Found problems: 649
2016 IFYM, Sozopol, 5
Prove that for an arbitrary $\Delta ABC$ the following inequality holds:
$\frac{l_a}{m_a}+\frac{l_b}{m_b}+\frac{l_c}{m_c} >1$,
Where $l_a,l_b,l_c$ and $m_a,m_b,m_c$ are the lengths of the bisectors and medians through $A$, $B$, and $C$.
2011 Swedish Mathematical Competition, 2
Given a triangle $ABC$, let $P$ be a point inside the triangle such that $| BP | > | AP |, | BP | > | CP |$. Show that $\angle ABC <90^o$
2014 BMT Spring, 12
Suppose four coplanar points $A, B, C$, and $D$ satisfy $AB = 3$, $BC = 4$, $CA = 5$, and $BD = 6$. Determine the maximal possible area of $\vartriangle ACD$.
2003 Bosnia and Herzegovina Junior BMO TST, 4
In the trapezoid $ABCD$ ($AB \parallel DC$) the bases have lengths $a$ and $c$ ($c < a$), while the other sides have lengths $b$ and $d$. The diagonals are of lengths $m$ and $n$. It is known that $m^2 + n^2 = (a + c)^2$.
a) Find the angle between the diagonals of the trapezoid.
b) Prove that $a + c < b + d$.
c) Prove that $ac < bd$.
2018 Balkan MO Shortlist, G3
Let $P$ be an interior point of triangle $ABC$. Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of
$$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right)$$
taking into consideration all possible choices of triangle $ABC$ and of point $P$.
by Elton Bojaxhiu, Albania
1989 IMO Shortlist, 21
Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$
2009 Junior Balkan Team Selection Tests - Romania, 4
Consider $K$ a polygon in plane, such that the distance between any two vertices is not greater than $1$. Let $X$ and $Y$ be two points inside $K$. Show that there exist a point $Z$, lying on the border of K, such that $XZ + Y Z \le 1$
2016 China Team Selection Test, 2
Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.
1994 Tournament Of Towns, (401) 3
Let $O$ be a point inside a convex polygon $A_1A_2... A_n$ such that $$\angle OA_1A_n \le \angle OA_1A_2, \angle OA_2A_1 \le \angle OA_2A_3, ..., \angle OA_{n-1}A_{n-2} \le \angle OA_{n-1}A_n, \angle OA_nA_{n-1} \le \angle OA_nA_1$$ and all of these angles are acute. Prove that $O$ is the centre of the circle inscribed in the polygon.
(V Proizvolov)
Durer Math Competition CD Finals - geometry, 2008.C2
Given a triangle with sides $a, b, c$ and medians $s_a, s_b, s_c$ respectively. Prove the following inequality:
$$a + b + c> s_a + s_b + s_c> \frac34 (a + b + c) $$
Kyiv City MO Juniors 2003+ geometry, 2014.7.41
The sides of triangles $ABC$ and $ACD$ satisfy the following conditions: $AB = AD = 3$ cm, $BC = 7$ cm, $DC = 11$ cm. What values can the side length $AC$ take if it is an integer number of centimeters, is the average in $\Delta ACD$ and the largest in $\Delta ABC$?
KoMaL A Problems 2022/2023, A.837
Let all the edges of tetrahedron \(A_1A_2A_3A_4\) be tangent to sphere \(S\). Let \(\displaystyle a_i\) denote the length of the tangent from \(A_i\) to \(S\). Prove that
\[\bigg(\sum_{i=1}^4 \frac 1{a_i}\bigg)^{\!\!2}> 2\bigg(\sum_{i=1}^4 \frac1{a_i^2}\bigg). \]
[i]Submitted by Viktor Vígh, Szeged[/i]
2023 German National Olympiad, 2
In a triangle, the edges are extended past both vertices by the length of the edge opposite to the respective vertex.
Show that the area of the resulting hexagon is at least $13$ times the area of the original triangle.
2024 Moldova EGMO TST, 5
$AD$ Is the angle bisector Of $\angle BAC$ Where $D$ lies on the The circumcircle of $\triangle ABC$. Show that $2AD>AB+AC$
1978 Chisinau City MO, 167
Prove that the largest area of a triangle with sides $a, b, c$ satisfying the relation $a^2 +b^2 c^2 = 3m^2$, equals to $\frac{\sqrt3}{4}m^2$.
2020 IMO Shortlist, G4
In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\]
(A disk is assumed to contain its boundary.)
2001 Kazakhstan National Olympiad, 6
Each interior point of an equilateral triangle with sides equal to $1$ lies in one of six circles of the same radius $ r $. Prove that $ r \geq \frac {{\sqrt 3}} {{10}} $.
1931 Eotvos Mathematical Competition, 3
Let $A$ and $B$ be two given points, distance $1 $ apart. Determine a point $P$ on the line $AB$ such that
$$\frac{1}{1 + AP}+\frac{1}{1 + BP}$$ is a maximum.
Indonesia Regional MO OSP SMA - geometry, 2009.3
Given triangle $ABC$ and point $D$ on the $AC$ side. Let $r_1, r_2$ and $r$ denote the radii of the incircle of the triangles $ABD, BCD$, and $ABC$, respectively. Prove that $r_1 + r_2> r$.
1989 Greece National Olympiad, 3
From a point $A$ not on line $\varepsilon$, we drop the perpendicular $AB$ on $\varepsilon$ and three other not perpendicular lines $AC$, $AD$,$AE $ which lie on the same semiplane defines by $AB$, such that $(AD )>\frac{1}{2}((AC)+(AE))$. Prove that $(CD )>(DE).$ (Points $B,C,D,,E$ lie on line $\varepsilon$ ) .
1998 Taiwan National Olympiad, 4
Let $I$ be the incenter of triangle $ABC$. Lines $AI$, $BI$, $CI$ meet the sides of $\triangle ABC$ at $D$, $E$, $F$ respectively. Let $X$, $Y$, $Z$ be arbitrary points on segments $EF$, $FD$, $DE$, respectively. Prove that $d(X, AB) + d(Y, BC) + d(Z, CA) \leq XY + YZ + ZX$, where $d(X, \ell)$ denotes the distance from a point $X$ to a line $\ell$.
1976 IMO, 1
In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.
2014 Sharygin Geometry Olympiad, 1
Let $ABCD$ be a cyclic quadrilateral. Prove that $AC > BD$ if and only if $(AD-BC)(AB- CD) > 0$.
(V. Yasinsky)
1902 Eotvos Mathematical Competition, 3
The area $T$ and an angle $\gamma$ of a triangle are given. Determine the lengths of the sides $a$ and $b$ so that the side $c$, opposite the angle $\gamma$, is as short as possible.
2014 Contests, 2
Let us consider a triangle $\Delta{PQR}$ in the co-ordinate plane. Show for every function $f: \mathbb{R}^2\to \mathbb{R}\;,f(X)=ax+by+c$ where $X\equiv (x,y) \text{ and } a,b,c\in\mathbb{R}$ and every point $A$ on $\Delta PQR$ or inside the triangle we have the inequality:
\begin{align*} & f(A)\le \text{max}\{f(P),f(Q),f(R)\} \end{align*}