Olympiad problems

1942 Putnam, A5

Tags: torus , ratio

A circle of radius $a$ is revolved through $180^{\circ}$ about a line in its plane, distant $b$ from the center of the circle, where $b>a$. For what value of the ratio $\frac{b}{a}$ does the center of gravity of the solid thus generated lie on the surface of the solid?

1962 AMC 12/AHSME, 14

Tags: ratio , geometric series

Let $ s$ be the limiting sum of the geometric series $ 4\minus{} \frac83 \plus{} \frac{16}{9} \minus{} \dots$, as the number of terms increases without bound. Then $ s$ equals: $ \textbf{(A)}\ \text{a number between 0 and 1} \qquad \textbf{(B)}\ 2.4 \qquad \textbf{(C)}\ 2.5 \qquad \textbf{(D)}\ 3.6 \qquad \textbf{(E)}\ 12$

2011 AMC 12/AHSME, 9

Tags: calculus , probability , tetrahedron , geometry , 3d geometry , limit , ratio

Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the product of those numbers is greater than zero? $ \textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3} $

1990 Tournament Of Towns, (248) 2

Tags: ratio , square , equal area , geometry

If a square is intersected by another square equal to it but rotated by $45^o$ around its centre, each side is divided into three parts in a certain ratio $a : b : a$ (which one can compute). Make the following construction for an arbitrary convex quadrilateral: divide each of its sides into three parts in this same ratio $a : b : a$, and draw a line through the two division points neighbouring each vertex. Prove that the new quadrilateral bounded by the four drawn lines has the same area as the original one. (A. Savin, Moscow)

2020 Tuymaada Olympiad, 4

Tags: ratio , geometry , perimeter

Points $D$ and $E$ lie on the lines $BC$ and $AC$ respectively so that $B$ is between $C$ and $D$, $C$ is between $A$ and $E$, $BC = BD$ and $\angle BAD = \angle CDE$. It is known that the ratio of the perimeters of the triangles $ABC$ and $ADE$ is $2$. Find the ratio of the areas of these triangles.

2008 China Team Selection Test, 1

Tags: ratio , trigonometry , circumcircle , projective geometry , geometry

Let $ ABC$ be a triangle, let $ AB > AC$. Its incircle touches side $ BC$ at point $ E$. Point $ D$ is the second intersection of the incircle with segment $ AE$ (different from $ E$). Point $ F$ (different from $ E$) is taken on segment $ AE$ such that $ CE \equal{} CF$. The ray $ CF$ meets $ BD$ at point $ G$. Show that $ CF \equal{} FG$.

2009 HMNT, 5

Tags: ratio

A polyhedron has faces that all either triangles or squares. No two square faces share an edge, and no two triangular faces share an edge. What is the ratio of the number of triangular faces to the number of square faces?

1966 AMC 12/AHSME, 2

Tags: ratio , percent , geometry

When the base of a triangle is increased $10\%$ and the altitude to this base is decreased $10\%$, the change in area is $\text{(A)} \ 1\%~ \text{increase} \qquad \text{(B)} \ \frac12 \%~ \text{increase} \qquad \text{(C)} \ 0\% \qquad \text{(D)} \ \frac12 \% ~\text{decrease} \qquad \text{(E)} \ 1\% ~\text{decrease}$

2012 BMT Spring, 3

Tags: geometry , ratio , square , area

Let $ABC$ be a triangle with side lengths $AB = 2011$, $BC = 2012$, $AC = 2013$. Create squares $S_1 =ABB'A''$, $S_2 = ACC''A'$ , and $S_3 = CBB''C'$ using the sides $AB$, $AC$, $BC$ respectively, so that the side $B'A''$ is on the opposite side of $AB$ from $C$, and so forth. Let square $S_4$ have side length $A''A' $, square $S_5$ have side length $C''C'$, and square $S_6$ have side length $B''B'$. Let $A(S_i)$ be the area of square $S_i$ . Compute $\frac{A(S_4)+A(S_5)+A(S_6)}{A(S_1)+A(S_2)+A(S_3)}$?

1991 Tournament Of Towns, (313) 3

Tags: geometry , obtuse , equal ratio , angle , ratio

Point $D$ lies on side $AB$ of triangle $ABC$, and $$\frac{AD}{DC} = \frac{AB}{BC}.$$ Prove that angle $C$ is obtuse. (Sergey Berlov)

2004 Romania Team Selection Test, 14

Tags: concurrency , circumcircle , geometry , ratio

Let $O$ be a point in the plane of the triangle $ABC$. A circle $\mathcal{C}$ which passes through $O$ intersects the second time the lines $OA,OB,OC$ in $P,Q,R$ respectively. The circle $\mathcal{C}$ also intersects for the second time the circumcircles of the triangles $BOC$, $COA$ and $AOB$ respectively in $K,L,M$. Prove that the lines $PK,QL$ and $RM$ are concurrent.

2009 Thailand Mathematical Olympiad, 4

Tags: ratio , equal segments , geometry , area

In triangle $\vartriangle ABC$, $D$ is the midpoint of $BC$. Points $E$ and $F$ are chosen on side $AC$ so that $AF = F E = EC$. Let $AD$ intersect $BE$ and $BF$ and $G$ and $H$, respectively. Find the ratio of the areas of $\vartriangle BGH$ and $\vartriangle ABC$.

2010 AIME Problems, 12

Tags: geometry , perimeter , ratio

Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $ 8: 7$. Find the minimum possible value of their common perimeter.

1992 China Team Selection Test, 1

Tags: ratio , rotation , trigonometry , geometry

A triangle $ABC$ is given in the plane with $AB = \sqrt{7},$ $BC = \sqrt{13}$ and $CA = \sqrt{19},$ circles are drawn with centers at $A,B$ and $C$ and radii $\frac{1}{3},$ $\frac{2}{3}$ and $1,$ respectively. Prove that there are points $A',B',C'$ on these three circles respectively such that triangle $ABC$ is congruent to triangle $A'B'C'.$

1999 Balkan MO, 1

Tags: geometric transformation , homothety , ratio , circumcircle , vector , trapezoid , geometry

Let $O$ be the circumcenter of the triangle $ABC$. The segment $XY$ is the diameter of the circumcircle perpendicular to $BC$ and it meets $BC$ at $M$. The point $X$ is closer to $M$ than $Y$ and $Z$ is the point on $MY$ such that $MZ = MX$. The point $W$ is the midpoint of $AZ$. a) Show that $W$ lies on the circle through the midpoints of the sides of $ABC$; b) Show that $MW$ is perpendicular to $AY$.

2022 AMC 10, 15

Tags: arithmetic sequence , ratio

Let $S_n$ be the sum of the first $n$ term of an arithmetic sequence that has a common difference of $2$. The quotient $\frac{S_{3n}}{S_n}$ does not depend on $n$. What is $S_{20}$? $\textbf{(A) } 340 \qquad \textbf{(B) } 360 \qquad \textbf{(C) } 380 \qquad \textbf{(D) } 400 \qquad \textbf{(E) } 420$

2010 Romania Team Selection Test, 2

Tags: ratio , geometry

Let $\ell$ be a line, and let $\gamma$ and $\gamma'$ be two circles. The line $\ell$ meets $\gamma$ at points $A$ and $B$, and $\gamma'$ at points $A'$ and $B'$. The tangents to $\gamma$ at $A$ and $B$ meet at point $C$, and the tangents to $\gamma'$ at $A'$ and $B'$ meet at point $C'$. The lines $\ell$ and $CC'$ meet at point $P$. Let $\lambda$ be a variable line through $P$ and let $X$ be one of the points where $\lambda$ meets $\gamma$, and $X'$ be one of the points where $\lambda$ meets $\gamma'$. Prove that the point of intersection of the lines $CX$ and $C'X'$ lies on a fixed circle. [i]Gazeta Matematica[/i]

2003 Baltic Way, 12

Tags: ratio , geometry , perpendicular bisector , circumcircle

Points $M$ and $N$ are taken on the sides $BC$ and $CD$ respectively of a square $ABCD$ so that $\angle MAN=45^{\circ}$. Prove that the circumcentre of $\triangle AMN$ lies on $AC$.

1990 AIME Problems, 6

Tags: ratio , percent

A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that $25\%$ of these fish are no longer in the lake on September 1 (because of death and emigrations), that $40\%$ of the fish were not in the lake May 1 (because of births and immigrations), and that the number of untagged fish and tagged fish in the September 1 sample are representative of the total population. What does the biologist calculate for the number of fish in the lake on May 1?

1977 Polish MO Finals, 2

Tags: trigonometry , ratio , geometric sequence , geometry

Let $s \geq 3$ be a given integer. A sequence $K_n$ of circles and a sequence $W_n$ of convex $s$-gons satisfy: \[ K_n \supset W_n \supset K_{n+1} \] for all $n = 1, 2, ...$ Prove that the sequence of the radii of the circles $K_n$ converges to zero.

2004 AMC 8, 22

Tags: probability , ratio

At a party there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\frac{2}{5}$. What fraction of the people in the room are married men? $\textbf{(A)}\ \frac{1}{3}\qquad \textbf{(B)}\ \frac{3}{8}\qquad \textbf{(C)}\ \frac{2}{5}\qquad \textbf{(D)}\ \frac{5}{12}\qquad \textbf{(E)}\ \frac{3}{5}$