Found problems: 25757
2010 Victor Vâlcovici, 3
$ A',B',C' $ are the feet of the heights of an acute-angled triangle $ ABC. $ Calculate
$$ \frac{\text{area} (ABC)}{\text{area}\left( A'B'C'\right)} , $$
knowing that $ ABC $ and $ A'B'C' $ have the same center of mass.
[i]Carmen[/i] and [i]Viorel Botea[/i]
2021 LMT Spring, B19
Kevin is at the point $(19,12)$. He wants to walk to a point on the ellipse $9x^2 + 25y^2 = 8100$, and then walk to $(-24, 0)$. Find the shortest length that he has to walk.
[i]Proposed by Kevin Zhao[/i]
Novosibirsk Oral Geo Oly VIII, 2022.1
A quadrilateral is given, in which the lengths of some two sides are equal to $1$ and $4$. Also, the diagonal of length $2$ divides it into two isosceles triangles. Find the perimeter of this quadrilateral.
2021 Romanian Master of Mathematics Shortlist, G4
Let $ABC$ be an acute triangle, let $H$ and $O$ be its orthocentre and circumcentre, respectively,
and let $S$ and $T$ be the feet of the altitudes from $B$ to $AC$ and from $C$ to $AB$, respectively.
Let $M$ be the midpoint of the segment $ST$, and let $N$ be the midpoint of the segment $AH$. The line
through $O$, parallel to $BC$, crosses the sides $AC$ and $AB$ at $F$ and $G$, respectively. The line $NG$
meets the circle $BGO$ again at $K$, and the line $NF$ meets the circle $CFO$ again at $L$. Prove that
the triangles $BCM$ and $KLN$ are similar.
2015 USA Team Selection Test, 3
Let $ABC$ be a non-equilateral triangle and let $M_a$, $M_b$, $M_c$ be the midpoints of the sides $BC$, $CA$, $AB$, respectively. Let $S$ be a point lying on the Euler line. Denote by $X$, $Y$, $Z$ the second intersections of $M_aS$, $M_bS$, $M_cS$ with the nine-point circle. Prove that $AX$, $BY$, $CZ$ are concurrent.
2008 Singapore Junior Math Olympiad, 3
In the quadrilateral $PQRS, A, B, C$ and $D$ are the midpoints of the sides $PQ, QR, RS$ and $SP$ respectively, and $M$ is the midpoint of $CD$. Suppose $H$ is the point on the line $AM$ such that $HC = BC$. Prove that $\angle BHM = 90^o$.
2018 Bosnia and Herzegovina Team Selection Test, 1
In acute triangle $ABC$ $(AB < AC)$ let $D$, $E$ and $F$ be foots of perpedicular from $A$, $B$ and $C$ to $BC$, $CA$ and $AB$, respectively. Let $P$ and $Q$ be points on line $EF$ such that $DP \perp EF$ and $BQ=CQ$. Prove that $\angle ADP = \angle PBQ$
1977 IMO Shortlist, 14
Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds:
(i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$
(ii) some plane contains exactly three points from $E.$
V Soros Olympiad 1998 - 99 (Russia), 10.9
Six cities are located at the vertices of a convex hexagon, all angles of which are equal. Three sides of this hexagon have length $a$, and the remaining three have length $b$ ($a \le b$). It is necessary to connect these cities with a network of roads so that from each city you can drive to any other (possibly through other cities). Find the shortest length of such a road network.
1990 IMO Longlists, 3
In coordinate plane, we call a point $(x, y)$ "lattice point" if both $x$ and $y$ are integers. Knowing that the vertices of triangle $ABC$ are all lattice points, and there exists exactly one lattice point interior to triangle $ABC$ (there might exist lattice points on the sides of $ABC$). Prove that the area of triangle $ABC$ is no larger than $\frac 92.$
1985 All Soviet Union Mathematical Olympiad, 416
Given big enough sheet of cross-lined paper with the side of the squares equal to $1$. We are allowed to cut it along the lines only. Prove that for every $m>12$ we can cut out a rectangle of the greater than $m$ area such, that it is impossible to cut out a rectangle of $m$ area from it.
1985 IMO, 5
A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.
2017 India PRMO, 20
Attached below:
2009 AMC 12/AHSME, 22
Parallelogram $ ABCD$ has area $ 1,\!000,\!000$. Vertex $ A$ is at $ (0,0)$ and all other vertices are in the first quadrant. Vertices $ B$ and $ D$ are lattice points on the lines $ y\equal{}x$ and $ y\equal{}kx$ for some integer $ k>1$, respectively. How many such parallelograms are there?
$ \textbf{(A)}\ 49\qquad
\textbf{(B)}\ 720\qquad
\textbf{(C)}\ 784\qquad
\textbf{(D)}\ 2009\qquad
\textbf{(E)}\ 2048$
Ukrainian TYM Qualifying - geometry, X.12
Inside the convex polygon $A_1A_2...A_n$ , there is a point $M$ such that $\sum_{k=1}^n \overrightarrow {A_kM} = \overrightarrow{0}$. Prove that $nP\ge 4d$, where $P$ is the perimeter of the polygon, and $d=\sum_{k=1}^n |\overrightarrow {A_kM}|$ . Investigate the question of the achievement of equality in this inequality.
2023 Bangladesh Mathematical Olympiad, P2
Let the points $A,B,C$ lie on a line in this order. $AB$ is the diameter of semicircle $\omega_1$, $AC$ is the diameter of semicircle $\omega_2$. Assume both $\omega_1$ and $\omega_2$ lie on the same side of $AC$. $D$ is a point on $\omega_2$ such that $BD\perp AC$. A circle centered at $B$ with radius $BD$ intersects $\omega_1$ at $E$. $F$ is on $AC$ such that $EF\perp AC$. Prove that $BC=BF$.
2005 International Zhautykov Olympiad, 2
The inner point $ X$ of a quadrilateral is [i]observable[/i] from the side $ YZ$ if the perpendicular to the line $ YZ$ meet it in the colosed interval $ [YZ].$ The inner point of a quadrilateral is a $ k\minus{}$point if it is observable from the exactly $ k$ sides of the quadrilateral. Prove that if a convex quadrilateral has a 1-point then it has a $ k\minus{}$point for each $ k\equal{}2,3,4.$
2010 CHKMO, 3
Let $ \triangle ABC$ be a right-angled triangle with $ \angle C\equal{}90^\circ$. $ CD$ is the altitude from $ C$ to $ AB$, with $ D$ on $ AB$. $ \omega$ is the circumcircle of $ \triangle BCD$. $ \omega_1$ is a circle situated in $ \triangle ACD$, it is tangent to the segments $ AD$ and $ AC$ at $ M$ and $ N$ respectively, and is also tangent to circle $ \omega$.
(i) Show that $ BD\cdot CN\plus{}BC\cdot DM\equal{}CD\cdot BM$.
(ii) Show that $ BM\equal{}BC$.
2010 Today's Calculation Of Integral, 628
(1) Evaluate the following definite integrals.
(a) $\int_0^{\frac{\pi}{2}} \cos ^ 2 x\sin x\ dx$
(b) $\int_0^{\frac{\pi}{2}} (\pi - 2x)\cos x\ dx$
(c) $\int_0^{\frac{\pi}{2}} x\cos ^ 3 x\ dx$
(2) Let $a$ be a positive constant. Find the area of the cross section cut by the plane $z=\sin \theta \ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ of the solid such that
\[x^2+y^2+z^2\leq a^2,\ \ x^2+y^2\leq ax,\ \ z\geq 0\]
, then find the volume of the solid.
[i]1984 Yamanashi Medical University entrance exam[/i]
Please slove the problem without multi integral or arcsine function for Japanese high school students aged 17-18 those who don't study them.
Thanks in advance.
kunny
2007 Romania Team Selection Test, 3
Let $ABCDE$ be a convex pentagon, such that $AB=BC$, $CD=DE$, $\angle B+\angle D=180^{\circ}$, and it's area is $\sqrt2$.
a) If $\angle B=135^{\circ}$, find the length of $[BD]$.
b) Find the minimum of the length of $[BD]$.
2013 NIMO Problems, 4
Let $a,b,c$ be the answers to problems $4$, $5$, and $6$, respectively. In $\triangle ABC$, the measures of $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, $c$ in degrees, respectively. Let $D$ and $E$ be points on segments $AB$ and $AC$ with $\frac{AD}{BD} = \frac{AE}{CE} = 2013$. A point $P$ is selected in the interior of $\triangle ADE$, with barycentric coordinates $(x,y,z)$ with respect to $\triangle ABC$ (here $x+y+z=1$). Lines $BP$ and $CP$ meet line $DE$ at $B_1$ and $C_1$, respectively. Suppose that the radical axis of the circumcircles of $\triangle PDC_1$ and $\triangle PEB_1$ pass through point $A$. Find $100x$.
[i]Proposed by Evan Chen[/i]
2012 ELMO Problems, 5
Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$.
[i]Calvin Deng.[/i]
2007 Today's Calculation Of Integral, 193
For $a>0$, let $l$ be the line created by rotating the tangent line to parabola $y=x^{2}$, which is tangent at point $A(a,a^{2})$, around $A$ by $-\frac{\pi}{6}$.
Let $B$ be the other intersection of $l$ and $y=x^{2}$. Also, let $C$ be $(a,0)$ and let $O$ be the origin.
(1) Find the equation of $l$.
(2) Let $S(a)$ be the area of the region bounded by $OC$, $CA$ and $y=x^{2}$. Let $T(a)$ be the area of the region bounded by $AB$ and $y=x^{2}$. Find $\lim_{a \to \infty}\frac{T(a)}{S(a)}$.
2020 June Advanced Contest, 3
Let a [i]lattice tetrahedron[/i] denote a tetrahedron whose vertices have integer coordinates. Given a lattice tetrahedron, a [i]move[/i] consists of picking some vertex and moving it parallel to one of the three edges of the face opposite the vertex so that it lands on a different point with integer coordinates. Prove that any two lattice tetrahedra with the same volume can be transformed into each other by a series of moves
2023 Indonesia TST, 3
Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.