AMC 10 Diagnostic

Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays \(40\) coins in toll to Rabbit after each crossing. The payment is made after the doubling, Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning?

Ximena lists the whole numbers \(1\) through \(30\) once. Emilio copies Ximena's numbers, replacing each occurrence of the digit \(2\) by the digit \(1\). Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena's sum than Emilio's?

Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths \(3\) and \(4\) units. In the corner where those sides meet at a right angle, he leaves a small unplanted square \(S\) so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from \(S\) to the hypotenuse is \(2\) units. What fraction of the field is planted?

The weight of \(\dfrac{1}{3}\) of a large pizza together with \(3 \dfrac{1}{2}\) cups of orange slices is the same weight of \(\dfrac{3}{4}\) of a large pizza together with \(\dfrac{1}{2}\) cups of orange slices. A cup of orange slices weigh \(\dfrac{1}{4}\) of a pound. What is the weight, in pounds, of a large pizza?

A rhombic dodecahedron is a solid with \(12\) congruent rhombus faces. At every vertex, \(3\) or \(4\) edges meet, depending on the vertex. How many vertices have exactly \(3\) edges meet?

An arithmetic sequence of positive integers has \(n \geq 3\) terms, initial term \(a\), and common difference \(d > 1\). Carl wrote down all the terms in this sequence correctly except for one term, which was off by \(1\). The sum of the terms he wrote down was \(222\). What is \(a + d + n\)?

For every dollar Ben spent on bagels, David spent \(25\) cents less. Ben paid \$12.50 more than David. How much did they spend in the bagel store together?

When a student multiplied the number \(66\) by the repeating decimal, \( \underline{1} . \underline{a} \underline{b} \underline{a} \underline{b} \cdots = \underline{1} . \overline{\underline{a} \underline{b}} , \) where \(a\) and \(b\) are digits, he did not notice the notation and just multiplied \(66\) times \(\underline{1} . \underline{a} \underline{b} .\) Later he found that his answer is \(0.5\) less than the correct answer. What is the \(2\)-digit number \(\underline{a} \underline{b} ?\)

Which of the following conditions is sufficient to guarantee that integers \(x\), \(y\), and \(z\) satisfy the equation \( x ( x - y ) + y ( y - z ) + z ( z - x ) = 1 ? \) and \(y = z \) bold("(B)") x = y - 1\( and \)y = z - 1 \( \mathbf{\text{(C)}} x = z + 1\) and \(y = x + 1 \) bold("(D)") x = z\( and \)y - 1 = x \( \mathbf{\text{(E)}} x + y + z = 1\)

Janet rolls a standard \(6\)-sided die \(4\) times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal \(3 ?\)