Found problems: 25757
2000 Turkey MO (2nd round), 2
A positive real number $a$ and two rays wich intersect at point $A$ are given. Show that all the circles which pass through $A$ and intersect these rays at points $B$ and $C$ where $|AB|+|AC|=a$ have a common point other than $A$.
2019 South East Mathematical Olympiad, 6
In $\triangle ABC$, $AB>AC$, the bisectors of $\angle ABC, \angle ACB$ meet sides $AC,AB$ at $D,E$ respectively. The tangent at $A$ to the circumcircle of $\triangle ABC$ intersects $ED$ extended at $P$. Suppose that $AP=BC$. Prove that $BD\parallel CP$.
1971 IMO Longlists, 15
Let $ABCD$ be a convex quadrilateral whose diagonals intersect at $O$ at an angle $\theta$. Let us set $OA = a, OB = b, OC = c$, and $OD = d, c > a > 0$, and $d > b > 0.$
Show that if there exists a right circular cone with vertex $V$, with the properties:
[b](1)[/b] its axis passes through $O$, and
[b](2)[/b] its curved surface passes through $A,B,C$ and $D,$ then
\[OV^2=\frac{d^2b^2(c + a)^2 - c^2a^2(d + b)^2}{ca(d - b)^2 - db(c - a)^2}.\]
Show also that if $\frac{c+a}{d+b}$ lies between $\frac{ca}{db}$ and $\sqrt{\frac{ca}{db}},$ and $\frac{c-a}{d-b}=\frac{ca}{db},$ then for a suitable choice of $\theta$, a right circular cone exists with properties [b](1) [/b]and [b](2).[/b]
1978 Germany Team Selection Test, 2
Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.
2007 Thailand Mathematical Olympiad, 3
Two circles intersect at $X$ and $Y$ . The line through the centers of the circles intersect the first circle at $A$ and $C$, and intersect the second circle at $B$ and $D$ so that $A, B, C, D$ lie in this order. The common chord $XY$ cuts $BC$ at $P$, and a point $O$ is arbitrarily chosen on segment $XP$. Lines $CO$ and $BO$ are extended to intersect the first and second circles at $M$ and $N$, respectively. If lines $AM$ and $DN$ intersect at $Z$, prove that $X, Y$ and $Z$ lie on the same line.
2011 China Western Mathematical Olympiad, 3
In triangle $ABC$ with $AB>AC$ and incenter $I$, the incircle touches $BC,CA,AB$ at $D,E,F$ respectively. $M$ is the midpoint of $BC$, and the altitude at $A$ meets $BC$ at $H$. Ray $AI$ meets lines $DE$ and $DF$ at $K$ and $L$, respectively. Prove that the points $M,L,H,K$ are concyclic.
2010 Postal Coaching, 1
Let $A, B, C, D$ be four distinct points in the plane such that the length of the six line segments $AB, AC, AD, BC, BD, CD$ form a $2$-element set ${a, b}$. If $a > b$, determine all the possible values of $\frac ab$.
2004 Thailand Mathematical Olympiad, 20
Two pillars of height $a$ and $b$ are erected perpendicular to the ground. On each pillar, a straight cable is placed connecting the top of the pillar to the base of the other pillar; the two lines of cable intersect at a point above ground. What is the height of this point?
1995 Polish MO Finals, 2
The diagonals of a convex pentagon divide it into a small pentagon and ten triangles. What is the largest number of the triangles that can have the same area?
Kyiv City MO Juniors 2003+ geometry, 2018.9.5
Given a triangle $ABC$, the perpendicular bisector of the side $AC$ intersects the angle bisector of the triangle $AK$ at the point $P$, $M$ - such a point that $\angle MAC = \angle PCB$, $\angle MPA = \angle CPK$, and points $M$ and $K$ lie on opposite sides of the line $AC$. Prove that the line $AK$ bisects the segment $BM$.
(Anton Trygub)
2013 Saudi Arabia BMO TST, 4
$ABCDEF$ is an equiangular hexagon of perimeter $21$. Given that $AB = 3, CD = 4$, and $EF = 5$, compute the area of hexagon $ABCDEF$.
IV Soros Olympiad 1997 - 98 (Russia), 11.11
An arbitrary point $M$ is taken on the basis of a regular triangular pyramid. Let $K$, $L$, $N$ be the projections of $M$ onto the lateral faces of this pyramid, and $P$ be the intersection point of the medians of the triangle $KLN$. Prove that the straight line passing through the points $M$ and$ P$ intersects the height of the pyramid (or its extension). Let us denote this intersection point by $E$. Find $MP: PE$ if the dihedral angles at the base of the pyramid are equal to $a$.
1981 AMC 12/AHSME, 9
In the adjoining figure, $PQ$ is a diagonal of the cube. If $PQ$ has length $a$, then the surface area of the cube is
$\text{(A)}\ 2a^2 \qquad \text{(B)}\ 2\sqrt{2}a^2 \qquad \text{(C)}\ 2\sqrt{3}a^2 \qquad \text{(D)}\ 3\sqrt{3}a^2 \qquad \text{(E)}\ 6a^2$
1963 All Russian Mathematical Olympiad, 029
a) Each diagonal of the quadrangle halves its area. Prove that it is a parallelogram.
b) Three main diagonals of the hexagon halve its area. Prove that they intersect in one point.
1999 Balkan MO, 3
Let $ABC$ be an acute-angled triangle of area 1. Show that the triangle whose vertices are the feet of the perpendiculars from the centroid $G$ to
$AB$, $BC$, $CA$ has area between $\frac 4{27}$ and $\frac 14$.
2020 BMT Fall, 16
Let $T$ be the answer to question $18$. Rectangle $ZOMR$ has $ZO = 2T$ and $ZR = T$. Point $B$ lies on segment $ZO$, $O'$ lies on segment $OM$, and $E$ lies on segment $RM$ such that $BR = BE = EO'$, and $\angle BEO' = 90^o$. Compute $2(ZO + O'M + ER)$.
PS. You had better calculate it in terms of $T$.
1991 Czech And Slovak Olympiad IIIA, 2
A museum has the shape of a (not necessarily convex) 3$n$-gon.
Prove that $n$ custodians can be positioned so as to control all of the museum’s space.
2011 Sharygin Geometry Olympiad, 3
Restore the isosceles triangle $ABC$ ($AB = AC$) if the common points $I, M, H$ of bisectors, medians and altitudes respectively are given.
2017 District Olympiad, 2
Let $ ABCDA’B’C’D’ $ a cube. $ M,P $ are the midpoints of $ AB, $ respectively, $ DD’. $
[b]a)[/b] Show that $ MP, A’C $ are perpendicular, but not coplanar.
[b]b)[/b] Calculate the distance between the lines above.
2002 Putnam, 2
Consider a polyhedron with at least five faces such that exactly three edges emerge from each of its vertices. Two players play the following game: Each, in turn, signs his or her name on a previously unsigned face. The winner is the player who first succeeds in signing three faces that share a common vertex. Show that the player who signs first will always win by playing as well as possible.
1985 ITAMO, 12
Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.
2007 Korea National Olympiad, 2
$ ABC$ is a triangle which is not isosceles. Let the circumcenter and orthocenter of $ ABC$ be $ O$, $ H$, respectively,
and the altitudes of $ ABC$ be $ AD$, $ BC$, $ CF$. Let $ K\neq A$ be the intersection of $ AD$ and circumcircle of triangle $ ABC$, $ L$ be the intersection of $ OK$ and $ BC$, $ M$ be the midpoint of $ BC$, $ P$ be the intersection of $ AM$ and the line that passes $ L$ and perpendicular to $ BC$, $ Q$ be the intersection of $ AD$ and the line that passes $ P$ and parallel to $ MH$, $ R$ be the intersection of line $ EQ$ and $ AB$, $ S$ be the intersection of $ FD$ and $ BE$.
If $ OL \equal{} KL$, then prove that two lines $ OH$ and $ RS$ are orthogonal.
KoMaL A Problems 2021/2022, A. 828
Triangle $ABC$ has incenter $I$ and excircles $\Omega_A$, $\Omega_B$, and $\Omega_C$. Let $\ell_A$ be the line through the feet of the tangents from $I$ to $\Omega_A$, and define lines $\ell_B$ and $\ell_C$ similarly. Prove that the orthocenter of the triangle formed by lines $\ell_A$, $\ell_B$, and $\ell_C$ coincides with the Nagel point of triangle $ABC$.
(The Nagel point of triangle $ABC$ is the intersection of segments $AT_A$, $BT_B$, and $CT_C$, where $T_A$ is the tangency point of $\Omega_A$ with side $BC$, and points $T_B$ and $T_C$ are defined similarly.)
Proposed by [i]Nikolai Beluhov[/i], Bulgaria
2024 Macedonian TST, Problem 4
Let $\triangle ABC$ be an acute scalene triangle. Denote by $k_A$ the circle with diameter $BC$, and let $B_A,C_A$ be the contact points of the tangents from $A$ to $k_A$, chosen so that $B$ and $B_A$ lie on opposite sides of $AC$ and $C$ and $C_A$ lie on opposite sides of $AB$. Similarly, let $k_B$ be the circle with diameter $CA$, with tangents from $B$ touching at $C_B,A_B$, and $k_C$ the circle with diameter $AB$, with tangents from $C$ touching at $A_C,B_C$.
Prove that the lines $B_AC_A, C_BA_B, A_CB_C$ are concurrent.
2021 Sharygin Geometry Olympiad, 8.5
Points $A_1,A_2,A_3,A_4$ are not concyclic, the same for points $B_1,B_2,B_3,B_4$. For all $i, j, k$ the circumradii of triangles $A_iA_jA_k$ and $B_iB_jB_k$ are equal. Can we assert that $A_iA_j=B_iB_j$ for all $i, j$'?