Olympiad problems

1999 Argentina National Olympiad, 4

Coins of diameter $1$ have been placed on a square of side $11$, without overlapping or protruding from the square. Can there be $126$ coins? and $127$? and $128$?

2011 Purple Comet Problems, 2

Tags: geometry

The target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. The area of the dark region is $n\pi$. Find $n$. [asy] size(150); defaultpen(linewidth(0.8)); int i; for(i=5;i>=1;i=i-1) { if (floor(i/2)==i/2) { filldraw(circle(origin,4*i),white); } else { filldraw(circle(origin,4*i),red); } } [/asy]

Kvant 2024, M2800

Tags: parallelogram , geometry

Let $ABCD$ be a parallelogram. Let $M$ be the midpoint of the arc $AC$ containing $B$ of the circumcircle of $ABC$ . Let $E$ be a point on segment $AD$ and $F$ a point on segment $CD$ such that $ME=MD=MF$. Show that $BMEF$ is cyclic. [i]Proposed by A. Tereshin[/i]

2021 Alibaba Global Math Competition, 14

Tags: metric space , injectivity , geometry , analysis

Let $f$ be a smooth function on $\mathbb{R}^n$, denote by $G_f=\{(x,f(x)) \in \mathbb{R}^{n+1}: x \in \mathbb{R}^n\}$. Let $g$ be the restriction of the Euclidean metric on $G_f$. (1) Prove that $g$ is a complete metric. (2) If there exists $\Lambda>0$, such that $-\Lambda I_n \le \text{Hess}(f) \le \Lambda I_n$, where $I_n$ is the unit matrix of order $n$, and $\text{Hess}8f)$ is the Hessian matrix of $f$, then the injectivity radius of $(G_f,g)$ is at least $\frac{\pi}{2\Lambda}$.

2023 Girls in Mathematics Tournament, 3

Tags: geometry

Let $ABC$ an acute triangle and $D$ and $E$ the feet of heights by $A$ and $B$, respectively, and let $M$ be the midpoint of $AC$. The circle that passes through $D$ and $B$ and is tangent to $BE$ in $B$ intersects the line $BM$ in $F, F\neq B$. Show that $FM$ is the angle bisector of $\angle AFD$.

2011 Today's Calculation Of Integral, 759

Tags: calculus , integration , 3d geometry , tetrahedron , geometric transformation , rotation , geometry

Given a regular tetrahedron $PQRS$ with side length $d$. Find the volume of the solid generated by a rotation around the line passing through $P$ and the midpoint $M$ of $QR$.

1966 IMO Longlists, 55

Tags: geometry , centroid , locus , locus problems

Given the vertex $A$ and the centroid $M$ of a triangle $ABC$, find the locus of vertices $B$ such that all the angles of the triangle lie in the interval $[40^\circ, 70^\circ].$

2011 JHMT, 1

Tags: geometry

Let $D_{x,y}$ denote the half-disk of radius $1$ with its curved boundary externally tangent to the unit circle at the point $(x, y)$, such that the straight boundary of the disk is parallel to the tangent line (so the point of tangency is the middle of the curved boundary). Find the area of the union of the $D_{x,y}$ over all $(x, y)$ with $x^2 + y^2 = 1$ (that is, $(x,y)$ is on the unit circle).

2011 Iran MO (3rd Round), 1

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

We have $4$ circles in plane such that any two of them are tangent to each other. we connect the tangency point of two circles to the tangency point of two other circles. Prove that these three lines are concurrent. [i]proposed by Masoud Nourbakhsh[/i]

2021 Czech and Slovak Olympiad III A, 6

Tags: geometry , fixed point , fixed

An acute triangle $ABC$ is given. Let us denote $X$ for each of its inner points $X_a, X_b, X_c$ its images in axial symmetries sequentially along the lines $BC, CA, AB$. Prove that all $X_aX_bX_c$ triangles have a common interior point. (Josef Tkadlec)

2010 Today's Calculation Of Integral, 606

Tags: calculus , integration , geometry , calculus computations

Find the area of the part bounded by two curves $y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $x$-axis. 1956 Tokyo Institute of Technology entrance exam

May Olympiad L1 - geometry, 2018.3

Tags: geometry , computational , polygon

Let $ABCDEFGHIJ$ be a regular $10$-sided polygon that has all its vertices in one circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at $P$ and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the measure of the segment $PQ$.

2005 Indonesia MO, 7

Tags: geometry

Let $ ABCD$ be a convex quadrilateral. Square $ AB_1A_2B$ is constructed such that the two vertices $ A_2,B_1$ is located outside $ ABCD$. Similarly, we construct squares $ BC_1B_2C$, $ CD_1C_2D$, $ DA_1D_2A$. Let $ K$ be the intersection of $ AA_2$ and $ BB_1$, $ L$ be the intersection of $ BB_2$ and $ CC_1$, $ M$ be the intersection of $ CC_2$ and $ DD_1$, and $ N$ be the intersection of $ DD_2$ and $ AA_1$. Prove that $ KM$ is perpendicular to $ LN$.

2015 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: geometry , circumcircle , incenter

Let $O$ and $I$ be circumcenter and incenter of triangle $ABC$. Let incircle of $ABC$ touches sides $BC$, $CA$ and $AB$ in points $D$, $E$ and $F$, respectively. Lines $FD$ and $CA$ intersect in point $P$, and lines $DE$ and $AB$ intersect in point $Q$. Furthermore, let $M$ and $N$ be midpoints of $PE$ and $QF$. Prove that $OI \perp MN$

Kyiv City MO 1984-93 - geometry, 1984.8.1

Tags: area , fixed , symmetric , geometry

Inside the convex quadrilateral $ABCD$ lies the point $'M$. Reflect it symmetrically with respect to the midpoints of the sides of the quadrilateral and connect the obtained points so that they form a convex quadrilateral. Prove that the area of this quadrilateral does not depend on the choice of the point $M$.

1996 Turkey MO (2nd round), 1

Tags: geometry

A circle is tangent to sides $AD,\text{ }DC,\text{ }CB$ of a convex quadrilateral $ABCD$ at $\text{K},\text{ L},\text{ M}$ respectively. A line $l$, passing through $L$ and parallel to $AD$, meets $KM$ at $N$ and $KC$ at $P$. Prove that $PL=PN$.

1981 Putnam, A6

Tags: geometry , lattice points

Suppose that each of the vertices of $ABC$ is a lattice point in the $xy$-plane and that there is exactly one lattice point $P$ in the interior of the triangle. The line $AP$ is extended to meet $BC$ at $E$. Determine the largest possible value for the ratio of lengths of segments $$\frac{|AP|}{|PE|}.$$

2015 Mathematical Talent Reward Programme, MCQ: P 6

Tags: geometry

Let $A C$ and $C E$ be perpendicular line segments, each of length $18 .$ Suppose $B$ and $D$ are the midpoints of $A C$ and $C E$ respectively. If $F$ be the point of intersection of $E B$ and $A D,$ then the area of $\triangle B D F$ is? [list=1] [*] $27\sqrt{2}$ [*] $18\sqrt{2}$ [*] 13.5 [*] 18 [/list]

1985 Yugoslav Team Selection Test, Problem 2

Tags: geometry , parallelogram

Let $ABCD$ be a parallelogram and let $E$ be a point in the plane such that $AE\perp AB$ and $BC\perp EC$. Show that either $\angle AED=\angle BEC$ or $\angle AED+\angle BEC=180^\circ$.

Russian TST 2022, P1

Tags: geometry , incenter

In triangle $ABC$, a point $M$ is the midpoint of $AB$, and a point $I$ is the incentre. Point $A_1$ is the reflection of $A$ in $BI$, and $B_1$ is the reflection of $B$ in $AI$. Let $N$ be the midpoint of $A_1B_1$. Prove that $IN > IM$.

2009 Sharygin Geometry Olympiad, 3

Tags: geometry , distance , angle bisector

The cirumradius and the inradius of triangle $ABC$ are equal to $R$ and $r, O, I$ are the centers of respective circles. External bisector of angle $C$ intersect $AB$ in point $P$. Point $Q$ is the projection of $P$ to line $OI$. Find distance $OQ.$ (A.Zaslavsky, A.Akopjan)

2023 Yasinsky Geometry Olympiad, 6

Tags: tangent , geometry

Let $ABC$ be an isosceles triangle with $\angle BAC = 108^o$. The angle bisector of the $\angle ABC$ intersects the circumcircle of a triangle $ABC$ at the point $D$. Let $E$ be a point on segment $CB$ such that $AB =BE$. Prove that the perpendicular bisector of $CD$ is tangent to circumcircle of triangle $ABE$ . (Bohdan Zheliabovskyi)

2020 Princeton University Math Competition, A2/B4

Tags: geometry

Hexagon $ABCDEF$ has an inscribed circle $\Omega$ that is tangent to each of its sides. If $AB = 12$, $\angle FAB = 120^o$, and $\angle ABC = 150^o$, and if the radius of $\Omega$ can be written as $m +\sqrt{n}$ for positive integers $m, n$, find $m + n$.

2016 Thailand Mathematical Olympiad, 8

Tags: geometry , incenter , orthocenter , parallel

Let $\vartriangle ABC$ be an acute triangle with incenter $I$. The line passing through $I$ parallel to $AC$ intersects $AB$ at $M$, and the line passing through $I$ parallel to $AB$ intersects $AC$ at $N$. Let the line $MN$ intersect the circumcircle of $\vartriangle ABC$ at $X$ and $Y$ . Let $Z$ be the midpoint of arc $BC$ (not containing $A$). Prove that $I$ is the orthocenter of $\vartriangle XY Z$

2021 Kyiv City MO Round 1, 7.2

Tags: game , combinatorics , cutting , geometry , rectangle

Andriy and Olesya take turns (Andriy starts) in a $2 \times 1$ rectangle, drawing horizontal segments of length $2$ or vertical segments of length $1$, as shown in the figure below. [img]https://i.ibb.co/qWqWxgh/Kyiv-MO-2021-Round-1-7-2.png[/img] After each move, the value $P$ is calculated - the total perimeter of all small rectangles that are formed (i.e., those inside which no other segment passes). The winner is the one after whose move $P$ is divisible by $2021$ for the first time. Who has a winning strategy? [i]Proposed by Bogdan Rublov[/i]