Olympiad problems

2014 Math Prize For Girls Problems, 17

Tags: algebra , quadratic , quadratic formula , area of a triangle , geometry , analytic geometry , ratio

Let $ABC$ be a triangle. Points $D$, $E$, and $F$ are respectively on the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ of $\triangle ABC$. Suppose that \[ \frac{AE}{AC} = \frac{CD}{CB} = \frac{BF}{BA} = x \] for some $x$ with $\frac{1}{2} < x < 1$. Segments $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ cut the triangle into 7 nonoverlapping regions: 4 triangles and 3 quadrilaterals. The total area of the 4 triangles equals the total area of the 3 quadrilaterals. Compute the value of $x$.

Estonia Open Senior - geometry, 1993.5

Tags: equilateral , projection , ratio , geometry

Within an equilateral triangle $ABC$, take any point $P$. Let $L, M, N$ be the projections of $P$ on sides $AB, BC, CA$ respectively. Prove that $\frac{AP}{NL}=\frac{BP}{LM}=\frac{CP}{MN}$.

2012 Sharygin Geometry Olympiad, 4

Tags: ratio , congruent triangles , geometry , perpendicular bisector , analytic geometry , parallelogram

Given triangle $ABC$. Point $M$ is the midpoint of side $BC$, and point $P$ is the projection of $B$ to the perpendicular bisector of segment $AC$. Line $PM$ meets $AB$ in point $Q$. Prove that triangle $QPB$ is isosceles.

2010 AMC 12/AHSME, 18

Tags: geometry , symmetry , integration , parallelogram , ratio , probability , calculus

A frog makes $ 3$ jumps, each exactly $ 1$ meter long. The directions of the jumps are chosen independently and at random. What is the probability the frog's final position is no more than $ 1$ meter from its starting position? $ \textbf{(A)}\ \frac {1}{6} \qquad \textbf{(B)}\ \frac {1}{5} \qquad \textbf{(C)}\ \frac {1}{4} \qquad \textbf{(D)}\ \frac {1}{3} \qquad \textbf{(E)}\ \frac {1}{2}$

2005 Junior Balkan Team Selection Tests - Romania, 11

Tags: homothety , ratio , geometry , reflection , geometric transformation , circumcircle

Three circles $\mathcal C_1(O_1)$, $\mathcal C_2(O_2)$ and $\mathcal C_3(O_3)$ share a common point and meet again pairwise at the points $A$, $B$ and $C$. Show that if the points $A$, $B$, $C$ are collinear then the points $Q$, $O_1$, $O_2$ and $O_3$ lie on the same circle.

2013 NIMO Problems, 2

Tags: ratio , trigonometry , geometry

In $\triangle ABC$, points $E$ and $F$ lie on $\overline{AC}, \overline{AB}$, respectively. Denote by $P$ the intersection of $\overline{BE}$ and $\overline{CF}$. Compute the maximum possible area of $\triangle ABC$ if $PB = 14$, $PC = 4$, $PE = 7$, $PF = 2$. [i]Proposed by Eugene Chen[/i]

2007 Paraguay Mathematical Olympiad, 3

Tags: graphing lines , slope , analytic geometry , ratio

Let $ABCD$ be a square, $E$ and $F$ midpoints of $AB$ and $AD$ respectively, and $P$ the intersection of $CF$ and $DE$. a) Show that $DE \perp CF$. b) Determine the ratio $CF : PC : EP$

2002 Croatia National Olympiad, Problem 3

Tags: 3d geometry , prism , geometry , ratio

Points $E$ and $F$ are taken on the diagonals $AB_1$ and $CA_1$ of the lateral faces $ABB_1A_1$ and $CAA_1C_1$ of a triangular prism $ABCA_1B_1C_1$ so that $EF\parallel BC_1$. Find the ratio of the lengths of $EF$ and $BC_1$.

2002 Mongolian Mathematical Olympiad, Problem 5

Tags: geometry , ratio

Let $A$ be the ratio of the product of sides to the product of diagonals in a circumscribed pentagon. Find the maximum possible value of $A$.

2018 AMC 12/AHSME, 18

Tags: geometry , ratio

Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? $ \textbf{(A) }60 \qquad \textbf{(B) }65 \qquad \textbf{(C) }70 \qquad \textbf{(D) }75 \qquad \textbf{(E) }80 \qquad $

2005 Austrian-Polish Competition, 2

Tags: polynomial , ratio , algebra

Determine all polynomials $P$ with integer coefficients satisfying \[P(P(P(P(P(x)))))=x^{28}\cdot P(P(x))\qquad \forall x\in\mathbb{R}\]

1995 Denmark MO - Mohr Contest, 3

Tags: median , ratio , geometry

From the vertex $C$ in triangle $ABC$, draw a straight line that bisects the median from $A$. In what ratio does this line divide the segment $AB$? [img]https://1.bp.blogspot.com/-SxWIQ12DIvs/XzcJv5xoV0I/AAAAAAAAMY4/Ezfe8bd7W-Mfp2Qi4qE_gppbh9Fzvb4XwCLcBGAsYHQ/s0/1995%2BMohr%2Bp3.png[/img]

2009 AIME Problems, 1

Tags: ratio

Call a $ 3$-digit number [i]geometric[/i] if it has $ 3$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.

V Soros Olympiad 1998 - 99 (Russia), 9.6

Tags: geometry , ratio

On side $AB$ of triangle $ABC$, points $M$ and $K$ are taken ($M$ on segment $AK$). It is known that $AM: MK: MB = a: b: c$. Straight lines $CM$ and $CK$ intersect for the second time the circumscribed circle of the triangle $ABC$ at points $E$ and $F$, respectively. In what ratio does the circumscribed circle of the triangle $BMF$ divide the segment $BE$?

1997 Mexico National Olympiad, 5

Tags: ratio , geometry , area

Let $P,Q,R$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$. Suppose that $BQ$ and $CR$ meet at $A', AP$ and $CR$ meet at $B'$, and $AP$ and $BQ$ meet at $C'$, such that $AB' = B'C', BC' =C'A'$, and $CA'= A'B'$. Compute the ratio of the area of $\triangle PQR$ to the area of $\triangle ABC$.

2011 Korea Junior Math Olympiad, 5

Tags: midpoint , circumcircle , geometry , ratio

In triangle $ABC$, ($AB \ne AC$), let the orthocenter be $H$, circumcenter be $O$, and the midpoint of $BC$ be $M$. Let $HM \cap AO = D$. Let $P,Q,R,S$ be the midpoints of $AB,CD,AC,BD$. Let $X = PQ\cap RS$. Find $AH/OX$.

2005 Taiwan TST Round 1, 1

Tags: geometric sequence , geometry , trigonometry , ratio

Consider a circle $O_1$ with radius $R$ and a point $A$ outside the circle. It is known that $\angle BAC=60^\circ$, where $AB$ and $AC$ are tangent to $O_1$. We construct infinitely many circles $O_i$ $(i=1,2,\dots\>)$ such that for $i>1$, $O_i$ is tangent to $O_{i-1}$ and $O_{i+1}$, that they share the same tangent lines $AB$ and $AC$ with respect to $A$, and that none of the $O_i$ are larger than $O_1$. Find the total area of these circles. I know this problem was easy, but it still appeared in the TST, and so I posted it. It was kind of a disappointment for me.

2000 Iran MO (3rd Round), 1

Tags: trigonometry , reflection , parallelogram , ratio , circumcircle , geometric transformation , geometry

Two circles intersect at two points $A$ and $B$. A line $\ell$ which passes through the point $A$ meets the two circles again at the points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ (which do not contain the point $A$) on the respective circles. Let $K$ be the midpoint of the segment $CD$. Prove that $\measuredangle MKN = 90^{\circ}$.

2011 AMC 10, 11

Tags: ratio , geometry

Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $\overline{AB}$ with $AE=7\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$? $\textbf{(A)}\,\frac{49}{64} \qquad\textbf{(B)}\,\frac{25}{32} \qquad\textbf{(C)}\,\frac78 \qquad\textbf{(D)}\,\frac{5\sqrt{2}}{8} \qquad\textbf{(E)}\,\frac{\sqrt{14}}{4} $

2005 All-Russian Olympiad, 3

Tags: circumcircle , geometric transformation , parallelogram , trigonometry , homothety , geometry , ratio

We have an acute-angled triangle $ABC$, and $AA',BB'$ are its altitudes. A point $D$ is chosen on the arc $ACB$ of the circumcircle of $ABC$. If $P=AA'\cap BD,Q=BB'\cap AD$, show that the midpoint of $PQ$ lies on $A'B'$.

1995 AIME Problems, 15

Tags: relatively prime , ratio , number theory , probability

Let $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p$ can be written in the form $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

2012 Online Math Open Problems, 1

Tags: ratio

The average of two positive real numbers is equal to their difference. What is the ratio of the larger number to the smaller one? [i]Author: Ray Li[/i]

2010 Indonesia MO, 8

Tags: geometry , ratio , circumcircle

Given an acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$. Let $K$ be a point inside $ABC$ which is not $O$ nor $H$. Point $L$ and $M$ are located outside the triangle $ABC$ such that $AKCL$ and $AKBM$ are parallelogram. At last, let $BL$ and $CM$ intersects at $N$, and let $J$ be the midpoint of $HK$. Show that $KONJ$ is also a parallelogram. [i]Raja Oktovin, Pekanbaru[/i]

2018 AMC 10, 24

Tags: geometry , ratio

Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? $ \textbf{(A) }60 \qquad \textbf{(B) }65 \qquad \textbf{(C) }70 \qquad \textbf{(D) }75 \qquad \textbf{(E) }80 \qquad $

2006 Vietnam Team Selection Test, 1

Tags: circumcircle , parallelogram , geometry , trigonometry , ratio

Given an acute angles triangle $ABC$, and $H$ is its orthocentre. The external bisector of the angle $\angle BHC$ meets the sides $AB$ and $AC$ at the points $D$ and $E$ respectively. The internal bisector of the angle $\angle BAC$ meets the circumcircle of the triangle $ADE$ again at the point $K$. Prove that $HK$ is through the midpoint of the side $BC$.