Olympiad problems

2019 JHMT, 6

Tags: geometry

Circles $C_1$ and $C_2$ intersect at exactly two points $I_1$ and $I_2$. A point $J$ on $C_1$ outside of $C_2$ is chosen such that $\overline{JI_2}$ is tangent to $C_2$ and $\overline{JI_2} = 3$. A line segment is drawn from $J$ through $I_1$ and intersects $C_2$ at point $K$ and $\overline{JK} = 6$. $\angle JI_2I_1 = \angle I_2KI_1 = \frac12 \angle I_1I_2K$. Let $\overline{I_1I_2} = a$, and let $a$ equal the fraction$ \frac{m\sqrt{p}}{n}$ , where $m$ and $n$ are coprime and $p$ is a positive integer not divisible by the square of any prime. Find $100m + 10p + n$.

1967 Putnam, B1

Tags: geometry , euclidean geometry , hexagon

Let $ABCDEF$ be a hexagon inscribed in a circle of radius $r.$ Show that if $AB=CD=EF=r,$ then the midpoints of $BC, DE$ and $FA$ are the vertices of an equilateral triangle.

2019 AMC 12/AHSME, 3

Tags: rotation , geometry , geometric transformation

Which one of the following rigid transformations (isometries) maps the line segment $\overline{AB}$ onto the line segment $\overline{A'B'}$ so that the image of $A(-2,1)$ is $A'(2,-1)$ and the image of $B(-1,4)$ is $B'(1,-4)?$ $\textbf{(A) } $ reflection in the $y$-axis $\textbf{(B) } $ counterclockwise rotation around the origin by $90^{\circ}$ $\textbf{(C) } $ translation by 3 units to the right and 5 units down $\textbf{(D) } $ reflection in the $x$-axis $\textbf{(E) } $ clockwise rotation about the origin by $180^{\circ}$

MMPC Part II 1958 - 95, 1958

Tags: number theory , geometry , 3d geometry , combinatorics , algebra

[b]p1.[/b] Show that $9x + 5y$ is a multiple of$ 17$ whenever $2x + 3y$ is a multiple of $17$. [b]p2.[/b] Express the five distinct fifth roots of $1$ in terms of radicals. [b]p3.[/b] Prove that the three perpendiculars dropped to the three sides of an equilateral triangle from any point inside the triangle have a constant sum. [b]p4.[/b] Find the volume of a sphere which circumscribes a regular tetrahedron of edge $a$. [b]p5.[/b] For any integer $n$ greater than $1$, show that $n^2-2n + 1$ is a factor at $n^{n-1}-1$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

1981 Poland - Second Round, 2

Tags: angle bisector , circles , geometry

Two circles touch internally at point $P$. A line tangent to one of the circles at point $A$ intersects the other circle at points $B$ and $C$. Prove that the line $ PA $ is the bisector of the angle $ BPC $.

2024 Princeton University Math Competition, B2

Tags: geometry

Let $ABCDE$ be a fixed regular pentagon of side length $1.$ An equilateral triangle $\triangle PQR$ with side length $1$ is initially placed with $P$ at $A, Q$ at $B,$ and $R$ outside the pentagon. The triangle then rolls without slipping around the perimeter of pentagon until it comes back to its starting point. The total area covered by any part of the triangle during its journey can be written as $\tfrac{a\sqrt{b}+c\pi}{d}$ for positive integers $a,b,c,d$ with $b$ square-free and $\gcd(a,c,d)=1.$ Find $a+b+c+d.$

Geometry Mathley 2011-12, 2.2

Tags: concurrency , circumcircle , geometry , circles

Let $ABC$ be a scalene triangle. A circle $(O)$ passes through $B,C$, intersecting the line segments $BA,CA$ at $F,E$ respectively. The circumcircle of triangle $ABE$ meets the line $CF$ at two points $M,N$ such that $M$ is between $C$ and $F$. The circumcircle of triangle $ACF$ meets the line $BE$ at two points $P,Q$ such that $P$ is betweeen $B$ and $E$. The line through $N$ perpendicular to $AN$ meets $BE$ at $R$, the line through $Q$ perpendicular to $AQ$ meets $CF$ at $S$. Let $U$ be the intersection of $SP$ and $NR, V$ be the intersection of $RM$ and $QS$. Prove that three lines $NQ,UV$ and $RS$ are concurrent. Trần Quang Hùng

1988 All Soviet Union Mathematical Olympiad, 481

Tags: minimum , cube , geometry , 3d geometry , broken line

A polygonal line connects two opposite vertices of a cube with side $2$. Each segment of the line has length $3$ and each vertex lies on the faces (or edges) of the cube. What is the smallest number of segments the line can have?

2012 Grigore Moisil Intercounty, 4

Tags: geometry , vector

Let $ \Delta ABC$ be a triangle with $M$ the middle of the side $[BC]$. On the line $BC$, to the left and to the right of the point $M,$ at the same distance from $M,$ let us consider $d_1$ and $d_2,$ which are perpendicular to the line BC. The perpendicular line from $M$ to $AB$ intersects $d_1$ in $P,$ and the perpendicular line from $M$ to $AC$ intersects $d_2$ in $Q.$ Prove that \[AM\perp PQ.\] [b]Author: Marin Bancoș Regional Mathematical Contest GRIGORE MOISIL, Romania, Baia Mare, 2012, 9th grade[/b]

2016 India IMO Training Camp, 1

Tags: geometry , triangle

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

1949-56 Chisinau City MO, 43

Tags: geometry , arc

On the radius $OA$ of a certain circle, as on the diameter, a circle is constructed. A ray is drawn from the center $O$, intersecting the larger and smaller circles at points $B$ and $C$, respectively. Show that the lengths of arcs $AB$ and $AC$ are equal.

2008 Hungary-Israel Binational, 3

Tags: reflection , geometry , angle bisector , conic , hyperbola , geometric transformation

P and Q are 2 points in the area bounded by 2 rays, e and f, coming out from a point O. Describe how to construct, with a ruler and a compass only, an isosceles triangle ABC, such that his base AB is on the ray e, the point C is on the ray f, P is on AC, and Q on BC.

2003 Estonia National Olympiad, 3

Tags: geometry , rectangle , right triangle , isosceles

In the rectangle $ABCD$ with $|AB|<2 |AD|$, let $E$ be the midpoint of $AB$ and $F$ a point on the chord $CE$ such that $\angle CFD = 90^o$. Prove that $FAD$ is an isosceles triangle.

2013 Iran MO (3rd Round), 4

Tags: geometry , rhombus , combinatorics

We have constructed a rhombus by attaching two equal equilateral triangles. By putting $n-1$ points on all 3 sides of each triangle we have divided the sides to $n$ equal segments. By drawing line segements between correspounding points on each side of the triangles we have divided the rhombus into $2n^2$ equal triangles. We write the numbers $1,2,\dots,2n^2$ on these triangles in a way no number appears twice. On the common segment of each two triangles we write the positive difference of the numbers written on those triangles. Find the maximum sum of all numbers written on the segments. (25 points) [i]Proposed by Amirali Moinfar[/i]

2010 Argentina Team Selection Test, 2

Tags: trig identities , trigonometry , geometry , symmetry , law of sines

Let $ABC$ be a triangle with $AB = AC$. The incircle touches $BC$, $AC$ and $AB$ at $D$, $E$ and $F$ respectively. Let $P$ be a point on the arc $\overarc{EF}$ that does not contain $D$. Let $Q$ be the second point of intersection of $BP$ and the incircle of $ABC$. The lines $EP$ and $EQ$ meet the line $BC$ at $M$ and $N$, respectively. Prove that the four points $P, F, B, M$ lie on a circle and $\frac{EM}{EN} = \frac{BF}{BP}$.

1997 Croatia National Olympiad, Problem 4

Tags: combinatorial geometry , geometry , rectangle , combinatorics

An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\times11$ rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.)

2011 Today's Calculation Of Integral, 737

Tags: geometry , integration , inequalities , calculus , logarithm , calculus computations

Let $a,\ b$ real numbers such that $a>1,\ b>1.$ Prove the following inequality. \[\int_{-1}^1 \left(\frac{1+b^{|x|}}{1+a^{x}}+\frac{1+a^{|x|}}{1+b^{x}}\right)\ dx<a+b+2\]

2014 Portugal MO, 5

Tags: geometry

Let $[ABCD]$ be a convex quadrilateral with area $2014$, and let $P$ be a point on $[AB]$ and $Q$ a point on $[AD]$ such that triangles $[ABQ]$ and $[ADP]$ have area $1$. Let $R$ be the intersection of $[AC]$ and $[PQ]$. Determine $\frac{\overline{RC}}{\overline{RA}}$.

1995 Tournament Of Towns, (459) 4

Tags: geometry , combinatorial geometry , lattice points , combinatorics

Some points with integer coordinates in the plane are marked. It is known that no four of them lie on a circle. Show that there exists a circle of radius 1995 without any marked points inside. (AV Shapovelov)

1966 AMC 12/AHSME, 6

Tags: trig identities , law of sines , trigonometry , geometry , law of cosines , pythagorean theorem

$AB$ is the diameter of a circle centered at $O$. $C$ is a point on the circle such that angle $BOC$ is $60^\circ$. If the diameter of the circle is $5$ inches, the length of chord $AC$, expressed in inches, is: $\text{(A)} \ 3 \qquad \text{(B)} \ \frac{5\sqrt{2}}{2} \qquad \text{(C)} \frac{5\sqrt3}{2} \ \qquad \text{(D)} \ 3\sqrt3 \qquad \text{(E)} \ \text{none of these}$

1981 Romania Team Selection Tests, 3.

Tags: geometry , 3d geometry , sphere

Consider three fixed spheres $S_1, S_2, S_3$ with pairwise disjoint interiors. Determine the locus of the centre of the sphere intersecting each $S_i$ along a great circle of $S_i$. [i]Stere Ianuș[/i]

2020 CCA Math Bonanza, T8

Tags: geometry

Call an [i]ordered[/i] triple $(a,b,c)$ [i]$d$-tall[/i] if there exists a triangle with side lengths $a,b,c$ and the height to the side with length $a$ is $d$. Suppose that for some positive integer $k$, there are exactly $210$ $k$-tall ordered triples of positive integers. How many $k$-tall ordered triples $(a,b,c)$ are there such that a triangle $ABC$ with $BC=a,CA=b,AB=c$ satisfies both $\angle{B}<90^\circ$ and $\angle{C}<90^\circ$? [i]2020 CCA Math Bonanza Team Round #8[/i]

May Olympiad L1 - geometry, 2001.2

Tags: geometry , rectangle , trapezoid

Let's take a $ABCD$ rectangle of paper; the side $AB$ measures $5$ cm and the side $BC$ measures $9$ cm. We do three folds: 1.We take the $AB$ side on the $BC$ side and call $P$ the point on the $BC$ side that coincides with $A$. A right trapezoid $BCDQ$ is then formed. 2. We fold so that $B$ and $Q$ coincide. A $5$-sided polygon $RPCDQ$ is formed. 3. We fold again by matching $D$ with $C$ and $Q$ with $P$. A new right trapezoid $RPCS$. After making these folds, we make a cut perpendicular to $SC$ by its midpoint $T$, obtaining the right trapezoid $RUTS$. Calculate the area of the figure that appears as we unfold the last trapezoid $RUTS$.

2011 Tuymaada Olympiad, 2

Tags: geometric transformation , geometry , reflection

Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$, and $M$ is the midpoint of $AB$. Points $S_1$ and $S_2$ lie on the line $AB$ (but not between $A$ and $B$). The tangents drawn from $S_1$ to $\omega_1$ touch it at $X_1$ and $Y_1$, and the tangents drawn from $S_2$ to $\omega_2$ touch it at $X_2$ and $Y_2$. Prove that if the line $X_1X_2$ passes through $M$, then line $Y_1Y_2$ also passes through $M$.

1999 IMO Shortlist, 2

Tags: combinatorial geometry , circles , geometry , point set

A circle is called a [b]separator[/b] for a set of five points in a plane if it passes through three of these points, it contains a fourth point inside and the fifth point is outside the circle. Prove that every set of five points such that no three are collinear and no four are concyclic has exactly four separators.