problem stringlengths 64 926 | answer stringlengths 1 3 |
|---|---|
A bag contains 20 lavender marbles, 12 emerald marbles, and some number of orange marbles. If the probability of drawing an orange marble in one try is $\frac{1}{y}$, compute the sum of all possible integer values of $y$. | 69 |
A function $f$ is defined so that
- $f(1)=1$,
- if $n$ is an even positive integer, then $f(n)=f\left(\frac{1}{2} n\right)$, and
- if $n$ is an odd positive integer with $n>1$, then $f(n)=f(n-1)+1$.
For example, $f(34)=f(17)$ and $f(17)=f(16)+1$.
Determine the value of $f(50)$. | 3 |
Compute the sum of all positive two-digit factors of $2^{32}-1$. | 168 |
The taxicab distance between points $A=\left(x_{A}, y_{A}\right)$ and $B=\left(x_{B}, y_{B}\right)$ is defined as $d(A, B)=$ $\left|x_{A}-x_{B}\right|+\left|y_{A}-y_{B}\right|$. Given some $s>0$ and points $A=\left(x_{A}, y_{A}\right)$ and $B=\left(x_{B}, y_{B}\right)$, define the taxicab ellipse with foci $A=\left(x_{... | 96 |
Rectangle $A R M L$ has length 125 and width 8. The rectangle is divided into 1000 squares of area 1 by drawing in gridlines parallel to the sides of $A R M L$. Diagonal $\overline{A M}$ passes through the interior of exactly $n$ of the 1000 unit squares. Compute $n$. | 132 |
At present, the sum of the ages of a husband and wife, $P$, is six times the sum of the ages of their children, $C$. Two years ago, the sum of the ages of the husband and wife was ten times the sum of the ages of the same children. Six years from now, it will be three times the sum of the ages of the same children. Det... | 3 |
Let $T=8$. At a party, everyone shakes hands with everyone else exactly once, except Ed, who leaves early. A grand total of $20 T$ handshakes take place. Compute the number of people at the party who shook hands with Ed. | 7 |
A computer is programmed to choose an integer between 1 and 99, inclusive, so that the probability that it selects the integer $x$ is equal to $\log _{100}\left(1+\frac{1}{x}\right)$. Suppose that the probability that $81 \leq x \leq 99$ is equal to 2 times the probability that $x=n$ for some integer $n$. What is the v... | 9 |
Compute the number of ordered quadruples of integers $(a, b, c, d)$ satisfying the following system of equations:
$$
\left\{\begin{array}{l}
a b c=12,000 \\
b c d=24,000 \\
c d a=36,000
\end{array}\right.
$$ | 12 |
Let T be a rational number. Compute $\sin ^{2} \frac{T \pi}{2}+\sin ^{2} \frac{(5-T) \pi}{2}$. | 1 |
Let $T=20$. The product of all positive divisors of $2^{T}$ can be written in the form $2^{K}$. Compute $K$. | 210 |
Let $T=\sqrt{26}$. Compute the number of ordered pairs of positive integers $(a, b)$ with the property that $a b=T^{20} \cdot 210^{12}$, and the greatest common divisor of $a$ and $b$ is 1 . | 32 |
Polygon $A_{1} A_{2} \ldots A_{n}$ is a regular $n$-gon. For some integer $k<n$, quadrilateral $A_{1} A_{2} A_{k} A_{k+1}$ is a rectangle of area 6 . If the area of $A_{1} A_{2} \ldots A_{n}$ is 60 , compute $n$. | 40 |
Let $T=8$. The parabola $y=x^{2}+T x$ is tangent to the parabola $y=-(x-2 T)^{2}+b$. Compute $b$. | 184 |
Lucy starts by writing $s$ integer-valued 2022-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=\left(v_{1}, \ldots, v_{2022}\right)$ and $\mathbf{w}=\left(w_{1}, \ldots, w_{2022}\right)$ that she has already written, and apply one of the following operations ... | 3 |
In an infinite array with two rows, the numbers in the top row are denoted $\ldots, A_{-2}, A_{-1}, A_{0}, A_{1}, A_{2}, \ldots$ and the numbers in the bottom row are denoted $\ldots, B_{-2}, B_{-1}, B_{0}, B_{1}, B_{2}, \ldots$ For each integer $k$, the entry $A_{k}$ is directly above the entry $B_{k}$ in the array, a... | 6 |
Xenia and Sergey play the following game. Xenia thinks of a positive integer $N$ not exceeding 5000. Then she fixes 20 distinct positive integers $a_{1}, a_{2}, \ldots, a_{20}$ such that, for each $k=1,2, \ldots, 20$, the numbers $N$ and $a_{k}$ are congruent modulo $k$. By a move, Sergey tells Xenia a set $S$ of posit... | 2 |
$\quad$ Let $T=40$. If $x+9 y=17$ and $T x+(T+1) y=T+2$, compute $20 x+14 y$. | 8 |
Let $T=$ 4. Pyramid $L E O J S$ is a right square pyramid with base $E O J S$, whose area is $T$. Given that $L E=5 \sqrt{2}$, compute $[L E O]$. | 7 |
Derek starts at the point $(0,0)$, facing the point $(0,1)$, and he wants to get to the point $(1,1)$. He takes unit steps parallel to the coordinate axes. A move consists of either a step forward, or a $90^{\circ}$ right (clockwise) turn followed by a step forward, so that his path does not contain any left turns. His... | 529 |
The sum of the interior angles of an $n$-gon equals the sum of the interior angles of a pentagon plus the sum of the interior angles of an octagon. Compute $n$. | 11 |
Let $T=5$. Hannah flips two fair coins, while Otto flips $T$ fair coins. Let $p$ be the probability that the number of heads showing on Hannah's coins is greater than the number of heads showing on Otto's coins. If $p=q / r$, where $q$ and $r$ are relatively prime positive integers, compute $q+r$. | 17 |
The solutions to the equation $x^{2}-180 x+8=0$ are $r_{1}$ and $r_{2}$. Compute
$$
\frac{r_{1}}{\sqrt[3]{r_{2}}}+\frac{r_{2}}{\sqrt[3]{r_{1}}}
$$ | 508 |
Let $N=\left\lfloor(3+\sqrt{5})^{34}\right\rfloor$. Compute the remainder when $N$ is divided by 100 . | 47 |
A multiplicative partition of a positive integer $n \geq 2$ is a way of writing $n$ as a product of one or more integers, each greater than 1. Note that we consider a positive integer to be a multiplicative partition of itself. Also, the order of the factors in a partition does not matter; for example, $2 \times 3 \tim... | 31 |
To morph a sequence means to replace two terms $a$ and $b$ with $a+1$ and $b-1$ if and only if $a+1<b-1$, and such an operation is referred to as a morph. Compute the least number of morphs needed to transform the sequence $1^{2}, 2^{2}, 3^{2}, \ldots, 10^{2}$ into an arithmetic progression. | 56 |
Compute the smallest positive integer $n$ such that $n^{n}$ has at least 1,000,000 positive divisors. | 84 |
For positive integers $a$ and $b$, define $f(a, b)=\frac{a}{b}+\frac{b}{a}+\frac{1}{a b}$.
For example, the value of $f(1,2)$ is 3 .
Determine the value of $f(2,5)$. | 3 |
Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_{1}, A_{2}, \ldots, A_{k}$ such that for all integers $n \geq 15$ and all $i \in\{1,2, \ldots, k\}$ there exist two distinct elements of $A_{i}$ whose sum is $n$. | 3 |
Compute the base $b$ for which $253_{b} \cdot 341_{b}=\underline{7} \underline{4} \underline{X} \underline{Y} \underline{Z}_{b}$, for some base- $b$ digits $X, Y, Z$. | 20 |
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