Olympiad problems

2013 Romania National Olympiad, 3

Given $P$ a point m inside a triangle acute-angled $ABC$ and $DEF$ intersections of lines with that $AP$, $BP$, $CP$ with$\left[ BC \right],\left[ CA \right],$respective $\left[ AB \right]$ a) Show that the area of the triangle $DEF$ is at most a quarter of the area of the triangle $ABC$ b) Show that the radius of the circle inscribed in the triangle $DEF$ is at most a quarter of the radius of the circle circumscribed on triangle $4ABC.$

1992 Poland - Second Round, 1

Tags: geometry , analytic geometry , lattice , combinatorial geometry

Every vertex of a polygon has both integer coordinates; the length of each side of this polygon is a natural number. Prove that the perimeter of the polygon is an even number.

2005 Taiwan TST Round 2, 2

Tags: trigonometry , analytic geometry , geometry

In $\triangle ABC$, $AD$ is the bisector of $\angle A$, and $E$, $F$ are the feet of the perpendiculars from $D$ to $AC$ and $AB$, respectively. $H$ is the intersection of $BE$ and $CF$, and $G$, $I$ are the feet of the perpendiculars from $D$ to $BE$ and $CF$, respectively. Prove that both $AFEH$ and $AEIH$ are cyclic quadrilaterals.

2000 CentroAmerican, 3

Tags: geometry , parallelogram , trigonometry , analytic geometry , graphing lines , slope , complex number

Let $ ABCDE$ be a convex pentagon. If $ P$, $ Q$, $ R$ and $ S$ are the respective centroids of the triangles $ ABE$, $ BCE$, $ CDE$ and $ DAE$, show that $ PQRS$ is a parallelogram and its area is $ 2/9$ of that of $ ABCD$.

2005 AMC 12/AHSME, 24

Tags: analytic geometry , parabola , graphing lines , slope , trigonometry , number theory , conic

All three vertices of an equilateral triangle are on the parabola $ y \equal{} x^2$, and one of its sides has a slope of 2. The x-coordinates of the three vertices have a sum of $ m/n$, where $ m$ and $ n$ are relatively prime positive integers. What is the value of $ m \plus{} n$? $ \textbf{(A)}\ 14\qquad \textbf{(B)}\ 15\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 17\qquad \textbf{(E)}\ 18$

2000 USAMO, 5

Tags: geometry , circumcircle , analytic geometry

Let $A_1A_2A_3$ be a triangle and let $\omega_1$ be a circle in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\omega_2, \omega_3, \dots, \omega_7$ such that for $k = 2, 3, \dots, 7,$ $\omega_k$ is externally tangent to $\omega_{k-1}$ and passes through $A_k$ and $A_{k+1},$ where $A_{n+3} = A_{n}$ for all $n \ge 1$. Prove that $\omega_7 = \omega_1.$

2007 Princeton University Math Competition, 8

Tags: geometry , analytic geometry , quadratic , calculus , integration , quadratic formula , algebra

For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2+4px-p+1$ an integer?

2004 Austrian-Polish Competition, 5

Tags: induction , analytic geometry , algebra

Determine all $n$ for which the system with of equations can be solved in $\mathbb{R}$: \[\sum^{n}_{k=1} x_k = 27\] and \[\prod^{n}_{k=1} x_k = \left( \frac{3}{2} \right)^{24}.\]

2009 IberoAmerican Olympiad For University Students, 2

Tags: linear algebra , vector , invariant , matrix , analytic geometry , induction

Let $x_1,\cdots, x_n$ be nonzero vectors of a vector space $V$ and $\varphi:V\to V$ be a linear transformation such that $\varphi x_1 = x_1$, $\varphi x_k = x_k - x_{k-1}$ for $k = 2, 3,\ldots,n$. Prove that the vectors $x_1,\ldots,x_n$ are linearly independent.

2014 Tuymaada Olympiad, 6

Tags: geometric transformation , analytic geometry , rectangle , combinatorics , geometry

Each of $n$ black squares and $n$ white squares can be obtained by a translation from each other. Every two squares of different colours have a common point. Prove that ther is a point belonging at least to $n$ squares. [i](V. Dolnikov)[/i]

2023-24 IOQM India, 23

Tags: analytic geometry , geometry , incenter

In the coordinate plane, a point is called a $\text{lattice point}$ if both of its coordinates are integers. Let $A$ be the point $(12,84)$. Find the number of right angled triangles $ABC$ in the coordinate plane $B$ and $C$ are lattice points, having a right angle at vertex $A$ and whose incenter is at the origin $(0,0)$.

1991 Iran MO (2nd round), 1

Tags: analytic geometry , algebra

Prove that there exist at least six points with rational coordinates on the curve of the equation \[y^3=x^3+x+1370^{1370}\]

2011 ELMO Shortlist, 1

Tags: analytic geometry , geometry

Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$, $BC$, $CD$, $DA$, respectively, and let $P$ be the intersection of $EG$ and $FH$. Given that quadrilaterals $HAEP$, $EBFP$, $FCGP$, $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle. [i]Evan O'Dorney.[/i]

2021 Princeton University Math Competition, 5

Tags: analytic geometry , geometry

Given a real number $t$ with $0 < t < 1$, define the real-valued function $f(t, \theta) = \sum^{\infty}_{n=-\infty} t^{|n|}\omega^n$, where $\omega = e^{i \theta} = \cos \theta + i\sin \theta$. For $\theta \in [0, 2\pi)$, the polar curve $r(\theta) = f(t, \theta)$ traces out an ellipse $E_t$ with a horizontal major axis whose left focus is at the origin. Let $A(t)$ be the area of the ellipse $E_t$. Let $A\left( \frac12 \right) = \frac{a\pi}{b}$ , where $a, b$ are relatively prime positive integers. Find $100a +b$ .

2015 AMC 10, 13

Tags: analytic geometry

The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle? $\textbf{(A) } 20 \qquad\textbf{(B) } \dfrac{360}{17} \qquad\textbf{(C) } \dfrac{107}{5} \qquad\textbf{(D) } \dfrac{43}{2} \qquad\textbf{(E) } \dfrac{281}{13} $

1967 IMO Longlists, 22

Tags: analytic geometry , geometry , geometric inequality , circles

Let $k_1$ and $k_2$ be two circles with centers $O_1$ and $O_2$ and equal radius $r$ such that $O_1O_2 = r$. Let $A$ and $B$ be two points lying on the circle $k_1$ and being symmetric to each other with respect to the line $O_1O_2$. Let $P$ be an arbitrary point on $k_2$. Prove that \[PA^2 + PB^2 \geq 2r^2.\]

2015 AIME Problems, 14

Tags: geometry , analytic geometry , inequalities

For each integer $n \ge 2$, let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1\le x \le n$ and $0\le y \le x \left\lfloor \sqrt x \right\rfloor$, where $\left\lfloor \sqrt x \right\rfloor$ is the greatest integer not exceeding $\sqrt x$. Find the number of values of $n$ with $2\le n \le 1000$ for which $A(n)$ is an integer.

2018 Harvard-MIT Mathematics Tournament, 10

Tags: geometry , rectangle , analytic geometry

Let $n$ and $m$ be positive integers in the range $[1, 10^{10}]$. Let $R$ be the rectangle with corners at $(0, 0), (n, 0), (n, m), (0, m)$ in the coordinate plane. A simple non-self-intersecting quadrilateral with vertices at integer coordinates is called [i]far-reaching[/i] if each of its vertices lie on or inside $R$, but each side of $R$ contains at least one vertex of the quadrilateral. Show that there is a far-reaching quadrilateral with area at most $10^6$.

2018 Kürschák Competition, 2

Tags: vector , analytic geometry , number theory , prime number

Given a prime number $p$ and let $\overline{v_1},\overline{v_2},\dotsc ,\overline{v_n}$ be $n$ distinct vectors of length $p$ with integer coordinates in an $\mathbb{R}^3$ Cartesian coordinate system. Suppose that for any $1\leqslant j<k\leqslant n$, there exists an integer $0<\ell <p$ such that all three coordinates of $\overline{v_j} -\ell \cdot \overline{v_k} $ is divisible by $p$. Prove that $n\leqslant 6$.

2014 Vietnam Team Selection Test, 2

Tags: analytic geometry , combinatorics

In the Cartesian plane is given a set of points with integer coordinate \[ T=\{ (x;y)\mid x,y\in\mathbb{Z} ; \ |x|,|y|\leq 20 ; \ (x;y)\ne (0;0)\} \] We colour some points of $ T $ such that for each point $ (x;y)\in T $ then either $ (x;y) $ or $ (-x;-y) $ is coloured. Denote $ N $ to be the number of couples $ {(x_1;y_1),(x_2;y_2)} $ such that both $ (x_1;y_1) $ and $ (x_2;y_2) $ are coloured and $ x_1\equiv 2x_2 \pmod {41}, y_1\equiv 2y_2 \pmod {41} $. Find the all possible values of $ N $.

2001 AIME Problems, 7

Tags: inradius , incenter , analytic geometry , similar triangles , geometry

Let $\triangle{PQR}$ be a right triangle with $PQ=90$, $PR=120$, and $QR=150$. Let $C_{1}$ be the inscribed circle. Construct $\overline{ST}$ with $S$ on $\overline{PR}$ and $T$ on $\overline{QR}$, such that $\overline{ST}$ is perpendicular to $\overline{PR}$ and tangent to $C_{1}$. Construct $\overline{UV}$ with $U$ on $\overline{PQ}$ and $V$ on $\overline{QR}$ such that $\overline{UV}$ is perpendicular to $\overline{PQ}$ and tangent to $C_{1}$. Let $C_{2}$ be the inscribed circle of $\triangle{RST}$ and $C_{3}$ the inscribed circle of $\triangle{QUV}$. The distance between the centers of $C_{2}$ and $C_{3}$ can be written as $\sqrt{10n}$. What is $n$?

2013 Tuymaada Olympiad, 8

Tags: inequalities , perimeter , analytic geometry , trigonometry , geometry

The point $A_1$ on the perimeter of a convex quadrilateral $ABCD$ is such that the line $AA_1$ divides the quadrilateral into two parts of equal area. The points $B_1$, $C_1$, $D_1$ are defined similarly. Prove that the area of the quadrilateral $A_1B_1C_1D_1$ is greater than a quarter of the area of $ABCD$. [i]L. Emelyanov [/i]

2020 Saint Petersburg Mathematical Olympiad, 6.

Tags: combinatorics , invariant , analytic geometry , number theory , algebra

The points $(1,1),(2,3),(4,5)$ and $(999,111)$ are marked in the coordinate system. We continue to mark points in the following way : [list] [*]If points $(a,b)$ are marked then $(b,a)$ and $(a-b,a+b)$ can be marked [*]If points $(a,b)$ and $(c,d)$ are marked then so can be $(ad+bc, 4ac-4bd)$. [/list] Can we, after some finite number of these steps, mark a point belonging to the line $y=2x$.

1998 All-Russian Olympiad, 1

Tags: analytic geometry , vieta , algebra

Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.

2003 Tournament Of Towns, 5

Tags: geometry , rectangle , analytic geometry , geometric transformation , reflection , complex number , combinatorics

Is it possible to tile $2003 \times 2003$ board by $1 \times 2$ dominoes placed horizontally and $1 \times 3$ rectangles placed vertically?