Found problems: 25757
1966 IMO Longlists, 17
Let $ABCD$ and $A^{\prime }B^{\prime}C^{\prime }D^{\prime }$ be two arbitrary parallelograms in the space, and let $M,$ $N,$ $P,$ $Q$ be points dividing the segments $AA^{\prime },$ $BB^{\prime },$ $CC^{\prime },$ $DD^{\prime }$ in equal ratios.
[b]a.)[/b] Prove that the quadrilateral $MNPQ$ is a parallelogram.
[b]b.)[/b] What is the locus of the center of the parallelogram $MNPQ,$ when the point $M$ moves on the segment $AA^{\prime }$ ?
(Consecutive vertices of the parallelograms are labelled in alphabetical order.
2025 Turkey EGMO TST, 5
In triangle \( ABC \), the incircle is tangent to side \( BC \) at point \( D \), the excircle opposite vertex \( B \) is tangent to line \( AB \) at point \( X \), and the excircle opposite vertex \( C \) is tangent to line \( AC \) at point \( Y \). If \( T \) is the midpoint of segment \( [AD] \) and \( U \) is the circumcenter of triangle \( AXY \), show that \( UT \perp BC \).
1994 IMO Shortlist, 4
Let $ ABC$ be an isosceles triangle with $ AB \equal{} AC$. $ M$ is the midpoint of $ BC$ and $ O$ is the point on the line $ AM$ such that $ OB$ is perpendicular to $ AB$. $ Q$ is an arbitrary point on $ BC$ different from $ B$ and $ C$. $ E$ lies on the line $ AB$ and $ F$ lies on the line $ AC$ such that $ E, Q, F$ are distinct and collinear. Prove that $ OQ$ is perpendicular to $ EF$ if and only if $ QE \equal{} QF$.
2010 CHMMC Winter, 3
Assume that the earth is a perfect sphere. A plane flies between $30^o N$ $45^o W$ and $30^o N$ $45^o E$ along the shortest possible route. Let $\theta$ be the northernmost latitude that the plane flies over. Compute $\sin \theta$.
2004 AIME Problems, 12
Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6$, $BC=5=DA$, and $CD=4$. Draw circles of radius 3 centered at $A$ and $B$, and circles of radius 2 centered at $C$ and $D$. A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\frac{-k+m\sqrt{n}}p$, where $k$, $m$, $n$, and $p$ are positive integers, $n$ is not divisible by the square of any prime, and $k$ and $p$ are relatively prime. Find $k+m+n+p$.
1994 Poland - Second Round, 3
A plane passing through the center of a cube intersects the cube in a cyclic hexagon. Show that this hexagon is regular.
1995 Poland - Second Round, 5
The incircles of the faces $ABC$ and $ABD$ of a tetrahedron $ABCD$ are tangent to the edge $AB$ in the same point. Prove that the points of tangency of these incircles to the edges $AC,BC,AD,BD$ are concyclic.
2021 Iran MO (3rd Round), 3
Given triangle $ABC$ variable points $X$ and $Y$ are chosen on segments $AB$ and $AC$, respectively. Point $Z$ on line $BC$ is chosen such that $ZX=ZY$. The circumcircle of $XYZ$ cuts the line $BC$ for the second time at $T$. Point $P$ is given on line $XY$ such that $\angle PTZ = 90^ \circ$. Point $Q$ is on the same side of line $XY$ with $A$ furthermore $\angle QXY = \angle ACP$ and $\angle QYX = \angle ABP$. Prove that the circumcircle of triangle $QXY$ passes through a fixed point (as $X$ and $Y$ vary).
2012 China Team Selection Test, 3
In some squares of a $2012\times 2012$ grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle $B$ flies to the centre of the square on which it lands is called the [i]translation vector[/i] of beetle $B$.
For all possible starting and ending configurations, find the maximum length of the sum of the [i]translation vectors[/i] of all beetles.
2004 Kurschak Competition, 1
Given is a triangle $ABC$, its circumcircle $\omega$, and a circle $k$ that touches $\omega$ from the outside, and also touches rays $AB$ and $AC$ in $P$ and $Q$, respectively. Prove that the $A$-excenter of $\triangle ABC$ is the midpoint of $\overline{PQ}$.
2013 Harvard-MIT Mathematics Tournament, 8
Let $ABCD$ be a convex quadrilateral. Extend line $CD$ past $D$ to meet line $AB$ at $P$ and extend line $CB$ past $B$ to meet line $AD$ at $Q$. Suppose that line $AC$ bisects $\angle BAD$. If $AD = \frac{7}{4}$, $AP = \frac{21}{2}$, and $AB = \frac{14}{11}$ , compute $AQ$.
2014 China Team Selection Test, 1
$ABCD$ is a cyclic quadrilateral, with diagonals $AC,BD$ perpendicular to each other. Let point $F$ be on side $BC$, the parallel line $EF$ to $AC$ intersect $AB$ at point $E$, line $FG$ parallel to $BD$ intersect $CD$ at $G$. Let the projection of $E$ onto $CD$ be $P$, projection of $F$ onto $DA$ be $Q$, projection of $G$ onto $AB$ be $R$. Prove that $QF$ bisects $\angle PQR$.
2018 PUMaC Geometry A, 8
Let $\omega$ be a circle. Let $E$ be on $\omega$ and $S$ outside $\omega$ such that line segment $SE$ is tangent to $\omega$. Let $R$ be on $\omega$. Let line $SR$ intersect $\omega$ at $B$ other than $R$, such that $R$ is between $S$ and $B$. Let $I$ be the intersection of the bisector of $\angle ESR$ with the line tangent to $\omega$ at $R$; let $A$ be the intersection of the bisector of $\angle ESR$ with $ER$. If the radius of the circumcircle of $\triangle EIA$ is $10$, the radius of the circumcircle of $\triangle SAB$ is $14$, and $SA = 18$, then $IA$ can be expressed in simplest form as $\frac{m}{n}$. Find $m + n$.
2000 USA Team Selection Test, 6
Let $ ABC$ be a triangle inscribed in a circle of radius $ R$, and let $ P$ be a point in the interior of triangle $ ABC$. Prove that
\[ \frac {PA}{BC^{2}} \plus{} \frac {PB}{CA^{2}} \plus{} \frac {PC}{AB^{2}}\ge \frac {1}{R}.
\]
[i]Alternative formulation:[/i] If $ ABC$ is a triangle with sidelengths $ BC\equal{}a$, $ CA\equal{}b$, $ AB\equal{}c$ and circumradius $ R$, and $ P$ is a point inside the triangle $ ABC$, then prove that
$ \frac {PA}{a^{2}} \plus{} \frac {PB}{b^{2}} \plus{} \frac {PC}{c^{2}}\ge \frac {1}{R}$.
2023 Indonesia TST, G
Incircle of triangle $ABC$ tangent to $AB$ and $AC$ on $E$ and $F$ respectively. If $X$ is the midpoint of $EF$, prove $\angle BXC > 90^{\circ}$
2019 JHMT, 6
Circles $C_1$ and $C_2$ intersect at exactly two points $I_1$ and $I_2$. A point $J$ on $C_1$ outside of $C_2$ is chosen such that $\overline{JI_2}$ is tangent to $C_2$ and $\overline{JI_2} = 3$. A line segment is drawn from $J$ through $I_1$ and intersects $C_2$ at point $K$ and $\overline{JK} = 6$. $\angle JI_2I_1 = \angle I_2KI_1 = \frac12 \angle I_1I_2K$. Let $\overline{I_1I_2} = a$, and let $a$ equal the fraction$ \frac{m\sqrt{p}}{n}$ , where $m$ and $n$ are coprime and $p$ is a positive integer not divisible by the square of any prime. Find $100m + 10p + n$.
2022 239 Open Mathematical Olympiad, 2
Point $I{}$ is the center of the circle inscribed in the quadrilateral $ABCD$. Prove that there is a point $K{}$ on the ray $CI$ such that $\angle KBI=\angle KDI=\angle BAI$.
2022 Bolivia Cono Sur TST, P4
Find all right triangles with integer sides and inradius 6.
1973 Putnam, A6
Prove that it is impossible for seven distinct straight lines to be situated in the euclidean plane so as to have at least six points where exactly three of these lines intersect and at least four points where exactly two of these lines intersect.
2017 Rioplatense Mathematical Olympiad, Level 3, 2
One have $n$ distinct circles(with the same radius) such that for any $k+1$ circles there are (at least) two circles that intersects in two points. Show that for each line $l$ one can make $k$ lines, each one parallel with $l$, such that each circle has (at least) one point of intersection with some of these lines.
2009 Today's Calculation Of Integral, 493
In the $ x \minus{} y$ plane, let $ l$ be the tangent line at the point $ A\left(\frac {a}{2},\ \frac {\sqrt {3}}{2}b\right)$ on the ellipse $ \frac {x^2}{a^2} \plus{} \frac {y^2}{b^2}\equal{}1\ (0 < b < 1 < a)$. Let denote $ S$ be the area of the figure bounded by $ l,$ the $ x$ axis and the ellipse.
(1) Find the equation of $ l$.
(2) Express $ S$ in terms of $ a,\ b$.
(3) Find the maximum value of $ S$ with the constraint $ a^2 \plus{} 3b^2 \equal{} 4$.
1948 Putnam, B6
Answer wither (i) or (ii):
(i) Let $V, V_1 , V_2$ and $V_3$ denote four vertices of a cube such that $V_1 , V_2 , V_3 $ are adjacent to $V.$ Project the cube orthogonally on a plane of which the points are marked with complex numbers. Let the projection of $V$ fall in the origin and the projections of $V_1 , V_2 , V_3 $ in points marked with the complex numbers $z_1 , z_2 , z_3$, respectively. Show that $z_{1}^{2} +z_{2}^{2} +z_{3}^{2}=0.$
(ii) Let $(a_{ij})$ be a matrix such that
$$|a_{ii}| > |a_{i1}| + |a_{i2}|+\ldots +|a_{i i-1}|+ |a_{i i+1}| +\ldots +|a_{in}|$$
for all $i.$ Show that the determinant is not equal to $0.$
2009 Math Prize For Girls Problems, 13
The figure below shows a right triangle $ \triangle ABC$.
[asy]unitsize(15);
pair A = (0, 4);
pair B = (0, 0);
pair C = (4, 0);
draw(A -- B -- C -- cycle);
pair D = (2, 0);
real p = 7 - 3sqrt(3);
real q = 4sqrt(3) - 6;
pair E = p + (4 - p)*I;
pair F = q*I;
draw(D -- E -- F -- cycle);
label("$A$", A, N);
label("$B$", B, S);
label("$C$", C, S);
label("$D$", D, S);
label("$E$", E, NE);
label("$F$", F, W);[/asy]
The legs $ \overline{AB}$ and $ \overline{BC}$ each have length $ 4$. An equilateral triangle $ \triangle DEF$ is inscribed in $ \triangle ABC$ as shown. Point $ D$ is the midpoint of $ \overline{BC}$. What is the area of $ \triangle DEF$?
1959 Poland - Second Round, 2
What relationship between the sides of a triangle makes it similar to the triangle formed by its medians?
2016 Tournament Of Towns, 4
A designer took a wooden cube $5 \times 5 \times 5$, divided each face into unit squares and painted each square black, white or red so that any two squares with a common side have different colours. What is the least possible number of black squares? (Squares with a common side may belong to the same face of the cube or to two different faces.)
[i](8 points)[/i]
[i]Mikhail Evdokimov[/i]