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: 25757

1998 India National Olympiad, 4
Tags: rectangle , geometric inequality , trigonometry , cyclic quadrilateral , geometry
Suppose $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius one unit. If $AB \cdot BC \cdot CD \cdot DA \geq 4$, prove that $ABCD$ is a square.

2025 Thailand Mathematical Olympiad, 4
Tags: geometry , circumcircle , complex bash , ceva , incenter , trigonometry
Let $D,E$ and $F$ be touch points of the incenter of $\triangle ABC$ at $BC, CA$ and $AB$, respectively. Let $P,Q$ and $R$ be the circumcenter of triangles $AFE, BDF$ and $CED$, respectively. Show that $DP, EQ$ and $FR$ concurrent.

2017 Romanian Master of Mathematics, 6
Tags: cyclic quadrilateral , geometry
Let $ABCD$ be any convex quadrilateral and let $P, Q, R, S$ be points on the segments $AB, BC, CD$, and $DA$, respectively. It is given that the segments $PR$ and $QS$ dissect $ABCD$ into four quadrilaterals, each of which has perpendicular diagonals. Show that the points $P, Q, R, S$ are concyclic.

Swiss NMO - geometry, 2014.8
Tags: altitude , concyclic , geometry
In the acute-angled triangle $ABC$, let $M$ be the midpoint of the atlitude $h_b$ through $B$ and $N$ be the midpoint of the height $h_c$ through $C$. Further let $P$ be the intersection of $AM$ and $h_c$ and $Q$ be the intersection of $AN$ and $h_b$. Show that $M, N, P$ and $Q$ lie on a circle.

2007 Czech and Slovak Olympiad III A, 2
Tags: geometry , cyclic quadrilateral , incenter
In a cyclic quadrilateral $ABCD$, let $L$ and $M$ be the incenters of $ABC$ and $BCD$ respectively. Let $R$ be a point on the plane such that $LR\bot AC$ and $MR\bot BD$.Prove that triangle $LMR$ is isosceles.

2009 Today's Calculation Of Integral, 470
Tags: conic , calculus , integration , parabola , calculus computations , geometry
Determin integers $ m,\ n\ (m>n>0)$ for which the area of the region bounded by the curve $ y\equal{}x^2\minus{}x$ and the lines $ y\equal{}mx,\ y\equal{}nx$ is $ \frac{37}{6}$.

2016 CCA Math Bonanza, L4.2
Tags: rectangle , geometry
Consider the $2\times3$ rectangle below. We fill in the small squares with the numbers $1,2,3,4,5,6$ (one per square). Define a [i]tasty[/i] filling to be one such that each row is [b]not[/b] in numerical order from left to right and each column is [b]not[/b] in numerical order from top to bottom. If the probability that a randomly selected filling is tasty is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, what is $m+n$? \begin{tabular}{|c|c|c|c|} \hline & & \\ \hline & & \\ \hline \end{tabular} [i]2016 CCA Math Bonanza Lightning #4.2[/i]

2005 Mediterranean Mathematics Olympiad, 4
Tags: algebra , function , polynomial , geometry , number theory , modular arithmetic , geometric transformation
Let $A$ be the set of all polynomials $f(x)$ of order $3$ with integer coefficients and cubic coefficient $1$, so that for every $f(x)$ there exists a prime number $p$ which does not divide $2004$ and a number $q$ which is coprime to $p$ and $2004$, so that $f(p)=2004$ and $f(q)=0$. Prove that there exists a infinite subset $B\subset A$, so that the function graphs of the members of $B$ are identical except of translations

2010 Contests, 3
Tags: geometry , rectangle , exterior angle
An angle is given in a plane. Using only a compass, one must find out $(a)$ if this angle is acute. Find the minimal number of circles one must draw to be sure. $(b)$ if this angle equals $31^{\circ}$.(One may draw as many circles as one needs).

2015 CCA Math Bonanza, L1.2
Tags: rectangle , geometry
Let $ABCDEF$ be a regular hexagon with side length $2$. Calculate the area of $ABDE$. [i]2015 CCA Math Bonanza Lightning Round #1.2[/i]

2024 Azerbaijan IMO TST, 6
Tags: geometry
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

1985 Greece National Olympiad, 3
Tags: analytic geometry , number theory , distance , lattice points , geometry
Consider the line (E): $5x-10y+3=0$ . Prove that: a) Line $(E)$ doesn't pass through points with integer coordinates. b) There is no point $A(a_1,a_2)$ with $ a_1,a_2 \in \mathbb{Z}$ with distance from $(E)$ less then $\frac{\sqrt3}{20}$.

2019 Jozsef Wildt International Math Competition, W. 69
Tags: angle bisector , triangle , geometric inequality , circumradius , inradius , inequalities , geometry
Denote $\overline{w_a}, \overline{w_b}, \overline{w_c}$ the external angle-bisectors in triangle $ABC$, prove that $$\sum \limits_{cyc} \frac{1}{w_a}\leq \sqrt{\frac{(s^2 - r^2 - 4Rr)(8R^2 - s^2 - r^2 - 2Rr)}{8s^2R^2r}}$$

2021 Romanian Master of Mathematics Shortlist, G3
Tags: geometry
Let $\Omega$ be the circumcircle of a triangle $ABC$ with $\angle BAC > 90^{\circ}$ and $AB > AC$. The tangents of $\Omega$ at $B$ and $C$ cross at $D$ and the tangent of $\Omega$ at $A$ crosses the line $BC$ at $E$. The line through $D$, parallel to $AE$, crosses the line $BC$ at $F$. The circle with diameter $EF$ meets the line $AB$ at $P$ and $Q$ and the line $AC$ at $X$ and $Y$. Prove that one of the angles $\angle AEB$, $\angle PEQ$, $\angle XEY$ is equal to the sum of the other two.

2003 Turkey Junior National Olympiad, 1
Tags: trig identities , quadratic , law of cosines , geometry , trigonometry , cyclic quadrilateral
Let $ABCD$ be a cyclic quadrilateral, and $E$ be the intersection of its diagonals. If $m(\widehat{ADB}) = 22.5^\circ$, $|BD|=6$, and $|AD|\cdot|CE|=|DC|\cdot|AE|$, find the area of the quadrilateral $ABCD$.

Ukrainian From Tasks to Tasks - geometry, 2012.4
Tags: trapezoid , angle , geometry
Let $ABCD$ be an isosceles trapezoid ($AD\parallel BC$), $\angle BAD = 80^o$, $\angle BDA = 60^o$. Point $P$ lies on $CD$ and $\angle PAD = 50^o$. Find $\angle PBC$

1980 Canada National Olympiad, 3
Tags: geometry , inradius , perimeter
Among all triangles having (i) a fixed angle $A$ and (ii) an inscribed circle of fixed radius $r$, determine which triangle has the least minimum perimeter.

2006 Baltic Way, 13
Tags: geometry , circumcircle
In a triangle $ABC$, points $D,E$ lie on sides $AB,AC$ respectively. The lines $BE$ and $CD$ intersect at $F$. Prove that if $\color{white}\ .\quad\ \color{black}\ \quad BC^2=BD\cdot BA+CE\cdot CA,$ then the points $A,D,F,E$ lie on a circle.

Denmark (Mohr) - geometry, 2007.4
Tags: circles , geometry , angle , tangent circle
The figure shows a $60^o$ angle in which are placed $2007$ numbered circles (only the first three are shown in the figure). The circles are numbered according to size. The circles are tangent to the sides of the angle and to each other as shown. Circle number one has radius $1$. Determine the radius of circle number $2007$. [img]https://1.bp.blogspot.com/-1bsLIXZpol4/Xzb-Nk6ospI/AAAAAAAAMWk/jrx1zVYKbNELTWlDQ3zL9qc_22b2IJF6QCLcBGAsYHQ/s0/2007%2BMohr%2Bp4.png[/img]

1986 Bundeswettbewerb Mathematik, 2
Tags: right triangle , geometry , excircle , exradius
A triangle has sides $a, b,c$, radius of the incircle $r$ and radii of the excircles $r_a, r_b, r_c$: Prove that: a) The triangle is right-angled if and only if: $r + r_a + r_b + r_c = a + b + c$. b) The triangle is right-angled if and only if: $r^2 + r^2_a + r^2_b + r^2_c = a^2 + b^2 + c^2$.

2006 National Olympiad First Round, 17
Tags: geometry , trigonometry
Let $D$ be a point on the side $[BC]$ of $\triangle ABC$ such that $|BD|=2$ and $|DC|=6$. If $|AB|=4$ and $m(\widehat{ACB})=20^\circ$, then what is $m(\widehat {BAD})$? $ \textbf{(A)}\ 10^\circ \qquad\textbf{(B)}\ 18^\circ \qquad\textbf{(C)}\ 20^\circ \qquad\textbf{(D)}\ 22^\circ \qquad\textbf{(E)}\ 25^\circ $

2018 IFYM, Sozopol, 5
Tags: geometry
On the sides $AB$,$BC$, and $CA$ of $\triangle ABC$ are chosen points $C_1$, $A_1$, and $B_1$ respectively, in such way that $AA_1$, $BB_1$, and $CC_1$ intersect in one point $X$. If $\angle A_1C_1B = \angle B_1C_1A$, prove that $CC_1$ is perpendicular to $AB$.

2010 Saint Petersburg Mathematical Olympiad, 7
Tags: geometry , invariant , rectangle , combinatorics , chessboard
$200 \times 200$ square is colored in chess order. In one move we can take every $2 \times 3$ rectangle and change color of all its cells. Can we make all cells of square in same color ?

2006 Pre-Preparation Course Examination, 3
Tags: geometry , conic , hyperbola
There is a right angle whose vertex moves on a fixed circle and one of it's sides passes a fixed point. What is the curve that the other side of the angle is always tangent to it.

2008 Putnam, B3
Tags: geometry , symmetry , 3d geometry , sphere , inequalities , trigonometry
What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?