Olympiad problems

2022 Yasinsky Geometry Olympiad, 5

Let $ABC$ be a right triangle with leg $CB = 2$ and hypotenuse $AB= 4$. Point $K$ is chosen on the hypotenuse $AB$, and point $L$ is chosen on the leg $AC$. a) Describe and justify how to construct such points $K$ and $ L$ so that the sum of the distances $CK+KL$ is the smallest possible. b) Find the smallest possible value of $CK+KL$. (Olexii Panasenko)

1984 All Soviet Union Mathematical Olympiad, 388

Tags: collinear , geometric inequality , geometry

The $A,B,C$ and $D$ points (from left to right) belong to the straight line. Prove that every point $E$, that doesn't belong to the line satisfy: $$|AE| + |ED| + | |AB| - |CD| | > |BE| + |CE|$$

Kyiv City MO 1984-93 - geometry, 1986.10.5

Tags: geometry , max , geometric inequality , area

Let $E$ be a point on the side $AD$ of the square $ABCD$. Find such points $M$ and $K$ on the sides $AB$ and $BC$ respectively, such that the segments $MK$ and $EC$ are parallel, and the quadrilateral $MKCE$ has the largest area.

the 6th XMO, 2

Tags: complex number , algebra , inequalities , geometric inequality

Assume that complex numbers $z_1,z_2,...,z_n$ satisfy $|z_i-z_j| \le 1$ for any $1 \le i <j \le n$. Let $$S= \sum_{1 \le i <j \le n} |z_i-z_j|^2.$$ (1) If $n = 6063$, find the maximum value of $S$. (2) If $n= 2021$, find the maximum value of $S$.

Ukrainian TYM Qualifying - geometry, 2015.20

Tags: ratio , inradii , geometric inequality , geometry

What is the smallest value of the ratio of the lengths of the largest side of the triangle to the radius of its inscribed circle?

2014 Federal Competition For Advanced Students, P2, 3

Tags: inequalities , triangle inequality , geometric inequality , exponential

(i) For which triangles with side lengths $a, b$ and $c$ apply besides the triangle inequalities $a + b> c, b + c> a$ and $c + a> b$ also the inequalities $a^2 + b^2> c^2, b^2 + c^2> a^2$ and $a^2 + c^2> b^2$ ? (ii) For which triangles with side lengths $a, b$ and $c$ apply besides the triangle inequalities $a + b> c, b + c> a$ and $c + a> b$ also for all positive natural $n$ the inequalities $a^n + b^n> c^n, b^n + c^n> a^n$ and $a^n + c^n> b^n$ ?

2016 BMT Spring, 10

Tags: perimeter , geometric inequality , geometry

What is the smallest possible perimeter of a triangle with integer coordinate vertices, area $\frac12$, and no side parallel to an axis?

2017 Czech And Slovak Olympiad III A, 2

Tags: inequalities , geometric inequality

Find all pairs of real numbers $k, l$ such that inequality $ka^2 + lb^2> c^2$ applies to the lengths of sides $a, b, c$ of any triangle.

2001 Mongolian Mathematical Olympiad, Problem 2

Tags: geometric inequality , inequality , geometry

In an acute-angled triangle $ABC$, $a,b,c$ are sides, $m_a,m_b,m_c$ the corresponding medians, $R$ the circumradius and $r$ the inradius. Prove the inequality $$\frac{a^2+b^2}{a+b}\cdot\frac{b^2+c^2}{b+c}\cdot\frac{a^2+c^2}{a+c}\ge16R^2r\frac{m_a}a\cdot\frac{m_b}b\cdot\frac{m_c}c.$$

1976 IMO Shortlist, 3

Tags: geometry , trigonometry , inequalities , area of a triangle , geometric inequality

In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.

2006 All-Russian Olympiad Regional Round, 9.6

Tags: geometry , geometric inequality , angle

In an acute triangle $ABC$, the angle bisector$AD$ and altitude $BE$ are drawn. Prove that angle $CED$ is greater than $45^o$.

IV Soros Olympiad 1997 - 98 (Russia), 11.6

Tags: geometry , 3d geometry , geometric inequality

On the planet Brick, which has the shape of a rectangular parallelepiped with edges of $1$ km,$ 2$ km and $4$ km, the Little Prince built a house in the center of the largest face. What is the distance from the house to the most remote point on the planet? (The distance between two points on the surface of a planet is defined as the length of the shortest path along the surface connecting these points.)

2003 Bosnia and Herzegovina Junior BMO TST, 4

Tags: geometry , trapezoid , geometric inequality

In the trapezoid $ABCD$ ($AB \parallel DC$) the bases have lengths $a$ and $c$ ($c < a$), while the other sides have lengths $b$ and $d$. The diagonals are of lengths $m$ and $n$. It is known that $m^2 + n^2 = (a + c)^2$. a) Find the angle between the diagonals of the trapezoid. b) Prove that $a + c < b + d$. c) Prove that $ac < bd$.

1967 IMO Longlists, 23

Tags: algebra , vector , inequality , geometric inequality , 3d geometry

Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality \[af^2 + bfg +cg^2 \geq 0\] holds if and only if the following conditions are fulfilled: \[a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.\]

2016 IFYM, Sozopol, 5

Tags: geometric inequality , geometry

Prove that for an arbitrary $\Delta ABC$ the following inequality holds: $\frac{l_a}{m_a}+\frac{l_b}{m_b}+\frac{l_c}{m_c} >1$, Where $l_a,l_b,l_c$ and $m_a,m_b,m_c$ are the lengths of the bisectors and medians through $A$, $B$, and $C$.

1952 Moscow Mathematical Olympiad, 215

Tags: geometry , incircle , inradius , geometric inequality

$\vartriangle ABC$ is divided by a straight line $BD$ into two triangles. Prove that the sum of the radii of circles inscribed in triangles $ABD$ and $DBC$ is greater than the radius of the circle inscribed in $\vartriangle ABC$.

2021 Tuymaada Olympiad, 8

Tags: geometric inequality , inequalities , geometry

An acute triangle $ABC$ is given, $AC \not= BC$. The altitudes drawn from $A$ and $B$ meet at $H$ and intersect the external bisector of the angle $C$ at $Y$ and $X$ respectively. The external bisector of the angle $AHB$ meets the segments $AX$ and $BY$ at $P$ and $Q$ respectively. If $PX = QY$, prove that $AP + BQ \ge 2CH$.

2015 Czech-Polish-Slovak Junior Match, 3

Tags: geometry , geometric inequality , equal segments , perpendicular

Different points $A$ and $D$ are on the same side of the line $BC$, with $|AB| = | BC|= |CD|$ and lines $AD$ and $BC$ are perpendicular. Let $E$ be the intersection point of lines $AD$ and $BC$. Prove that $||BE| - |CE|| < |AD| \sqrt3$

1988 All Soviet Union Mathematical Olympiad, 486

Tags: sphere , tetrahedron , radical , geometric inequality , 3d geometry , geometry

Prove that for any tetrahedron the radius of the inscribed sphere $r <\frac{ ab}{ 2(a + b)}$, where $a$ and $b$ are the lengths of any pair of opposite edges.

Kyiv City MO 1984-93 - geometry, 1986.7.5

Tags: geometry , geometric inequality

Prove that the sum of the lengths of the diagonals of an arbitrary quadrilateral is less than the sum of the lengths of its sides.

2010 Sharygin Geometry Olympiad, 3

Tags: convex polygon , inequality , geometric inequality , diagonal , geometry

All sides of a convex polygon were decreased in such a way that they formed a new convex polygon. Is it possible that all diagonals were increased?

2008 IMAC Arhimede, 2

Tags: geometry , geometric inequality , inequalities , angle bisector , circumcircle

In the $ ABC$ triangle, the bisector of $A $ intersects the $ [BC] $ at the point $ A_ {1} $ , and the circle circumscribed to the triangle $ ABC $ at the point $ A_ {2} $. Similarly are defined $ B_ {1} $ and $ B_ {2} $ , as well as $ C_ {1} $ and $ C_ {2} $. Prove that $$ \frac {A_{1}A_{2}}{BA_{2} + A_{2}C} + \frac {B_{1}B_{2}}{CB_{2} + B_{2}A} + \frac {C_{1}C_{2}}{AC_{2} + C_{2}B} \geq \frac {3}{4}$$

2020 Ukrainian Geometry Olympiad - April, 5

Tags: geometry , geometric inequality , angle

Inside the convex quadrilateral $ABCD$ there is a point $M$ such that $\angle AMB = \angle ADM + \angle BCM$ and $\angle AMD = \angle ABM + \angle DCM$. Prove that $AM \cdot CM + BM \cdot DM \ge \sqrt{AB \cdot BC\cdot CD \cdot DA}$

1986 Poland - Second Round, 3

Tags: 3d geometry , sphere , tetrahedron , geometric inequality , geometry

Let S be a sphere cirucmscribed on a regular tetrahedron with an edge length greater than 1. The sphere $ S $ is represented as the sum of four sets. Prove that one of these sets includes points $ P $, $ Q $ such that the length of the segment $ PQ $ exceeds 1.

1969 Spain Mathematical Olympiad, 7

Tags: geometric inequality , geometry , regular polygon

A convex polygon $A_1A_2 . . .A_n$ of $n$ sides and inscribed in a circle, has its sides that satisfy the inequalities $$A_nA_1 > A_1A_2 > A_2A_3 >...> A_{n-1}A_n$$ Show that its interior angles satisfy the inequalities $$\angle A_1 < \angle A_2 < \angle A_3 < ... < \angle A_{n-1}, \angle A_{n-1} > \angle A_n> \angle A_1.$$