This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 649

1978 IMO Longlists, 9

Let $T_1$ be a triangle having $a, b, c$ as lengths of its sides and let $T_2$ be another triangle having $u, v,w$ as lengths of its sides. If $P,Q$ are the areas of the two triangles, prove that \[16PQ \leq a^2(-u^2 + v^2 + w^2) + b^2(u^2 - v^2 + w^2) + c^2(u^2 + v^2 - w^2).\] When does equality hold?

1950 Moscow Mathematical Olympiad, 176

Let $a, b, c$ be the lengths of the sides of a triangle and $A, B, C$, the opposite angles. Prove that $$Aa + Bb + Cc \ge \frac{Ab + Ac + Ba + Bc + Ca + Cb}{2}$$

the 6th XMO, 2

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 $S$.

Kyiv City MO 1984-93 - geometry, 1989.8.2

Let $h_a,h_b,h_c$ be the altitudes, and let $m_a,m_b,m_c$ be the medians of the acute triangle drawn to the sides $a, b, c$ respectively. Let $r$ and $R$ be the radii of the inscribed and circumscribed circles. Prove that $$\frac{m_a}{h_a}+\frac{m_b}{h_b}+\frac{m_c}{h_c} <1+\frac{R}{r}.$$

2021 Estonia Team Selection Test, 3

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

1989 Greece National Olympiad, 4

A trapezoid with bases $a,b$ and altitude $h$ is circumscribed around a circl.. Prove that $h^2\le ab$.

2015 BMT Spring, 9

Find the side length of the largest square that can be inscribed in the unit cube.

Novosibirsk Oral Geo Oly IX, 2019.7

Denote $X,Y$ two convex polygons, such that $X$ is contained inside $Y$. Denote $S (X)$, $P (X)$, $S (Y)$, $P (Y)$ the area and perimeter of the first and second polygons, respectively. Prove that $$ \frac{S(X)}{P(X)}<2 \frac{S(Y)}{P(Y)}.$$

1995 Tournament Of Towns, (476) 4

Three different points $A$, $B$ and $C$ are placed in the plane. Construct a line $m$ through $C$ so that the product of the distances from $A$ and $B$ to $m$ has the maximal value. Is $m$ unique for every triple $A$, $B$ and $C$? (NB Vassiliev)

II Soros Olympiad 1995 - 96 (Russia), 9.7

Through a point located on a side of a triangle of area $1$, two straight lines are drawn parallel to the two remaining sides. They divided the triangle into three parts. Let $s$ be the largest of the areas of these parts. Find the smallest possible value of $s$.

2018 Swedish Mathematical Competition, 5

In a triangle $ABC$, two lines are drawn that together trisect the angle at $A$. These intersect the side $BC$ at points $P$ and $Q$ so that $P$ is closer to $B$ and $Q$ is closer to $C$. Determine the smallest constant $k$ such that $| P Q | \le k (| BP | + | QC |)$, for all such triangles. Determine if there are triangles for which equality applies.

1997 APMO, 3

Let $ABC$ be a triangle inscribed in a circle and let \[ l_a = \frac{m_a}{M_a} \ , \ \ l_b = \frac{m_b}{M_b} \ , \ \ l_c = \frac{m_c}{M_c} \ , \] where $m_a$,$m_b$, $m_c$ are the lengths of the angle bisectors (internal to the triangle) and $M_a$, $M_b$, $M_c$ are the lengths of the angle bisectors extended until they meet the circle. Prove that \[ \frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \geq 3 \] and that equality holds iff $ABC$ is an equilateral triangle.

Cono Sur Shortlist - geometry, 2021.G4

Let $ABC$ be a triangle and $\Gamma$ the $A$- exscribed circle whose center is $J$ . Let $D$ and $E$ be the touchpoints of $\Gamma$ with the lines $AB$ and $AC$, respectively. Let $S$ be the area of the quadrilateral $ADJE$, Find the maximum value that $\frac{S}{AJ^2}$ has and when equality holds.

2012 German National Olympiad, 5

Let $a,b$ be the lengths of two nonadjacent edges of a tetrahedron with inradius $r$. Prove that \[r<\frac{ab}{2(a+b)}.\]

2024 South Africa National Olympiad, 5

Consider three circles $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$, with centres $O_1$, $O_2$ and $O_3$, respectively, such that each pair of circles is externally tangent. Suppose we have another circle $\Gamma$ with centre $O$ on the line segment $O_1O_3$ such that $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$ are each internally tangent to $\Gamma$. Show that $\angle O_1O_2O_3$ measures less than $90^\circ$.

2022 Kosovo National Mathematical Olympiad, 4

Assume that in the $\triangle ABC$ there exists a point $D$ on $BC$ and a line $l$ passing through $A$ such that $l$ is tangent to $(ADC)$ and $l$ bisects $BD.$ Prove that $a\sqrt{2}\geq b+c.$

2020 BMT Fall, 12

A hollow box (with negligible thickness) shaped like a rectangular prism has a volume of $108$ cubic units. The top of the box is removed, exposing the faces on the inside of the box. What is the minimum possible value for the sum of the areas of the faces on the outside and inside of the box?

1967 IMO Shortlist, 1

The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only \[a\le\cos A+\sqrt3\sin A.\]

1983 Czech and Slovak Olympiad III A, 2

Given a triangle $ABC$, prove that for every inner point $P$ of the side $AB$ the inequality $$PC\cdot AB<PA\cdot BC+PB\cdot AC$$ holds.

2005 Oral Moscow Geometry Olympiad, 3

In triangle $ABC$, points $K ,P$ are chosen on the side $AB$ so that $AK = BL$, and points $M,N$ are chosen on the side $BC$ so that $CN = BM$. Prove that $KN + LM \ge AC$. (I. Bogdanov)

1992 IMO Longlists, 70

Let two circles $A$ and $B$ with unequal radii $r$ and $R$, respectively, be tangent internally at the point $A_0$. If there exists a sequence of distinct circles $(C_n)$ such that each circle is tangent to both $A$ and $B$, and each circle $C_{n+1}$ touches circle $C_{n}$ at the point $A_n$, prove that \[\sum_{n=1}^{\infty} |A_{n+1}A_n| < \frac{4 \pi Rr}{R+r}.\]

2013 Turkmenistan National Math Olympiad, 4

Let $ ABCD$ be a convex quadrilateral such that the sides $ AB, AD, BC$ satisfy $ AB \equal{} AD \plus{} BC.$ There exists a point $ P$ inside the quadrilateral at a distance $ h$ from the line $ CD$ such that $ AP \equal{} h \plus{} AD$ and $ BP \equal{} h \plus{} BC.$ Show that: \[ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} \plus{} \frac {1}{\sqrt {BC}} \]

2000 Rioplatense Mathematical Olympiad, Level 3, 2

In a triangle $ABC$, points $D, E$ and $F$ are considered on the sides $BC, CA$ and $AB$ respectively, such that the areas of the triangles $AFE, BFD$ and $CDE$ are equal. Prove that $$\frac{(DEF) }{ (ABC)} \ge \frac{1}{4}$$ Note: $(XYZ)$ is the area of triangle $XYZ$.

1939 Moscow Mathematical Olympiad, 047

Prove that for any triangle the bisector lies between the median and the height drawn from the same vertex.

Revenge ELMO 2023, 1

In cyclic quadrilateral $ABCD$ with circumcenter $O$ and circumradius $R$, define $X=\overline{AB}\cap\overline{CD}$, $Y=\overline{AC}\cap \overline{BD}$, and $Z=\overline{AD}\cap\overline{BC}$. Prove that \[OX^2+OY^2+OZ^2\ge 2R^2+2[ABCD].\] [i]Rohan Bodke[/i]