Found problems: 85335
2012 India PRMO, 2
A triangle with perimeter $7$ has integer sidelengths. What is the maximum possible area of such a triangle?
2006 Germany Team Selection Test, 2
In an acute triangle $ABC$, let $D$, $E$, $F$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $BC$, $CA$, $AB$, respectively, and let $P$, $Q$, $R$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $EF$, $FD$, $DE$, respectively.
Prove that $p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}$, where $p\left(T\right)$ denotes the perimeter of triangle $T$ .
[i]Proposed by Hojoo Lee, Korea[/i]
2017 CCA Math Bonanza, I1
Find the integer $n$ such that $6!\times7!=n!$.
[i]2017 CCA Math Bonanza Individual Round #1[/i]
1989 IMO Longlists, 5
The sequences $ a_0, a_1, \ldots$ and $ b_0, b_1, \ldots$ are defined for $ n \equal{} 0, 1, 2, \ldots$ by the equalities
\[ a_0 \equal{} \frac {\sqrt {2}}{2}, \quad a_{n \plus{} 1} \equal{} \frac {\sqrt {2}}{2} \cdot \sqrt {1 \minus{} \sqrt {1 \minus{} a^2_n}}
\]
and
\[ b_0 \equal{} 1, \quad b_{n \plus{} 1} \equal{} \frac {\sqrt {1 \plus{} b^2_n} \minus{} 1}{b_n}
\]
Prove the inequalities for every $ n \equal{} 0, 1, 2, \ldots$
\[ 2^{n \plus{} 2} a_n < \pi < 2^{n \plus{} 2} b_n.
\]
VMEO III 2006 Shortlist, A5
Find all continuous functions $f : (0,+\infty) \to (0,+\infty)$ such that if $a, b, c$ are the lengths of the sides of any triangle then it is satisfied that $$\frac{f(a+b-c)+f(b+c-a)+f(c+a-b)}{3}=f\left(\sqrt{\frac{ab+bc+ca}{3}}\right)$$
2003 Romania Team Selection Test, 17
A permutation $\sigma: \{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ is called [i]straight[/i] if and only if for each integer $k$, $1\leq k\leq n-1$ the following inequality is fulfilled
\[ |\sigma(k)-\sigma(k+1)|\leq 2. \]
Find the smallest positive integer $n$ for which there exist at least 2003 straight permutations.
[i]Valentin Vornicu[/i]
2016 Estonia Team Selection Test, 12
The circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects with the circle $k_1$ at points $A$ and $C$ and with circle $k_2$ at points $B$ and $D$, so that points $A, B, C$ and $D$ are on the line $\ell$ in that order. Let $X$ be a point on line $MN$ such that the point $M$ is between points $X$ and $N$. Lines $AX$ and $BM$ intersect at point $P$ and lines $DX$ and $CM$ intersect at point $Q$. Prove that $PQ \parallel \ell $.
2005 Slovenia National Olympiad, Problem 2
Let $(a_n)$ be a geometrical progression with positive terms. Define $S_n=\log a_1+\log a_2+\ldots+\log a_n$. Prove that if $S_n=S_m$ for some $m\ne n$, then $S_{n+m}=0$.
1971 IMO Longlists, 16
Knowing that the system
\[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\]
has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.
2012 HMNT, 9
Triangle $ABC$ satisfies $\angle B > \angle C$. Let $M$ be the midpoint of $BC$, and let the perpendicular bisector of $BC$ meet the circumcircle of $\vartriangle ABC$ at a point $D$ such that points $A$, $D$, $C$, and $B$ appear on the circle in that order. Given that $\angle ADM = 68^o$ and $\angle DAC = 64^o$ , find $\angle B$.
MathLinks Contest 6th, 7.3
A lattice point in the Carthesian plane is a point with both coordinates integers. A rectangle minor (respectively a square minor) is a set of lattice points lying inside or on the boundaries of a rectangle (respectively square) with vertices lattice points. We assign to each lattice point a real number, such that the sum of all the numbers in any square minor is less than $1$ in absolute value. Prove that the sum of all the numbers in any rectangle minor is less than $4$ in absolute value.
2019 IMO Shortlist, G6
Let $I$ be the incentre of acute-angled triangle $ABC$. Let the incircle meet $BC, CA$, and $AB$ at $D, E$, and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle DPA + \angle AQD =\angle QIP$.
(Slovakia)
1993 India National Olympiad, 9
Show that there exists a convex hexagon in the plane such that
(i) all its interior angles are equal;
(ii) its sides are $1,2,3,4,5,6$ in some order.
1983 IMO Longlists, 29
Let $O$ be a point outside a given circle. Two lines $OAB, OCD$ through $O$ meet the circle at $A,B,C,D$, where $A,C$ are the midpoints of $OB,OD$, respectively. Additionally, the acute angle $\theta$ between the lines is equal to the acute angle at which each line cuts the circle. Find $\cos \theta$ and show that the tangents at $A,D$ to the circle meet on the line $BC.$
1994 IMC, 4
Let $A$ be a $n\times n$ diagonal matrix with characteristic polynomial
$$(x-c_1)^{d_1}(x-c_2)^{d_2}\ldots (x-c_k)^{d_k}$$
where $c_1, c_2, \ldots, c_k$ are distinct (which means that $c_1$ appears $d_1$ times on the diagonal, $c_2$ appears $d_2$ times on the diagonal, etc. and $d_1+d_2+\ldots + d_k=n$).
Let $V$ be the space of all $n\times n$ matrices $B$ such that $AB=BA$. Prove that the dimension of $V$ is
$$d_1^2+d_2^2+\cdots + d_k^2$$
2010 Junior Balkan MO, 3
Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.
2009 ISI B.Math Entrance Exam, 4
Find the values of $x,y$ for which $x^2+y^2$ takes the minimum value where $(x+5)^2+(y-12)^2=14$.
2010 Dutch IMO TST, 5
The polynomial $A(x) = x^2 + ax + b$ with integer coefficients has the following property:
for each prime $p$ there is an integer $k$ such that $A(k)$ and $A(k + 1)$ are both divisible by $p$.
Proof that there is an integer $m$ such that $A(m) = A(m + 1) = 0$.
2016 Azerbaijan BMO TST, 4
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.
2011 Kyrgyzstan National Olympiad, 8
Given a sequence $x_1,x_2,...,x_n$ of real numbers with ${x_{n + 1}}^3 = {x_n}^3 - 3{x_n}^2 + 3{x_n}$, where $(n=1,2,3,...)$. What must be value of $x_1$, so that $x_{100}$ and $x_{1000}$ becomes equal?
2002 Junior Balkan MO, 4
Prove that for all positive real numbers $a,b,c$ the following inequality takes place
\[ \frac{1}{b(a+b)}+ \frac{1}{c(b+c)}+ \frac{1}{a(c+a)} \geq \frac{27}{2(a+b+c)^2} . \]
[i]Laurentiu Panaitopol, Romania[/i]
2012 Junior Balkan Team Selection Tests - Romania, 3
Consider the triangle $ABC$ and the points $D \in (BC)$ and $M \in (AD)$. Lines $BM$ and $AC$ meet at $E$, lines $CM$ and $AB$ meet at $F$, and lines $EF$ and $AD$ meet at $N$. Prove that $$\frac{AN}{DN}=\frac{1}{2}\cdot \frac{AM}{DM}$$
2009 Harvard-MIT Mathematics Tournament, 6
Let $ABC$ be a triangle in the coordinate plane with vertices on lattice points and with $AB = 1$. Suppose the perimeter of $ABC$ is less than $17$. Find the largest possible value of $1/r$, where $r$ is the inradius of $ABC$.
2014 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be an acute triangle and $D \in (BC) , E \in (AD)$ be mobile points. The circumcircle of triangle $CDE$ meets the median from $C$ of the triangle $ABC$ at $F$ Prove that the circumcenter of triangle $AEF$ lies on a fixed line.
2015 Putnam, B5
Let $P_n$ be the number of permutations $\pi$ of $\{1,2,\dots,n\}$ such that \[|i-j|=1\text{ implies }|\pi(i)-\pi(j)|\le 2\] for all $i,j$ in $\{1,2,\dots,n\}.$ Show that for $n\ge 2,$ the quantity \[P_{n+5}-P_{n+4}-P_{n+3}+P_n\] does not depend on $n,$ and find its value.