Found problems: 25757
V Soros Olympiad 1998 - 99 (Russia), 11.5
It is known that the distances from all the vertices of a cube and the centers of its faces to a certain plane ($14$ values in total) take two different values. The smallest is $1$. What can the edge of a cube be equal to?
1997 VJIMC, Problem 1
Decide whether it is possible to cover the $3$-dimensional Euclidean space with lines which are pairwise skew (i.e. not coplanar).
1997 Vietnam Team Selection Test, 1
Let $ ABCD$ be a given tetrahedron, with $ BC \equal{} a$, $ CA \equal{} b$, $ AB \equal{} c$, $ DA \equal{} a_1$, $ DB \equal{} b_1$, $ DC \equal{} c_1$. Prove that there is a unique point $ P$ satisfying
\[ PA^2 \plus{} a_1^2 \plus{} b^2 \plus{} c^2 \equal{} PB^2 \plus{} b_1^2 \plus{} c^2 \plus{} a^2 \equal{} PC^2 \plus{} c_1^2 \plus{} a^2 \plus{} b^2 \equal{} PD^2 \plus{} a_1^2 \plus{} b_1^2 \plus{} c_1^2
\]
and for this point $ P$ we have $ PA^2 \plus{} PB^2 \plus{} PC^2 \plus{} PD^2 \ge 4R^2$, where $ R$ is the circumradius of the tetrahedron $ ABCD$. Find the necessary and sufficient condition so that this inequality is an equality.
1998 Baltic Way, 13
In convex pentagon $ABCDE$, the sides $AE,BC$ are parallel and $\angle ADE=\angle BDC$. The diagonals $AC$ and $BE$ intersect at $P$. Prove that $\angle EAD=\angle BDP$ and $\angle CBD=\angle ADP$.
2014 AIME Problems, 15
In $ \triangle ABC $, $ AB = 3 $, $ BC = 4 $, and $ CA = 5 $. Circle $\omega$ intersects $\overline{AB}$ at $E$ and $B$, $\overline{BC}$ at $B$ and $D$, and $\overline{AC}$ at $F$ and $G$. Given that $EF=DF$ and $\tfrac{DG}{EG} = \tfrac{3}{4}$, length $DE=\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.
2006 All-Russian Olympiad, 4
Consider an isosceles triangle $ABC$ with $AB=AC$, and a circle $\omega$ which is tangent to the sides $AB$ and $AC$ of this triangle and intersects the side $BC$ at the points $K$ and $L$. The segment $AK$ intersects the circle $\omega$ at a point $M$ (apart from $K$). Let $P$ and $Q$ be the reflections of the point $K$ in the points $B$ and $C$, respectively. Show that the circumcircle of triangle $PMQ$ is tangent to the circle $\omega$.
2018 Romanian Master of Mathematics Shortlist, G2
Let $\triangle ABC$ be a triangle, and let $S$ and $T$ be the midpoints of the sides $BC$ and $CA$, respectively. Suppose $M$ is the midpoint of the segment $ST$ and the circle $\omega$ through $A, M$ and $T$ meets the line $AB$ again at $N$. The tangents of $\omega$ at $M$ and $N$ meet at $P$. Prove that $P$ lies on $BC$ if and only if the triangle $ABC$ is isosceles with apex at $A$.
[i]Proposed by Reza Kumara, Indonesia[/i]
2005 Sharygin Geometry Olympiad, 11.1
$A_1, B_1, C_1$ are the midpoints of the sides $BC,CA,BA$ respectively of an equilateral triangle $ABC$. Three parallel lines, passing through $A_1, B_1, C_1$ intersect, respectively, lines $B_1C_1, C_1A_1, A_1B_1$ at points $A_2, B_2, C_2$. Prove that the lines $AA_2, BB_2, CC_2$ intersect at one point lying on the circle circumscribed around the triangle $ABC$.
2004 Pre-Preparation Course Examination, 6
Let $ l,d,k$ be natural numbers. We want to prove that for large numbers $ n$, for each $ k$-coloring of the $ n$-dimensional cube with side length $ l$, there is a $ d$-dimensional subspace that all of its vertices have the same color. Let $ H(l,d,k)$ be the least number such that for $ n\geq H(l,d,k)$ the previus statement holds.
a) Prove that:
\[ H(l,d \plus{} 1,k)\leq H(l,1,k) \plus{} H(l,d,k^l)^{H(l,1,k)}
\]
b) Prove that
\[ H(l \plus{} 1,1,k \plus{} 1)\leq H(l,1 \plus{} H(l \plus{} 1,1,k),k \plus{} 1)
\]
c) Prove the statement of problem.
d) Prove Van der Waerden's Theorem.
2000 District Olympiad (Hunedoara), 3
Let $ \alpha $ be a plane and let $ ABC $ be an equilateral triangle situated on a parallel plane whose distance from $ \alpha $ is $ h. $ Find the locus of the points $ M\in\alpha $ for which
$$ \left|MA\right| ^2 +h^2 = \left|MB\right|^2 +\left|MC\right|^2. $$
1966 IMO Longlists, 27
Given a point $P$ lying on a line $g,$ and given a circle $K.$ Construct a circle passing through the point $P$ and touching the circle $K$ and the line $g.$
Estonia Open Senior - geometry, 2007.2.5
Consider triangles whose each side length squared is a rational number. Is it true
that
(a) the square of the circumradius of every such triangle is rational;
(b) the square of the inradius of every such triangle is rational?
1909 Eotvos Mathematical Competition, 3
Let $A_1, B_1, C_1$, be the feet of the altitudes of $\vartriangle ABC$ drawn from the vertices $A, B, C $ respectively, and let $M$ be the orthocenter (point of intersection of altitudes) of $\vartriangle ABC$. Assume that the orthic triangle (i.e. the triangle whose vertices are the feet of the altitudes of the original triangle) $A_1$,$B_1$,$C_1$ exists. Prove that each of the points $M$, $A$, $B$, and $C$ is the center of a circle tangent to all three sides (extended if necessary) of $\vartriangle A_1B_1C_1$. What is the difference in the behavior of acute and obtuse triangles $ABC$?
1995 South africa National Olympiad, 3
The circumcircle of $\triangle ABC$ has radius $1$ and centre $O$ and $P$ is a point inside the triangle such that $OP=x$. Prove that
\[AP\cdot BP\cdot CP\le(1+x)^2(1-x),\]
with equality if and only if $P=O$.
1997 Tournament Of Towns, (538) 3
A circle centred at $(a, b)$ contains the origin $(0,0)$. Denote by $S^+$ the total area of the parts of the circle in the first and third quadrants, and by $S^-$ the total area of the parts of the circle in the second and the fourth quadrants. Compute $S^+ -S^-$.
(G Galperin)
2023/2024 Tournament of Towns, 4
4. A triangle $A B C$ with angle $A$ equal to $60^{\circ}$ is given. Its incircle is tangent to side $A B$ at point $D$, while its excircle tangent to side $A C$, is tangent to the extension of side $A B$ at point $E$. Prove that the perpendicular to side $A C$, passing through point $D$, meets the incircle again at a point equidistant from points $E$ and $C$. (The excircle is the circle tangent to one side of the triangle and to the extensions of two other sides.)
Azamat Mardanov
2023 Malaysia IMONST 2, 4
Given a right angled triangle $ABC$ with $\angle BAC = 90^{\circ}$. The points $D,E,F$ lie on sides $BC,CA,AB$ respectively so that $AD$ is perpendicular to $BC$ and $EF$ is parallel to $BC$. A point $G$ lies on side $AC$ such that $AG=CE$. Prove that $\angle GDF = 90^{\circ}$.
1941 Moscow Mathematical Olympiad, 081
a) Prove that it is impossible to divide a rectangle into five squares of distinct sizes.
b) Prove that it is impossible to divide a rectangle into six squares of distinct sizes.
2019 Germany Team Selection Test, 3
A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$.
[i]Proposed by Mongolia[/i]
Estonia Open Senior - geometry, 2002.1.2
The sidelengths of a triangle and the diameter of its incircle, taken in some order, form an arithmetic progression. Prove that the triangle is right-angled.
2009 Indonesia TST, 4
Given triangle $ ABC$ with $ AB>AC$. $ l$ is tangent line of the circumcircle of triangle $ ABC$ at $ A$. A circle with center $ A$ and radius $ AC$, intersect $ AB$ at $ D$ and $ l$ at $ E$ and $ F$. Prove that the lines $ DE$ and $ DF$ pass through the incenter and excenter of triangle $ ABC$.
2023 LMT Spring, 7
In $\vartriangle ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be a point on $BC$ such that $BD = 6$. Let $E$ be a point on $CA$ such that $CE = 6$. Finally, let $F$ be a point on $AB$ such that $AF = 6$. Find the area of $\vartriangle DEF$.
2011 National Olympiad First Round, 25
Let $S_1$ be the area of the regular pentagon $ABCDE$. And let $S_2$ be the area of the regular pentagon whose sides lie on the lines $AC, CE, EB, BD, DA$. What is values of $\frac{S_1}{S_2}$ ?
$\textbf{(A)}\ \frac{41}{6} \qquad\textbf{(B)}\ \frac{3+5\sqrt5}{2} \qquad\textbf{(C)}\ 4+\sqrt5 \qquad\textbf{(D)}\ \frac{7+3\sqrt5}2 \qquad\textbf{(E)}\ \text{None}$
1951 Poland - Second Round, 6
The given points are $ A $ and $ B $ and the circle $ k $. Draw a circle passing through the points $ A $ and $ B $ and defining, at the intersection with the circle $ k $, a common chord of a given length $ d $.
2016 Irish Math Olympiad, 10
Let $AE$ be a diameter of the circumcircle of triangle $ABC$. Join $E$ to the orthocentre, $H$, of $\triangle ABC$ and extend $EH$ to meet the circle again at $D$. Prove that the nine point circle of $\triangle ABC$ passes through the midpoint of $HD$.
Note. The nine point circle of a triangle is a circle that passes through the midpoints of the sides, the feet of the altitudes and the midpoints of the line segments that join the orthocentre to the vertices.