A tripod has three legs each of length \(5\) feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is \(4\) feet from the ground. \[ \text{1. What is the area between the points where the legs meet the ground?} \] \[ \text{2. What angle does the slope make with each face?}\] In setting up the tripod, the lower 1 foot of one leg breaks off. \[ \text{1. What is the area between the points where the legs meet the ground?} \] \[ \text{2. (*) What is the new height of the tripod?} \]
Tags: geometry,3d-geometry,cosine-rule,volume,area,scale-factors
Difficulty: 3
Source: AIMEI 2006 Problem 14 - modified
EditA tripod has three legs each of length \(5\) feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is \(4\) feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let \(h\) be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then \(h\) can be written in the form \(\frac m{\sqrt{n}},\) where \(m\) and \(n\) are positive integers and \(n\) is not divisible by the square of any prime. Find \(\lfloor m+\sqrt{n}\rfloor.\) (The notation \(\lfloor x\rfloor\) denotes the greatest integer that is less than or equal to \(x.\))
Tags: geometry,3d-geometry,cosine-rule,volume,area,scale-factors
Difficulty: 4
Source: AIMEI 2006 Problem 14
EditIn \(\triangle ABC, AB = AC = 10\) and \(BC = 12\). Point \(D\) lies strictly between \(A\) and \(B\) on \(\overline{AB}\) and point \(E\) lies strictly between \(A\) and \(C\) on \(\overline{AC}\) so that \(AD = DE = EC\). Then \(AD\) can be expressed in the form \(\tfrac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p + q\).
Tags: geometry,cosine-rule
Difficulty: 4
Source: AIMEI 2018 Problem 4
Edit\(\sqrt{164}\) is $$\text{(A)}\ 42 \qquad \text{(B)}\ \text{less than }10 \qquad \text{(C)}\ \text{between }10\text{ and }11 \qquad \text{(D)}\ \text{between }11\text{ and }12 \qquad \text{(E)}\ \text{between }12\text{ and }13$$
Tags: square-root,approximation
Difficulty: 1
Source: AMC8 1988 Problem 11
EditChris's birthday is on a Thursday this year. What day of the week will it be $60$ days after her birthday? $$\text{(A)}\ \text{Monday} \qquad \text{(B)}\ \text{Wednesday} \qquad \text{(C)}\ \text{Thursday} \qquad \text{(D)}\ \text{Friday} \qquad \text{(E)}\ \text{Saturday}$$
Tags: date,modulo
Difficulty: 1
Source: AMC8 1988 Problem 10
EditBetty used a calculator to find the product \(0.075 \times 2.56\). She forgot to enter the decimal points. The calculator showed \(19200\). If Betty had entered the decimal points correctly, the answer would have been $$\text{(A)}\ .0192 \qquad \text{(B)}\ .192 \qquad \text{(C)}\ 1.92 \qquad \text{(D)}\ 19.2 \qquad \text{(E)}\ 192$$
Tags: number,place-value
Difficulty: 1
Source: AMC8 1988 Problem 8
Edit$$2.46\times 8.163\times (5.17+4.829)$$ is closest to $$\text{(A)}\ 100 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 300 \qquad \text{(D)}\ 400 \qquad \text{(E)}\ 500$$
Tags: number,approximation
Difficulty: 1
Source: AMC8 1988 Problem 7
Edit$$\frac{(.2)^3}{(.02)^2} =$$ $$\text{(A)}\ .2 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 20$$
Tags: number,fraction,indices
Difficulty: 1
Source: AMC8 1988 Problem 6
EditThe product \(8\times .25\times 2\times .125 =\) $$\text{(A)}\ \frac18 \qquad \text{(B)}\ \frac14 \qquad \text{(C)}\ \frac12 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 2$$
Tags: number,place-value,fraction
Difficulty: 1
Source: AMC8 1988 Problem 2
Edit$$\frac{1}{10}+\frac{2}{20}+\frac{3}{30} =$$ $$\text{(A)}\ .1 \qquad \text{(B)}\ .123 \qquad \text{(C)}\ .2 \qquad \text{(D)}\ .3 \qquad \text{(E)}\ .6$$
Tags: number,fraction,decimal
Difficulty: 1
Source: AMC8 1988 Problem 3
Edit