Olympiad problems

1995 Tournament Of Towns, (461) 6

Does there exist a nonconvex polyhedron such that not one of its vertices is visible from a point $M$ outside it? (The polyhedron is made out of an opaque material.) (AY Belov, S Markelov)

2010 Kazakhstan National Olympiad, 1

Tags: rectangle , ellipse , geometry , conic

Triangle $ABC$ is given. Consider ellipse $ \Omega _1$, passes through $C$ with focuses in $A$ and $B$. Similarly define ellipses $ \Omega _2 , \Omega _3$ with focuses $B,C$ and $C,A$ respectively. Prove, that if all ellipses have common point $D$ then $A,B,C,D$ lies on the circle. Ellipse with focuses $X,Y$, passes through $Z$- locus of point $T$, such that $XT+YT=XZ+YZ$

1993 Tournament Of Towns, (395) 3

Tags: geometry , fixed , hexagon , equilateral

Consider the hexagon which is formed by the vertices of two equilateral triangles (not necessarily equal) when the triangles intersect. Prove that the area of the hexagon is unchanged when one of the triangles is translated (without rotating) relative to the other in such a way that the hexagon continues to be defined. (V Proizvolov)

2010 Today's Calculation Of Integral, 626

Tags: limit , calculus , geometry , integration , logarithm , calculus computations

Find $\lim_{a\rightarrow +0} \int_a^1 \frac{x\ln x}{(1+x)^3}dx.$ [i]2010 Nara Medical University entrance exam[/i]

Russian TST 2018, P2

Tags: homothety , power of a point , geometry , excircle , tangent circle

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

1997 Israel National Olympiad, 8

Tags: geometry , parallel , tangent circle

Two equal circles are internally tangent to a larger circle at $A$ and $B$. Let $M$ be a point on the larger circle, and let lines $MA$ and $MB$ intersect the corresponding smaller circles at $A'$ and $B'$. Prove that $A'B'$ is parallel to $AB$.

2016 Philippine MO, 5

Tags: geometry

Pentagon $ABCDE$ is inscribed in a circle. Its diagonals $AC$ and $BD$ intersect at $F$. The bisectors of $\angle BAC$ and $\angle CDB$ intersect at $G$. Let $AG$ intersect $BD$ at $H$, let $DG$ intersect $AC$ at $I$, and let $EG$ intersect $AD$ at $J$. If $FHGI$ is cyclic and \[JA \cdot FC \cdot GH = JD \cdot FB \cdot GI,\] prove that $G$, $F$ and $E$ are collinear.

2014 ELMO Shortlist, 6

Tags: circumcircle , cyclic quadrilateral , power of a point , radical axis , geometry

Let $ABCD$ be a cyclic quadrilateral with center $O$. Suppose the circumcircles of triangles $AOB$ and $COD$ meet again at $G$, while the circumcircles of triangles $AOD$ and $BOC$ meet again at $H$. Let $\omega_1$ denote the circle passing through $G$ as well as the feet of the perpendiculars from $G$ to $AB$ and $CD$. Define $\omega_2$ analogously as the circle passing through $H$ and the feet of the perpendiculars from $H$ to $BC$ and $DA$. Show that the midpoint of $GH$ lies on the radical axis of $\omega_1$ and $\omega_2$. [i]Proposed by Yang Liu[/i]

III Soros Olympiad 1996 - 97 (Russia), 10.10

Tags: geometric inequality , combinatorial geometry , trigonometry , geometry

There are several triangles. From them a new triangle is obtained according to the following rule. The largest side of the new triangle is equal to the sum of the large sides of the data, the middle one is equal to the sum of the middle sides, and the smallest one is the sum of the smaller ones. Prove that if all the angles of these triangles were less than $a$, and $\phi$, where $\phi$ is the largest angle of the resulting triangle, then $\cos \phi \ge 1-\sin (a/2)$.

2009 Tournament Of Towns, 1

Tags: combinatorial geometry , rectangle , geometry

A rectangle is dissected into several smaller rectangles. Is it possible that for each pair of these rectangles, the line segment connecting their centers intersects some third rectangle?

1982 National High School Mathematics League, 10

Tags: geometry

Semi-circle $AB$ with diameter $AB$, and $AB=2r$. Given line $l$, satisfying that $l \perp BA, l \cap BA=T , |AT|=2a(2a<r)$. $M,N$ are two points on the semi-circle, such that $$d(M,l)=|AM|,d(N,l)=|AN|(M\neq N).$$ Prove: $|AM|+|AN|=|AB|$.

2023 HMNT, 10

Tags: geometry , trapezoid

Let $ABCD$ be a convex trapezoid such that $\angle ABC = \angle BCD = 90^o$, $AB = 3$, $BC = 6$, and $CD = 12$. Among all points $X$ inside the trapezoid satisfying $\angle XBC = \angle XDA$, compute the minimum possible value of $CX$.

2008 ISI B.Stat Entrance Exam, 3

Tags: calculus , function , derivative , geometry , 3d geometry , calculus computations

Study the derivatives of the function \[y=\sqrt{x^3-4x}\] and sketch its graph on the real line.

2020 Tournament Of Towns, 5

Tags: geometry , concurrency , cyclic quadrilateral , circles

Let $ABCD$ be an inscribed quadrilateral. Let the circles with diameters $AB$ and $CD$ intersect at two points $X_1$ and $Y_1$, the circles with diameters $BC$ and $AD$ intersect at two points $X_2$ and $Y_2$, the circles with diameters $AC$ and $BD$ intersect at two points $X_3$ and $Y_3$. Prove that the lines $X_1Y_1, X_2Y_2$ and $X_3Y_3$ are concurrent. Maxim Didin

1998 Federal Competition For Advanced Students, Part 2, 2

Tags: number theory , geometry , 3d geometry

Let $Q_n$ be the product of the squares of even numbers less than or equal to $n$ and $K_n$ equal to the product of cubes of odd numbers less than or equal to $n$. What is the highest power of $98$, that [b]a)[/b]$Q_n$, [b]b)[/b] $K_n$ or [b]c)[/b] $Q_nK_n$ divides? If one divides $Q_{98}K_{98}$ by the highest power of $98$, then one get a number $N$. By which power-of-two number is $N$ still divisible?

Kyiv City MO Juniors 2003+ geometry, 2013.8.5

Tags: concurrency , geometry , rhombus , circles

Let $ABCD$ be a convex quadrilateral. Prove that the circles inscribed in the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a common point if and only if $ABCD$ is a rhombus.

2014 Iranian Geometry Olympiad (junior), P2

Tags: incircle , ratio , geometry , perpendicular , area

The inscribed circle of $\triangle ABC$ touches $BC, AC$ and $AB$ at $D,E$ and $F$ respectively. Denote the perpendicular foots from $F, E$ to $BC$ by $K, L$ respectively. Let the second intersection of these perpendiculars with the incircle be $M, N$ respectively. Show that $\frac{{{S}_{\triangle BMD}}}{{{S}_{\triangle CND}}}=\frac{DK}{DL}$ by Mahdi Etesami Fard

2017 USA Team Selection Test, 2

Tags: circumcircle , inversion , euler line , geometry

Let $ABC$ be an acute scalene triangle with circumcenter $O$, and let $T$ be on line $BC$ such that $\angle TAO = 90^{\circ}$. The circle with diameter $\overline{AT}$ intersects the circumcircle of $\triangle BOC$ at two points $A_1$ and $A_2$, where $OA_1 < OA_2$. Points $B_1$, $B_2$, $C_1$, $C_2$ are defined analogously. [list=a][*] Prove that $\overline{AA_1}$, $\overline{BB_1}$, $\overline{CC_1}$ are concurrent. [*] Prove that $\overline{AA_2}$, $\overline{BB_2}$, $\overline{CC_2}$ are concurrent on the Euler line of triangle $ABC$. [/list][i]Evan Chen[/i]

1935 Eotvos Mathematical Competition, 2

Tags: geometry , symmetry

Prove that a finite point set cannot have more than one center of symmetry.

2013 AMC 10, 23

Tags: ratio , geometry , analytic geometry , cyclic quadrilateral , similar triangles

In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? ${ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 27\qquad\textbf{(E)}\ 30 $

2020 Serbia National Math Olympiad, 3

Tags: geometry , circumcircle , triangle , angle

We are given a triangle $ABC$. Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$, with the arrangment of points $D - A - B - E$. The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$, and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$. Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$.

2017 ITAMO, 4

Tags: geometry

Let $ABCD$ be a thetraedron with the following propriety: the four lines connecting a vertex and the incenter of opposite face are concurrent. Prove $AB \cdot CD= AC \cdot BD = AD\cdot BC$.

2019 Novosibirsk Oral Olympiad in Geometry, 4

Tags: isosceles , collinear , square , geometry

Two squares and an isosceles triangle are positioned as shown in the figure (the up left vertex of the large square lies on the side of the triangle). Prove that points $A, B$ and $C$ are collinear. [img]https://cdn.artofproblemsolving.com/attachments/d/c/03515e40f74ced1f8243c11b3e610ef92137ac.png[/img]

2009 Romania Team Selection Test, 3

Tags: circumcircle , geometry , analytic geometry , conic , trigonometry

Prove that pentagon $ ABCDE$ is cyclic if and only if \[\mathrm{d(}E,AB\mathrm{)}\cdot \mathrm{d(}E,CD\mathrm{)} \equal{} \mathrm{d(}E,AC\mathrm{)}\cdot \mathrm{d(}E,BD\mathrm{)} \equal{} \mathrm{d(}E,AD\mathrm{)}\cdot \mathrm{d(}E,BC\mathrm{)}\] where $ \mathrm{d(}X,YZ\mathrm{)}$ denotes the distance from point $ X$ ot the line $ YZ$.

2021 Brazil EGMO TST, 7

Tags: geometry

The incircle $\omega$ of a triangle $ABC$ touches the sides $BC, AC, AB$ in the points $D, E, F$ respectively. Two different points $K$ and $L$ are chosen in $\omega$ such that $\angle CKE+\angle BKF=\angle CLE+\angle BLF=180^{\circ}$. Prove that the line $KL$ is in the same distance to the point $D, E,$ and $F$.