided die other than the fact that it has $20$ faces instead of $6$)[/i]
2011 Abels Math Contest (Norwegian MO), 2b
The diagonals $AD, BE$, and $CF$ of a convex hexagon $ABCDEF$ intersect in a common point.
Show that $a(ABE) a(CDA) a(EFC) = a(BCE) a(DEA) a(FAC)$,
where $a(KLM)$ is the area of the triangle $KLM$.
[img]https://cdn.artofproblemsolving.com/attachments/0/a/bcbbddedde159150fe3c26b1f0a2bfc322aa1a.png[/img]
2014-2015 SDML (High School), 2
Sally is thinking of a positive four-digit integer. When she divides it by any one-digit integer greater than $1$, the remainder is $1$. How many possible values are there for Sally's four-digit number?
2014 Contests, 2
Let $A$ be the $n\times n$ matrix whose entry in the $i$-th row and $j$-th column is \[\frac1{\min(i,j)}\] for $1\le i,j\le n.$ Compute $\det(A).$