Found problems: 25757
2015 Czech-Polish-Slovak Junior Match, 1
In the right triangle $ABC$ with shorter side $AC$ the hypotenuse $AB$ has length $12$. Denote $T$ its centroid and $D$ the feet of altitude from the vertex $C$. Determine the size of its inner angle at the vertex $B$ for which the triangle $DTC$ has the greatest possible area.
2007 AMC 12/AHSME, 16
Each face of a regular tetrahedron is painted either red, white or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible?
$ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 54 \qquad \textbf{(E)}\ 81$
2017 Peru IMO TST, 11
Let $ABC$ be an acute and scalene of circumcircle $\Gamma$ and orthocenter $H$. Let $A_1,B_1,C_1$ be the second intersection points of the lines $AH, BH, CH$ with $\Gamma$, respectively. The lines that pass through $A_1,B_1,C_1$ and are parallel to $BC,CA, AB$ intersect again to $\Gamma$ at $A_2,B_2,C_2$, respectively. Let $M$ be the intersection point of $AC_2$ and $BC_1, N$ the intersection point of $BA_2$ and $CA_1$, and $P$ the intersection point of $CB_2$ and $AB_1$. Prove that $\angle MNB = \angle AMP$ .
Kharkiv City MO Seniors - geometry, 2019.11.5
In the acute-angled triangle $ABC$, let $CD, AE$ be the altitudes. Points $F$ and $G$ are the projections of $A$ and $C$ on the line $DE$, respectively, $H$ and $K$ are the projections of $D$ and $E$ on the line $AC$, respectively. The lines $HF$ and $KG$ intersect at point $P$. Prove that line $BP$ bisects the segment $DE$.
2006 Estonia National Olympiad, 3
Let $AG, CH$ be the angle bisectors of a triangle $ABC$. It is known that one of the intersections of the circles of triangles $ABG$ and $ACH$ lies on the side $BC$. Prove that the angle $BAC$ is $60 ^o$
2012 Balkan MO Shortlist, G5
$\boxed{\text{G5}}$ The incircle of a triangle $ABC$ touches its sides $BC$,$CA$,$AB$ at the points $A_1$,$B_1$,$C_1$.Let the projections of the orthocenter $H_1$ of the triangle $A_{1}B_{1}C_{1}$ to the lines $AA_1$ and $BC$ be $P$ and $Q$,respectively. Show that $PQ$ bisects the line segment $B_{1}C_{1}$
1981 USAMO, 4
The sum of the measures of all the face angles of a given complex polyhedral angle is equal to the sum of all its dihedral angles. Prove that the polyhedral angle is a trihedral angle.
$\mathbf{Note:}$ A convex polyhedral angle may be formed by drawing rays from an exterior point to all points of a convex polygon.
2015 IMO Shortlist, G4
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 $\frac{BT}{BM}$.
2002 Junior Balkan Team Selection Tests - Romania, 4
Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\{ A,B,C,D,M,N\}$. Find the least number $k$ such that $s(M,N)\le k$, for all points $M,N$.
[i]Dinu Șerbănescu[/i]
2018 Nepal National Olympiad, 3b
[b] Problem Section #3
NOTE: Neglect that HF and CD.
2006 Irish Math Olympiad, 5
Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible.
[i]Proposed by Horst Sewerin, Germany[/i]
2006 Irish Math Olympiad, 3
Prove that a square of side 2.1 units can be completely covered by seven squares of side 1 unit.
Extra: Try to prove that 7 is the minimal amount.
2001 National Olympiad First Round, 21
Let $b$ be the length of the largest diagonal and $c$ be the length of the smallest diagonal of a regular nonagon with side length $a$. Which one of the followings is true?
$
\textbf{(A)}\ b=\dfrac{a+c}2
\qquad\textbf{(B)}\ b=\sqrt {ac}
\qquad\textbf{(C)}\ b^2=\dfrac{a^2+c^2}2 \\
\textbf{(D)}\ c=a+b
\qquad\textbf{(E)}\ c^2=a^2+b^2
$
2012 Kyrgyzstan National Olympiad, 3
Prove that if the diagonals of a convex quadrilateral are perpendicular, then the feet of perpendiculars dropped from the intersection point of diagonals on the sides of this quadrilateral lie on one circle. Is the converse true?
2014 IMO Shortlist, G2
Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$.
[i]Proposed by Estonia[/i]
2014 Nordic, 2
Given an equilateral triangle, find all points inside the triangle such that the distance from the point to one of the sides is equal to the geometric mean of the distances from the point to the other two sides of the triangle.
2015 Singapore Senior Math Olympiad, 5
Let $A$ be a point on the circle $\omega$ centred at $B$ and $\Gamma$ a circle centred at $A$. For $i=1,2,3$, a chord $P_iQ_i$ of $\omega$ is tangent to $\Gamma$ at $S_i$ and another chord $P_iR_i$ of $\omega$ is perpendicular to $AB$ at $M_i$. Let $Q_iT_i$ be the other tangent from $Q_i$ to $\Gamma$ at $T_i$ and $N_i$ be the intersection of $AQ_i$ with $M_iT_i$. Prove that $N_1,N_2,N_3$ are collinear.
1962 Kurschak Competition, 2
Show that given any $n+1$ diagonals of a convex $n$-gon, one can always find two which have no common point.
1989 Chile National Olympiad, 2
We have a rectangle with integer sides $m, n$ that is subdivided into $mn$ squares of side $1$. Find the number of little squares that are crossed by the diagonal (without counting those that are touched only in one vertex)
2023 Adygea Teachers' Geometry Olympiad, 3
Three cevians are drawn in a triangle that do not intersect at one point. In this case, $4$ triangles and $3$ quadrangles were formed. Find the sum of the areas of the quadrilaterals if the area of each of the four triangles is $8$.
1997 All-Russian Olympiad Regional Round, 10.7
Points $O_1$ and $O_2$ are the centers of the circumscribed and inscribed circles of an isosceles triangle $ABC$ ($AB = BC$). The circumcircles of triangles $ABC$ and $O_1O_2A$ intersect at points $A$ and $D$. Prove that line $BD$ is tangent to the circumcircle of the triangle $O_1O_2A$.
2001 Mexico National Olympiad, 5
$ABC$ is a triangle with $AB < AC$ and $\angle A = 2 \angle C$. $D$ is the point on $AC$ such that $CD = AB$. Let L be the line through $B$ parallel to $AC$. Let $L$ meet the external bisector of $\angle A$ at $M$ and the line through $C$ parallel to $AB$ at $N$. Show that $MD = ND$.
2016 Bangladesh Mathematical Olympiad, 6
$\triangle ABC$ is an isosceles triangle with $AC = BC$ and $\angle ACB < 60^{\circ}$. $I$ and $O$ are the incenter and circumcenter of $\triangle ABC$. The circumcircle of $\triangle BIO$ intersects $BC$ at $D \neq B$.
(a) Do the lines $AC$ and $DI$ intersect? Give a proof.
(b) What is the angle of intersection between the lines $OD$ and $IB$?
1985 ITAMO, 11
An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?
2013 ELMO Shortlist, 7
A $2^{2014} + 1$ by $2^{2014} + 1$ grid has some black squares filled. The filled black squares form one or more snakes on the plane, each of whose heads splits at some points but never comes back together. In other words, for every positive integer $n$ greater than $2$, there do not exist pairwise distinct black squares $s_1$, $s_2$, \dots, $s_n$ such that $s_i$ and $s_{i+1}$ share an edge for $i=1,2, \dots, n$ (here $s_{n+1}=s_1$).
What is the maximum possible number of filled black squares?
[i]Proposed by David Yang[/i]