Found problems: 25757
2015 Romania National Olympiad, 3
In the convex quadrilateral $ABCD$ we have that $\angle BCD = \angle ADC \ge 90 ^o$. The bisectors of $\angle BAD$ and $\angle ABC$ intersect in $M$. Prove that if $M \in CD$, then $M$ is the middle of $CD$.
2021 Indonesia TST, G
Do there exist a rectangle that can be partitioned into a regular hexagon with side length $1$, and several right triangles with side lengths $1, \sqrt3 , 2$?
1989 IMO Longlists, 32
Given an acute triangle find a point inside the triangle such that the sum of the distances from this point to the three vertices is the least.
2021 Romania EGMO TST, P4
Consider a coordinate system in the plane, with the origin $O$. We call a lattice point $A{}$ [i]hidden[/i] if the open segment $OA$ contains at least one lattice point. Prove that for any positive integer $n$ there exists a square of side-length $n$ such that any lattice point lying in its interior or on its boundary is hidden.
2017 China Team Selection Test, 3
Let $ABCD$ be a quadrilateral and let $l$ be a line. Let $l$ intersect the lines $AB,CD,BC,DA,AC,BD$ at points $X,X',Y,Y',Z,Z'$ respectively. Given that these six points on $l$ are in the order $X,Y,Z,X',Y',Z'$, show that the circles with diameter $XX',YY',ZZ'$ are coaxal.
2022 Belarusian National Olympiad, 8.5
Inside a triangle $ABC$ three circles with radius $1$ are drawn. (Circles can be tangent to each other and to the sides of the triangle, but can not have any common internal points.)
Find the biggest value of $r$ for which one can state that he can always draw a fourth circle inside the triangle of radius $r$, which does not intersect three already drawn circles.
2014 India Regional Mathematical Olympiad, 4
let $ABC$ be a right angled triangle with inradius $1$
find the minimum area of triangle $ABC$
2005 Taiwan TST Round 1, 1
Consider a circle $O_1$ with radius $R$ and a point $A$ outside the circle. It is known that $\angle BAC=60^\circ$, where $AB$ and $AC$ are tangent to $O_1$. We construct infinitely many circles $O_i$ $(i=1,2,\dots\>)$ such that for $i>1$, $O_i$ is tangent to $O_{i-1}$ and $O_{i+1}$, that they share the same tangent lines $AB$ and $AC$ with respect to $A$, and that none of the $O_i$ are larger than $O_1$. Find the total area of these circles.
I know this problem was easy, but it still appeared in the TST, and so I posted it. It was kind of a disappointment for me.
2015 AMC 8, 1
How many square yards of carpet are required to cover a rectangular floor that is $12$ feet long and $9$ feet wide? (There are 3 feet in a yard.)
$\textbf{(A) }12\qquad\textbf{(B) }36\qquad\textbf{(C) }108\qquad\textbf{(D) }324\qquad \textbf{(E) }972$
2000 Iran MO (3rd Round), 1
Two circles intersect at two points $A$ and $B$. A line $\ell$ which passes through the point $A$ meets the two circles again at the points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ (which do not contain the point $A$) on the respective circles. Let $K$ be the midpoint of the segment $CD$. Prove that $\measuredangle MKN = 90^{\circ}$.
2022 Oral Moscow Geometry Olympiad, 4
In triangle $ABC$, angle $C$ is equal to $60^o$. Bisectors $AA'$ and $BB'$ intersect at point $I$. Point $K$ is symmetric to $I$ with respect to line $AB$. Prove that lines $CK$ and $A'B'$ are perpendicular.
(D. Shvetsov, A. Zaslavsky)
2013 Putnam, 1
Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all $20$ integers is $39.$ Show that there are two faces that share a vertex and have the same integer written on them.
1998 All-Russian Olympiad, 3
In scalene $\triangle ABC$, the tangent from the foot of the bisector of $\angle A$ to the incircle of $\triangle ABC$, other than the line $BC$, meets the incircle at point $K_a$. Points $K_b$ and $K_c$ are analogously defined. Prove that the lines connecting $K_a$, $K_b$, $K_c$ with the midpoints of $BC$, $CA$, $AB$, respectively, have a common point on the incircle.
2008 Romania Team Selection Test, 3
Show that each convex pentagon has a vertex from which the distance to the opposite side of the pentagon is strictly less than the sum of the distances from the two adjacent vertices to the same side.
[i]Note[/i]. If the pentagon is labeled $ ABCDE$, the adjacent vertices of $ A$ are $ B$ and $ E$, the ones of $ B$ are $ A$ and $ C$ etc.
2009 Dutch IMO TST, 5
Suppose that we are given an $n$-gon of which all sides have the same length, and of which all the vertices have rational coordinates. Prove that $n$ is even.
2020 LIMIT Category 2, 10
In a triangle $\triangle XYZ$, $\tan(x)\tan(z)=2$, $\tan(y)\tan(z)=18$. Then what is $\tan^2(z)$?
2011 AMC 10, 11
Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $\overline{AB}$ with $AE=7\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$?
$\textbf{(A)}\,\frac{49}{64} \qquad\textbf{(B)}\,\frac{25}{32} \qquad\textbf{(C)}\,\frac78 \qquad\textbf{(D)}\,\frac{5\sqrt{2}}{8} \qquad\textbf{(E)}\,\frac{\sqrt{14}}{4} $
2021 Novosibirsk Oral Olympiad in Geometry, 1
Cut the $19 \times 20$ grid rectangle along the grid lines into several squares so that there are exactly four of them with odd sidelengths.
2020 Turkey EGMO TST, 5
$A, B, C, D, E$ points are on $\Gamma$ cycle clockwise. $[AE \cap [CD = \{M\}$ and $[AB \cap [DC = \{N\}$. The line parallels to $EC$ and passes through $M$ intersects with the line parallels to $BC$ and passes through $N$ on $K$. Similarly, the line parallels to $ED$ and passes through $M$ intersects with the line parallels to $BD$ and passes through $N$ on $L$. Show that the lines $LD$ and $KC$ intersect on $\Gamma$.
2002 Tournament Of Towns, 2
A game is played on a $23\times 23$ board. The first player controls two white chips which start in the bottom left and top right corners. The second player controls two black ones which start in bottom right and top left corners. The players move alternately. In each move, a player moves one of the chips under control to a square which shares a side with the square the chip is currently in. The first player wins if he can bring the white chips to squares which share a side with each other. Can the second player prevent the first player from winning?
1980 Austrian-Polish Competition, 8
Let $S$ be a set of 1980 points in the plane such that the distance between every pair of them is at least 1. Prove that $S$ has a subset of 220 points such that the distance between every pair of them is at least $\sqrt{3}.$
2013 Online Math Open Problems, 30
Pairwise distinct points $P_1,P_2,\ldots, P_{16}$ lie on the perimeter of a square with side length $4$ centered at $O$ such that $\lvert P_iP_{i+1} \rvert = 1$ for $i=1,2,\ldots, 16$. (We take $P_{17}$ to be the point $P_1$.) We construct points $Q_1,Q_2,\ldots,Q_{16}$ as follows: for each $i$, a fair coin is flipped. If it lands heads, we define $Q_i$ to be $P_i$; otherwise, we define $Q_i$ to be the reflection of $P_i$ over $O$. (So, it is possible for some of the $Q_i$ to coincide.) Let $D$ be the length of the vector $\overrightarrow{OQ_1} + \overrightarrow{OQ_2} + \cdots + \overrightarrow{OQ_{16}}$. Compute the expected value of $D^2$.
[i]Ray Li[/i]
2011 Math Prize For Girls Problems, 8
In the figure below, points $A$, $B$, and $C$ are distance 6 from each other. Say that a point $X$ is [i]reachable[/i] if there is a path (not necessarily straight) connecting $A$ and $X$ of length at most 8 that does not intersect the interior of $\overline{BC}$. (Both $X$ and the path must lie on the plane containing $A$, $B$, and $C$.) Let $R$ be the set of reachable points. What is the area of $R$?
[asy]
unitsize(40);
pair A = dir(90);
pair B = dir(210);
pair C = dir(330);
dot(A);
dot(B);
dot(C);
draw(B -- C);
label("$A$", A, N);
label("$B$", B, W);
label("$C$", C, E);
[/asy]
2011 Stars Of Mathematics, 1
Let $ABC$ be an acute-angled triangle with $AB \neq BC$, $M$ the midpoint of $AC$, $N$ the point where the median $BM$ meets again the circumcircle of $\triangle ABC$, $H$ the orthocentre of $\triangle ABC$, $D$ the point on the circumcircle for which $\angle BDH = 90^{\circ}$, and $K$ the point that makes $ANCK$ a parallelogram.
Prove the lines $AC$, $KH$, $BD$ are concurrent.
(I. Nagel)
2005 AIME Problems, 1
Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle $C$ with radius $30$. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$.