Practice (41)

2019

In triangle

$ABC$ ,

$AB = 2BC$ . Given that

$M$ is the midpoint of

$AB$ and

$\angle MCA = 60^{\circ}$ , compute

$\frac{CM}{AC}$ .

2031

In rectangle

$ABCD$ , points

$E$ and

$F$ are on sides

$AB$ and

$CD$ , respectively, such that

$AE = CF > AD$ and

$\angle CED = 90^{\circ}$ . Lines

$AF$ ,

$BF$ ,

$CE$ and

$DE$ enclose a rectangle whose area is 24% of the area of

$ABCD$ . Compute

$\frac{BF}{CE}$ .

2034

Let

$ABCDE$ be a convex pentagon such that

$\angle ABC = \angle BCD = 108^{\circ}$ ,

$\angle CDE = 168^{\circ}$ and

$AB =BC = CD = DE$ . Find the measure of

$\angle AEB$ .

2045

Let

$ABCD$ be a trapezoid such that

$AB$ is parallel to

$CD$ ,

$BC$ = 10 and

$AD$ = 18. Given that the two circles with diameters

$BC$ and

$AD$ are tangent, find the perimeter of

$ABCD$ .

2048

Let

$A$ be the answer to problem 13, and let

$C$ be the answer to problem 15. In the interior of angle

$NOM$ =

$45^{\circ}$ , there is a point

$P$ such that

$\angle MOP = A^{\circ}$ and

$OP = C$ . Let

$X$ and

$Y$ be the reflections of

$P$ over

$MO$ and

$NO$ , respectively. Find

$(XY )^2$ .

2054

Let line segment

$AB$ have length 25 and let points

$C$ and

$D$ lie on the same side of line

$AB$ such that

$AC$ = 15,

$AD$ = 24,

$BC$ = 20, and

$BD$ = 7. Given that rays

$AC$ and

$BD$ intersect at point

$E$ , compute

$EA + EB$ .

2112

Let

$AB$ be the diameter of a semicircle.

$C$ ,

$D$ , and

$E$ are points on

$AB$ , in that order, such that

$AC=1$ ,

$CD=3$ ,

$DE=4$ , and

$EB=2$ . From

$C$ and

$E$ , draw perpendicular lines of

$AB$ , intersecting the semicircle at

$F$ and

$G$ , respectively. Find the measurement of

$\angle{FDG}$ .

2139

Given triangle

$ABC$ , construct equilateral triangle

$ABC_1$ ,

$BCA_1$ ,

$CAB_1$ on the outside of

$ABC$ . Let

$P, Q$ denote the midpoints of

$C_1A_1$ and

$C_1B_1$ respectively. Let

$R$ be the midpoint of

$AB$ . Prove that triangle

$PQR$ is isosceles.

2140

(Napolean's Triangle) Given triangle

$ABC$ , construct an equilateral triangle on the outside of each of the sides. Let

$P, Q, R$ be the centroids of these equilateral triangles, prove that triangle

$PQR$ is equilateral.

2143

Let

$W_1W_2W_3$ be a triangle with circumcircle

$S$ , and let

$A_1, A_2, A_3$ be the midpoints of

$W_2W_3, W_1W_3, W_1W_2$ respectively. From Ai drop a perpendicular to the line tangent to

$S$ at

$W_i$ . Prove that these perpendicular lines are concurrent and identify this point of concurrency.

2144

Let

$A_0A_1A_2A_3A_4A_5A_6$ be a regular 7-gon. Prove that

$\frac{1}{A_0A_1} = \frac{1}{A_0A_2}+\frac{1}{A_0A_3}$

2145

Given point

$P_0$ in the plane of triangle

$A_1A_2A_3$ . Denote

$A_s = A_{s-3}$ , for

$s > 3$ . Construct points

$P_1; P_2; \cdots$ sequentially such that point

$P_{k+1}$ is

$P_k$ rotated

$120^\circ$ counter-clockwise around

$A_{k+1}$ . Prove that if

$P_{1986} = P_0$ then triangle

$A_1A_2A_3$ is isosceles.

2146

Point

$H$ is the orthocenter of triangle

$ABC$ . Points

$D, E$ and

$F$ lie on the circumcircle of triangle

$ABC$ such that

$AD\parallel BE\parallel CF$ . Points

$S, T,$ and

$U$ are the respective reflections of

$D, E, F$ across the lines

$BC, CA$ and

$AB$ . Prove that

$S, T, U, H$ are cyclic.

2147

Let

$ABCD$ be a cyclic quadrilateral. Let

$P, Q, R$ be the feet of the perpendiculars from

$D$ to the lines

$BC, CA$ and

$AB$ respectively. Show that

$PQ = QR$ iff the bisectors of

$\angle ABC$ and

$\angle ADC$ meet on

$AC$ .

2148

Let

$O$ be the circumcentre of triangle

$ABC$ . A line through

$O$ intersects sides

$AB$ and

$AC$ at

$M$ and

$N$ respectively. Let

$S$ and

$R$ be the midpoints of

$BN$ and

$CM$ , respectively. Prove that

$\angle ROS = \angle BAC$ .

2149

Let

$ABCD$ be a convex quadrilateral for which

$AC = BD$ . Equilateral triangles are constructed on the sides of the quadrilateral and pointing outward. Let

$O_1, O_2, O_3, O_4$ be the centres of the triangles constructed on

$AB, BC, CD,$ and

$DA$ respectively. Prove that lines

$O_1O_3$ and

$O_2O_4$ are perpendicular.

2150

Let

$ABC$ be a triangle. Triangles

$PAB$ and

$QAC$ are constructed outside of

$ABC$ such that

$AP = AB$ and

$AQ = AC$ and

$\angle BAP = \angle CAQ$ . Segments

$BQ$ and

$CP$ meet at

$R$ . Let

$O$ be the circumcentre of triangle

$BCR$ . Prove that

$AO \perp PQ$ .

2215

In the diagram below, the circle with center $A$ is congruent to and tangent to the circle with center $B$ . A third circle is tangent to the circle with center $A$ at point $C$ and passes through point $B$ . Points $C, A$ , and $B$ are collinear. The line segment $\overline{CDEFG}$ intersects the circles at the indicated points. Suppose that $DE=6$ and $FG=9$ . Find $AG$ .

2247

In the diagram

$ABCDEFG$ is a regular heptagon (a 7 sided polygon). Shown is the star

$AEBFCGD$ . The degree measure of the obtuse angle formed by

$AE$ and

$CG$ is

$\frac{m}{n}$ where

$m$ and

$n$ are relatively prime positive integers. Find

$m + n$ . %

2269

$45^\circ$ arc of circle A is equal in length to a

$30^\circ$ arc of circle B. What is the ratio of circle A's area and circle B's area?

2270

Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let

$B$ be the total area of the blue triangles,

$W$ the total area of the white squares, and

$R$ the area of the red square. Which of the following is correct?

2275

Given a triangle with side lengths 15, 20, and 25, find the triangle's smallest height.

2282

Points

$A,B,C,D,E$ and

$F$ lie, in that order, on

$\overline{AF}$ , dividing it into five segments, each of length 1. Point

$G$ is not on line

$AF$ . Point

$H$ lies on

$\overline{GD}$ , and point

$J$ lies on

$\overline{GF}$ . The line segments

$\overline{HC}, \overline{JE},$ and

$\overline{AG}$ are parallel. Find

$HC/JE$ .

2285

Points

$A,B,C$ and

$D$ lie on a line, in that order, with

$AB = CD$ and

$BC = 12$ . Point

$E$ is not on the line, and

$BE = CE = 10$ . The perimeter of

$\triangle AED$ is twice the perimeter of

$\triangle BEC$ . Find

$AB$ .

2337

Let

$ABC$ be acute triangle. The circle with diameter

$AB$ intersects

$CA,\, CB$ at

$M,\, N,$ respectively. Draw

$CT\perp AB$ and intersects above circle at

$T$ , where

$C$ and

$T$ lie on the same side of

$AB$ .

$S$ is a point on

$AN$ such that

$BT = BS$ . Prove that

$BS\perp SC$ .

PlaneGeometry Triangle PythagoreanTheorem Circle AreaMethod RotationMethod CoordinatedGeometry ComplexNumberApplication

AMC10/12 USAMO Exeter IMO

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