Found problems: 85335
2002 Moldova National Olympiad, 1
The sequence $ (a_n)$ is defined by $ a_1\in (0,1)$ and $ a_{n\plus{}1}\equal{}a_n(1\minus{}a_n)$ for $ n\ge 1$.
Prove that $ \lim_{n\rightarrow \infty} na_n\equal{}1$
2011 India IMO Training Camp, 2
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
1964 AMC 12/AHSME, 32
If $\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}$, then:
$\textbf{(A)}\ a\text{ must equal }c \qquad
\textbf{(B)}\ a+b+c+d\text{ must equal zero }\qquad$
$
\textbf{(C)}\ \text{either }a=c\text{ or }a+b+c+d=0,\text{ or both} \qquad$
$
\textbf{(D)}\ a+b+c+d\neq 0\text{ if }a=c \qquad
\textbf{(E)}\ a(b+c+d)=c(a+b+d)$
2018 South East Mathematical Olympiad, 3
Let $O$ be the circumcenter of acute $\triangle ABC$($AB<AC$), the angle bisector of $\angle BAC$ meets $BC$ at $T$ and $M$ is the midpoint of $AT$. Point $P$ lies inside $\triangle ABC$ such that $PB\perp PC$. $D,E$ distinct from $P$ lies on the perpendicular to $AP$ through $P$ such that $BD=BP, CE=CP$. If $AO$ bisects segment $DE$, prove that $AO$ is tangent to the circumcircle of $\triangle AMP$.
2007 Tournament Of Towns, 2
[b](a)[/b] Each of Peter and Basil thinks of three positive integers. For each pair of his numbers, Peter writes down the greatest common divisor of the two numbers. For each pair of his numbers, Basil writes down the least common multiple of the two numbers. If both Peter and Basil write down the same three numbers, prove that these three numbers are equal to each other.
[b](b)[/b] Can the analogous result be proved if each of Peter and Basil thinks of four positive integers instead?
2011-2012 SDML (High School), 1
If $\left(0.67\right)^x=0.5$, then find the value of $16\cdot\left(0.67\right)^{3x}$.
$\text{(A) }2\qquad\text{(B) }8\qquad\text{(C) }16\qquad\text{(D) }64\qquad\text{(E) }128$
2019 Romania National Olympiad, 3
Let $ABC$ be a triangle in which $\angle ABC = 45^o$ and $\angle BAC > 90^o$. Let $O$ be the midpoint of the side $[BC]$. Consider the point $M \in (AC)$ such that $\angle COM =\angle CAB$. Perpendicular from $M$ on $AC$ intersects line $AB$ at point $P$.
a) Find the measure of the angle $\angle BCP$.
b) Show that if $\angle BAC = 105^o$, then $PB = 2MO$.
2022 Sharygin Geometry Olympiad, 20
Let $O$, $I$ be the circumcenter and the incenter of $\triangle ABC$; $R$,$r$ be the circumradius and the inradius; $D$ be the touching point of the incircle with $BC$; and $N$ be an arbitrary point of segment $ID$. The perpendicular to $ID$ at $N$ meets the circumcircle of $ABC$ at points $X$ and $Y$ . Let $O_{1}$ be the circumcircle of $\triangle XIY$.
Find the product $OO_{1}\cdot IN$.
2004 Iran MO (2nd round), 5
The interior bisector of $\angle A$ from $\triangle ABC$ intersects the side $BC$ and the circumcircle of $\Delta ABC$ at $D,M$, respectively. Let $\omega$ be a circle with center $M$ and radius $MB$. A line passing through $D$, intersects $\omega$ at $X,Y$. Prove that $AD$ bisects $\angle XAY$.
1986 Czech And Slovak Olympiad IIIA, 3
Prove that the entire space can be partitioned into “crosses” made of seven unit cubes as shown in the picture.
[img]https://cdn.artofproblemsolving.com/attachments/2/b/77c4a4309170e8303af321daceccc4010da334.png[/img]
2014 May Olympiad, 2
In a convex quadrilateral $ABCD$, let $M$, $N$, $P$, and $Q$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. If $MP$ and $NQ$ divide $ABCD$ in four quadrilaterals with the same area, prove that $ABCD$ is a parallelogram.
2007 France Team Selection Test, 1
Do there exist $5$ points in the space, such that for all $n\in\{1,2,\ldots,10\}$ there exist two of them at distance between them $n$?
2014 NZMOC Camp Selection Problems, 9
Let $AB$ be a line segment with midpoint $I$. A circle, centred at $I$ has diameter less than the length of the segment. A triangle $ABC$ is tangent to the circle on sides $AC$ and $BC$. On $AC$ a point $X$ is given, and on $BC$ a point $Y$ is given such that $XY$ is also tangent to the circle (in particular $X$ lies between the point of tangency of the circle with $AC$ and $C$, and similarly $Y$ lies between the point of tangency of the circle with $BC$ and $C$. Prove that $AX \cdot BY = AI \cdot BI$.
Estonia Open Senior - geometry, 2009.1.3
Three circles in a plane have the sides of a triangle as their diameters. Prove that there is a point that is in the interior of all three circles.
2008 Bulgaria Team Selection Test, 2
In the triangle $ABC$, $AM$ is median, $M \in BC$, $BB_{1}$ and $CC_{1}$ are altitudes, $C_{1} \in AB$, $B_{1} \in AC$. The line through $A$ which is perpendicular to $AM$ cuts the lines $BB_{1}$ and $CC_{1}$ at points $E$ and $F$, respectively. Let $k$ be the circumcircle of $\triangle EFM$. Suppose also that $k_{1}$ and $k_{2}$ are circles touching both $EF$ and the arc $EF$ of $k$ which does not contain $M$. If $P$ and $Q$ are the points at which $k_{1}$ intersects $k_{2}$, prove that $P$, $Q$, and $M$ are collinear.
2007 Today's Calculation Of Integral, 253
Evaluate $ \int_0^1 (1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{n \minus{} 1})\{1 \plus{} 3x \plus{} 5x^2 \plus{} \cdots \plus{} (2n \minus{} 3)x^{n \minus{} 2} \plus{} (2n \minus{} 1)x^{n \minus{} 1}\}\ dx.$
2000 Iran MO (3rd Round), 1
A sequence of natural numbers $c_1, c_2,\dots$ is called [i]perfect[/i] if every natural
number $m$ with $1\le m \le c_1 +\dots+ c_n$ can be represented as
$m =\frac{c_1}{a_1}+\frac{c_2}{a_2}+\dots+\frac{c_n}{a_n}$
Given $n$, find the maximum possible value of $c_n$ in a perfect sequence $(c_i)$.
2008 Germany Team Selection Test, 2
[b](i)[/b] Determine the smallest number of edges which a graph of $ n$ nodes may have given that adding an arbitrary new edge would give rise to a 3-clique (3 nodes joined pairwise by edges).
[b](ii)[/b] Determine the smallest number of edges which a graph of $ n$ nodes may have given that adding an arbitrary new edge would give rise to a 4-clique (4 nodes joined pairwise by edges).
2023 Junior Balkan Mathematical Olympiad, 4
Let $ABC$ be an acute triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $BC$ and let $M$ be the midpoint of $OD$. The points $O_b$ and $O_c$ are the circumcenters of triangles $AOC$ and $AOB$, respectively. If $AO=AD$, prove that points $A$, $O_b$, $M$ and $O_c$ are concyclic.
[i]Marin Hristov and Bozhidar Dimitrov, Bulgaria[/i]
2013 Harvard-MIT Mathematics Tournament, 27
Let $W$ be the hypercube $\{(x_1,x_2,x_3,x_4)\,|\,0\leq x_1,x_2,x_3,x_4\leq 1\}$. The intersection of $W$ and a hyperplane parallel to $x_1+x_2+x_3+x_4=0$ is a non-degenerate $3$-dimensional polyhedron. What is the maximum number of faces of this polyhedron?
I Soros Olympiad 1994-95 (Rus + Ukr), 9.6
A circle can be drawn around the quadrilateral $ABCD$. $K$ is a point on the diagonal $BD$ . The straight line $CK$ intersects the side $AD$ at the point $M$. Prove that the circles circumscribed around the triangles $BCK$ and $ACM$ are tangent.
LMT Speed Rounds, 2011.17
Let $ABC$ be a triangle with $AB = 15$, $AC = 20$, and right angle at $A$. Let $D$ be the point on $\overline{BC}$ such that $\overline{AD}$ is perpendicular to $\overline{BC}$, and let $E$ be the midpoint of $\overline{AC}$. If $F$ is the point on $\overline{BC}$ such that $\overline{AD} \parallel \overline{EF}$, what is the area of quadrilateral $ADFE$?
2016 Azerbaijan JBMO TST, 3
Find all the pime numbers $(p,q)$ such that :
$p^{3}+p=q^{2}+q$
2016-2017 SDML (Middle School), 1
What is the integer value of $\left(\sqrt{3}^{\sqrt{2}}\right)^{\sqrt{8}}$?
2007 Silk Road, 4
The set of polynomials $f_1, f_2, \ldots, f_n$ with real coefficients is called [i]special [/i], if for any different $i,j,k \in \{ 1,2, \ldots, n\}$ polynomial $\dfrac{2}{3}f_i + f_j + f_k$ has no real roots, but for any different $p,q,r,s \in \{ 1,2, \ldots, n\}$ of a polynomial $f_p + f_q + f_r + f_s$ there is a real root.
a) Give an example of a [i]special [/i] set of four polynomials whose sum is not a zero polynomial.
b) Is there a [i]special [/i] set of five polynomials?