Olympiad problems

1998 AIME Problems, 2

Tags: inequalities , geometry , perimeter , symmetry , parameterization

Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x\le 2y\le 60$ and $y\le 2x\le 60.$

1970 Poland - Second Round, 4

Tags: geometry , perimeter , geometric inequalities

Prove that if triangle $T_1$ contains triangle $T_2$, then the perimeter of triangle $T_1$ is not less than the perimeter of triangle $T_2$.

2015 AMC 10, 24

Tags: geometry , perimeter , calculus , rhombus , parabola , conic

For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible? $\textbf{(A) }30\qquad\textbf{(B) }31\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

2022 IOQM India, 3

Tags: number theory , prime number , arithmetic sequence , geometry , perimeter , trivial

Consider the set $\mathcal{T}$ of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let $\triangle \in \mathcal{T}$ be the triangle with least perimeter. If $a^{\circ}$ is the largest angle of $\triangle$ and $L$ is its perimeter, determine the value of $\frac{a}{L}$.

1950 Moscow Mathematical Olympiad, 183

Tags: geometry , perimeter , square , circles

A circle is inscribed in a triangle and a square is circumscribed around this circle so that no side of the square is parallel to any side of the triangle. Prove that less than half of the square’s perimeter lies outside the triangle.

2017 BMT Spring, 2

Tags: geometry , perimeter , equilateral , square , area

Barack is an equilateral triangle and Michelle is a square. If Barack and Michelle each have perimeter $ 12$, find the area of the polygon with larger area.

2002 AMC 10, 23

Tags: perimeter , geometry

Points $ A,B,C$ and $ D$ lie on a line, in that order, with $ AB\equal{}CD$ and $ BC\equal{}12$. Point $ E$ is not on the line, and $ BE\equal{}CE\equal{}10$. The perimeter of $ \triangle AED$ is twice the perimeter of $ \triangle BEC$. Find $ AB$. $ \text{(A)}\ 15/2 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 17/2 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 19/2$

2005 China Team Selection Test, 2

Tags: perimeter , cyclic quadrilateral , pythagorean theorem , geometry

Cyclic quadrilateral $ABCD$ has positive integer side lengths $AB$, $BC$, $CA$, $AD$. It is known that $AD=2005$, $\angle{ABC}=\angle{ADC} = 90^o$, and $\max \{ AB,BC,CD \} < 2005$. Determine the maximum and minimum possible values for the perimeter of $ABCD$.

2010 Today's Calculation Of Integral, 662

Tags: calculus , integration , geometric transformation , rotation , inequalities , perimeter , geometry

In $xyz$ space, let $A$ be the solid generated by a rotation of the figure, enclosed by the curve $y=2-2x^2$ and the $x$-axis about the $y$-axis. (1) When the solid is cut by the plane $x=a\ (|a|\leq 1)$, find the inequality which expresses the figure of the cross-section. (2) Denote by $L$ the distance between the point $(a,\ 0,\ 0)$ and the point on the perimeter of the cross-section found in (1), find the maximum value of $L$. (3) Find the volume of the solid by a rotation of the solid $A$ about the $x$-axis. [i]1987 Sophia University entrance exam/Science and Technology[/i]

1985 Traian Lălescu, 2.3

Tags: geometry , perimeter , pure geometry

Let $ ABC $ a triangle, and $ P\neq B,C $ be a point situated upon the segment $ BC $ such that $ ABP $ and $ APC $ have the same perimeter. $ M $ represents the middle of $ BC, $ and $ I, $ the center of the incircle of $ ABC. $ Prove that $ IM\parallel AP. $

2011 Purple Comet Problems, 2

Tags: geometry , perimeter

The diagram below shows a $12$-sided figure made up of three congruent squares. The figure has total perimeter $60$. Find its area. [asy] size(150); defaultpen(linewidth(0.8)); path square=unitsquare; draw(rotate(360-135)*square^^rotate(345)*square^^rotate(105)*square); [/asy]

2021 Novosibirsk Oral Olympiad in Geometry, 3

Tags: geometry , perimeter

In triangle $ABC$, side $AB$ is $1$. It is known that one of the angle bisectors of triangle $ABC$ is perpendicular to one of its medians, and some other angle bisector is perpendicular to the other median. What can be the perimeter of triangle $ABC$?

2021 Iranian Geometry Olympiad, 3

Tags: geometry , perimeter , semicircle , angle

As shown in the following figure, a heart is a shape consist of three semicircles with diameters $AB$, $BC$ and $AC$ such that $B$ is midpoint of the segment $AC$. A heart $\omega$ is given. Call a pair $(P, P')$ bisector if $P$ and $P'$ lie on $\omega$ and bisect its perimeter. Let $(P, P')$ and $(Q,Q')$ be bisector pairs. Tangents at points $P, P', Q$, and $Q'$ to $\omega$ construct a convex quadrilateral $XYZT$. If the quadrilateral $XYZT$ is inscribed in a circle, find the angle between lines $PP'$ and $QQ'$. [img]https://cdn.artofproblemsolving.com/attachments/3/c/8216889594bbb504372d8cddfac73b9f56e74c.png[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

1915 Eotvos Mathematical Competition, 2

Tags: geometry , perimeter , geometric inequalities

Triangle $ABC$ lies entirely inside a polygon. Prove that the perimeter of triangle $ABC$ is not greater than that of the polygon.

2015 Regional Olympiad of Mexico Southeast, 3

Tags: algebra , perimeter , geometry , inequalities

If $T(n)$ is the numbers of triangles with integers sizes(not congruent with each other) with it´s perimeter is equal to $n$, prove that: $$T(2012)<T(2015)$$ $$T(2013)=T(2016)$$

1957 AMC 12/AHSME, 7

Tags: geometry , perimeter , inradius

The area of a circle inscribed in an equilateral triangle is $ 48\pi$. The perimeter of this triangle is: $ \textbf{(A)}\ 72\sqrt{3} \qquad \textbf{(B)}\ 48\sqrt{3}\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 24\qquad \textbf{(E)}\ 72$

1990 IMO Longlists, 81

Tags: geometry , inradius , perimeter , exradius

A circle of radius $\rho$ is tangent to the sides $AB$ and $AC$ of the triangle $ABC$, and its center $K$ is at a distance $p$ from $BC$. [i](a)[/i] Prove that $a(p - \rho) = 2s(r - \rho)$, where $r$ is the inradius and $2s$ the perimeter of $ABC$. [i](b)[/i] Prove that if the circle intersect $BC$ at $D$ and $E$, then \[DE=\frac{4\sqrt{rr_1(\rho-r)(r_1-\rho)}}{r_1-r}\] where $r_1$ is the exradius corresponding to the vertex $A.$

1986 Balkan MO, 4

Tags: geometry , perimeter , hyperbola , parallelogram , rectangle , perpendicular bisector , conic

Let $ABC$ a triangle and $P$ a point such that the triangles $PAB, PBC, PCA$ have the same area and the same perimeter. Prove that if: a) $P$ is in the interior of the triangle $ABC$ then $ABC$ is equilateral. b) $P$ is in the exterior of the triangle $ABC$ then $ABC$ is right angled triangle.

Novosibirsk Oral Geo Oly VIII, 2020.2

Tags: geometry , perimeter , chessboard , combinatorics

Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?

2012 NIMO Problems, 10

Tags: geometry , incenter , trigonometry , circumcircle , perimeter , ratio , quadratic

In cyclic quadrilateral $ABXC$, $\measuredangle XAB = \measuredangle XAC$. Denote by $I$ the incenter of $\triangle ABC$ and by $D$ the projection of $I$ on $\overline{BC}$. If $AI = 25$, $ID = 7$, and $BC = 14$, then $XI$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$. [i]Proposed by Aaron Lin[/i]

1980 Canada National Olympiad, 3

Tags: perimeter , inradius , geometry

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.

1949 Moscow Mathematical Olympiad, 159

Tags: disc , geometry , perimeter , combinatorial geometry

Consider a closed broken line of perimeter $1$ on a plane. Prove that a disc of radius $\frac14$ can cover this line.

2015 AIME Problems, 4

Tags: geometry , trapezoid , perimeter

In an isosceles trapezoid, the parallel bases have lengths $\log3$ and $\log192$, and the altitude to these bases has length $\log16$. The perimeter of the trapezoid can be written in the form $\log2^p3^q$, where $p$ and $q$ are positive integers. Find $p+q$.

1999 Moldova Team Selection Test, 7

Tags: geometry , perimeter

Let $ABC$ be an equilateral triangle and $n{}, n>1$ an integer. Let $S{}$ be the set of the $n-1$ lines parallel with $BC$ that cut $ABC$ in $n{}$ figures with equal areas and $S^{'}$ be the set of the $n-1$ lines parallel with $BC$ that cut $ABC$ in $n{}$ figures with equal perimeters. Show that $S{}$ and $S^{'}$ are disjunctive.

2010 Philippine MO, 2

Tags: geometry , cyclic quadrilateral , perimeter

On a cyclic quadrilateral $ABCD$, there is a point $P$ on side $AD$ such that the triangle $CDP$ and the quadrilateral $ABCP$ have equal perimeters and equal areas. Prove that two sides of $ABCD$ have equal lengths.