Found problems: 85335
2007 Stars of Mathematics, 3
Let $ ABC $ be a triangle and $ A_1,B_1,C_1 $ the projections of $ A,B,C $ on their opposite sides. Let $ A_2,A_3 $ be the projection of $ A_1 $ on $ AB, $ respectively on $ AC. B_2,B_3,C_2,C_3 $ are defined analogously. Moreover, $ A_4 $ is the intersection of $ B_2B_3 $ with $ C_2C_3; B_4, $ the intersection of $C_2C_3 $ with $ A_2A_3; C_4, $ the intersection of $ A_2A_3 $ with $ B_2B_3. $
Show that $ AA_4,BB_4 $ and $ CC_4 $ are concurrent.
1996 Czech And Slovak Olympiad IIIA, 4
Points $A$ and $B$ on the rays $CX$ and $CY$ respectively of an acute angle $XCY$ are given so that $CX < CA = CB < CY$. Construct a line meeting the ray $CX$ and the segments $AB,BC$ at $K,L,M$, respectively, such that $KA \cdot YB = XA \cdot MB = LA\cdot LB \ne 0$.
2021 BMT, 9
Let $ABCD$ be a convex quadrilateral such that $\vartriangle ABC$ is equilateral. Let $P$ be a point inside the quadrilateral such that $\vartriangle AP D$ is equilateral and $\angle P CD = 30^o$ . Given that $CP = 2$ and $CD = 3$, compute the area of the triangle formed by $P$, the midpoint of segment $\overline{BC}$, and the midpoint of segment $\overline{AB}$.
2000 IMO Shortlist, 8
Let $ AH_1, BH_2, CH_3$ be the altitudes of an acute angled triangle $ ABC$. Its incircle touches the sides $ BC, AC$ and $ AB$ at $ T_1, T_2$ and $ T_3$ respectively. Consider the symmetric images of the lines $ H_1H_2, H_2H_3$ and $ H_3H_1$ with respect to the lines $ T_1T_2, T_2T_3$ and $ T_3T_1$. Prove that these images form a triangle whose vertices lie on the incircle of $ ABC$.
1949-56 Chisinau City MO, 60
Show that the sum of the distances from any point of a regular tetrahedron to its faces is equal to the height of this tetrahedron.
2015 Germany Team Selection Test, 3
Let $ABC$ be an acute triangle with $|AB| \neq |AC|$ and the midpoints of segments $[AB]$ and $[AC]$ be $D$ resp. $E$. The circumcircles of the triangles $BCD$ and $BCE$ intersect the circumcircle of triangle $ADE$ in $P$ resp. $Q$ with $P \neq D$ and $Q \neq E$.
Prove $|AP|=|AQ|$.
[i](Notation: $|\cdot|$ denotes the length of a segment and $[\cdot]$ denotes the line segment.)[/i]
1976 IMO Longlists, 25
We consider the following system
with $q=2p$:
\[\begin{matrix} a_{11}x_{1}+\ldots+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots+a_{2q}x_{q}=0,\\ \ldots ,\\ a_{p1}x_{1}+\ldots+a_{pq}x_{q}=0,\\ \end{matrix}\]
in which every coefficient is an element from the set $\{-1,0,1\}$$.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties:
[b]a.)[/b] all $x_{j}, j=1,\ldots,q$ are integers$;$
[b]b.)[/b] there exists at least one j for which $x_{j} \neq 0;$
[b]c.)[/b] $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$
2019 Oral Moscow Geometry Olympiad, 3
In the acute triangle $ABC, \angle ABC = 60^o , O$ is the center of the circumscribed circle and $H$ is the orthocenter. The angle bisector $BL$ intersects the circumscribed circle at the point $W, X$ is the intersection point of segments $WH$ and $AC$ . Prove that points $O, L, X$ and $H$ lie on the same circle.
1998 Estonia National Olympiad, 1
Solve the equation $x^2+1 = log_2(x+2)- 2x$.
V Soros Olympiad 1998 - 99 (Russia), 9.10
The bisector of angle $\angle BAC$ of triangle $ABC$ intersects arc $BC$ (not containing point $A$) of the circle circumscribed around this triangle at point $P$. Segment $AP$ is divided by side $BC$ in ratio $k$ (counting from vertex $A$). Find the perimeter of triangle $ABC$ if $BC = a$.
1974 IMO Longlists, 44
We are given $n$ mass points of equal mass in space. We define a sequence of points $O_1,O_2,O_3,\ldots $ as follows: $O_1$ is an arbitrary point (within the unit distance of at least one of the $n$ points); $O_2$ is the centre of gravity of all the $n$ given points that are inside the unit sphere centred at $O_1$;$O_3$ is the centre of gravity of all of the $n$ given points that are inside the unit sphere centred at $O_2$; etc. Prove that starting from some $m$, all points $O_m,O_{m+1},O_{m+2},\ldots$ coincide.
2021 Baltic Way, 1
Let $n$ be a positive integer. Find all functions $f\colon \mathbb{R}\rightarrow \mathbb{R}$ that satisfy the equation
$$
(f(x))^n f(x+y) = (f(x))^{n+1} + x^n f(y)
$$
for all $x ,y \in \mathbb{R}$.
2009 India National Olympiad, 5
Let $ ABC$ be an acute angled triangle and let $ H$ be its ortho centre. Let $ h_{max}$ denote the largest altitude of the triangle $ ABC$. Prove that:
$AH \plus{} BH \plus{} CH\leq2h_{max}$
2011 China Team Selection Test, 1
In $\triangle ABC$ we have $BC>CA>AB$. The nine point circle is tangent to the incircle, $A$-excircle, $B$-excircle and $C$-excircle at the points $T,T_A,T_B,T_C$ respectively. Prove that the segments $TT_B$ and lines $T_AT_C$ intersect each other.
2010 Bulgaria National Olympiad, 2
Each of two different lines parallel to the the axis $Ox$ have exactly two common points on the graph of the function $f(x)=x^3+ax^2+bx+c$. Let $\ell_1$ and $\ell_2$ be two lines parallel to $Ox$ axis which meet the graph of $f$ in points $K_1, K_2$ and $K_3, K_4$, respectively. Prove that the quadrilateral formed by $K_1, K_2, K_3$ and $ K_4$ is a rhombus if and only if its area is equal to $6$ units.
2021 Iranian Geometry Olympiad, 5
Given a triangle $ABC$ with incenter $I$. The incircle of triangle $ABC$ is tangent to $BC$ at $D$. Let $P$ and $Q$ be points on the side BC such that $\angle PAB = \angle BCA$ and $\angle QAC = \angle ABC$, respectively. Let $K$ and $L$ be the incenter of triangles $ABP$ and $ACQ$, respectively. Prove that $AD$ is the Euler line of triangle $IKL$.
[i]Proposed by Le Viet An, Vietnam[/i]
1989 Austrian-Polish Competition, 1
Show that $(\sum_{i=1}^{n}x_iy_iz_i)^2 \le (\sum_{i=1}^{n}x_i^3) (\sum_{i=1}^{n}y_i^3) (\sum_{i=1}^{n}z_i^3)$ for any positive reals $x_i, y_i, z_i$.
2001 Putnam, 1
Consider a set $S$ and a binary operation $*$, i.e. for each $a,b\in S$, $a*b\in S$. Assume $(a*b)*a=b$ for all $a,b\in S$. Prove that $a*(b*a)=b$ for all $a,b \in S$.
2009 Moldova Team Selection Test, 1
Let $ m,n\in \mathbb{N}^*$. Find the least $ n$ for which exists $ m$, such that rectangle $ (3m \plus{} 2)\times(4m \plus{} 3)$ can be covered with $ \dfrac{n(n \plus{} 1)}{2}$ squares, among which exist $ n$ squares of length $ 1$, $ n \minus{} 1$ of length $ 2$, $ ...$, $ 1$ square of length $ n$. For the found value of $ n$ give the example of covering.
2007 Indonesia MO, 4
A 10-digit arrangement $ 0,1,2,3,4,5,6,7,8,9$ is called [i]beautiful[/i] if (i) when read left to right, $ 0,1,2,3,4$ form an increasing sequence, and $ 5,6,7,8,9$ form a decreasing sequence, and (ii) $ 0$ is not the leftmost digit. For example, $ 9807123654$ is a beautiful arrangement. Determine the number of beautiful arrangements.
2005 MOP Homework, 4
Let $ABC$ be an obtuse triangle with $\angle A>90^{\circ}$, and let $r$ and $R$ denote its inradius and circumradius. Prove that \[\frac{r}{R} \le \frac{a\sin A}{a+b+c}.\]
2013 IFYM, Sozopol, 7
Let $O$ be the center of the inscribed circle of $\Delta ABC$ and point $D$ be the middle point of $AB$.
If $\angle AOD=90^\circ$, prove that $AB+BC=3AC$.
2012 Olympic Revenge, 3
Let $G$ be a finite graph. Prove that one can partition $G$ into two graphs $A \cup B=G$ such that if we erase all edges conecting a vertex from $A$ to a vertex from $B$, each vertex of the new graph has even degree.
2001 USA Team Selection Test, 9
Let $A$ be a finite set of positive integers. Prove that there exists a finite set $B$ of positive integers such that $A \subseteq B$ and
\[\prod_{x\in B} x = \sum_{x\in B} x^2.\]
2018 Moscow Mathematical Olympiad, 6
There are $2018$ peoples. We call the group of people as "club" if all members of same "club" are all friends, but not friends with a nonmember of "club". Prove, that we can divide peoples for $90$ rooms, such that no one room has all members of some "club".