problem stringlengths 10 7.44k | answer stringlengths 1 270 | difficulty stringclasses 8
values |
|---|---|---|
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Eric`
- The mothers' names in different house... | Holly | 3/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Bob`, `Alice`, `Eric`, `Arnold`
- People use unique phone models... | 2 | 0/8 |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Peter`, `Bob`, `Eric`, `Arnold`, `Carol`
- Each person has a uni... | swimming | 0/8 |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Carol`, `Peter`, `Bob`, `Eric`, `Arnold`, `Alice`
- Each person prefers a... | 2 | 2/8 |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Arnold`, `Alice`, `Peter`, `Eric`
- Each mother is accompanied by their c... | 4 | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Eric`, `Peter`, `Bob`, `Arnold`
- Each person has a favorite col... | romance | 0/8 |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Eric`, `Peter`, `Carol`, `Bob`, `Arnold`
- Everyone has a favori... | 5 | 0/8 |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
- The people are of nationalities: `bri... | 4 | 1/8 |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Eric`, `Bob`, `Arnold`, `Carol`, `Alice`
- Each person has a uni... | mar | 0/8 |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Peter`, `Eric`, `Arnold`, `Carol`, `Bob`
- People have unique he... | pizza | 0/8 |
There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Arnold`, `Eric`
- People have unique favorite music genres: `cla... | fish | 0/8 |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`
- Everyone has something unique for lun... | 1 | 3/8 |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Peter`, `Carol`, `Alice`, `Bob`, `Arnold`
- The people are of nat... | norwegian | 0/8 |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`
- Everyone has a favorite smoothie: `dr... | iphone 13 | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`, `Bob`
- Each person has a unique type ... | pall mall | 0/8 |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`
- Each person lives in a unique style o... | 2 | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Bob`, `Eric`, `Arnold`, `Alice`
- People have unique hair colors... | tea | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`, `Bob`
- Each person has a unique level... | mountain | 0/8 |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Eric`
- Each person lives in a unique style o... | 3 | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Alice`, `Peter`, `Bob`, `Arnold`
- Each person has a unique type ... | hamster | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Bob`, `Eric`
- Each person has a unique hobby... | tea | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Eric`, `Bob`, `Arnold`, `Peter`
- The people keep unique animals... | fish | 4/8 |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Arnold`, `Eric`, `Bob`, `Alice`, `Carol`, `Peter`
- The people are of nat... | hip hop | 0/8 |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Arnold`, `Eric`, `Alice`
- Each person prefers a unique type of ... | ranch | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Bob`, `Alice`
- People have unique hair colors... | 1 | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Bob`, `Peter`, `Alice`, `Eric`, `Arnold`
- Each person has a unique type ... | fish | 0/8 |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Eric`, `Arnold`, `Peter`
- Each person has a unique hobby: `pain... | master | 0/8 |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Peter`, `Arnold`, `Alice`
- Each person has a unique birthday mon... | Fred | 0/8 |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Alice`, `Peter`, `Arnold`
- Each person lives in a unique style o... | lilies | 0/8 |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Bob`, `Carol`, `Peter`, `Alice`, `Eric`, `Arnold`
- They all have a uniqu... | red | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Bob`, `Alice`, `Eric`, `Peter`, `Arnold`
- People have unique heights: `a... | feb | 0/8 |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Arnold`, `Peter`, `Alice`, `Eric`
- People have unique hair colors: `blac... | 4 | 2/8 |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Eric`
- Each person lives in a unique style o... | white | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Eric`, `Bob`
- Everyone has a unique favorite... | tennis | 1/8 |
There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Arnold`, `Eric`
- People have unique favorite book genres: `scie... | 3 | 4/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Alice`, `Arnold`, `Bob`, `Eric`
- Each person has a unique hobby... | 1 | 0/8 |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Carol`, `Arnold`, `Bob`, `Alice`, `Peter`, `Eric`
- People use unique pho... | roses | 0/8 |
Problem 8.6. Vasya thought of three natural numbers with a sum of 1003. Calculating their product, Vasya noticed that it ends with $N$ zeros. What is the maximum value that
$N$ can take? | 7 | 2/8 |
5. For a convex polyhedron, the internal dihedral angle at each edge is acute. How many faces can the polyhedron have? | 4 | 2/8 |
The sum of two numbers is 173,717. The prime factors of the four-digit difference between the two numbers do not include any single-digit numbers. One of the numbers is divisible by 1558. Which are these two numbers? | 91143 | 4/8 |
7. For what values of the parameter $a$ will the minimum value of the function
$$
f(x)=|7 x-3 a+8|+|5 x+4 a-6|+|x-a-8|-24
$$
be the smallest. | \dfrac{82}{43} | 4/8 |
Problem 3. Find all real numbers $a$ such that $4[a n]=n+[a[a n]]$ for any positive integer $n$ ( $[x]$ denotes the largest integer less than or equal to $x$ ).
Nikolai Nikolov | 2 + \sqrt{3} | 4/8 |
Let $f^1(x)=x^3-3x$. Let $f^n(x)=f(f^{n-1}(x))$. Let $\mathcal{R}$ be the set of roots of $\tfrac{f^{2022}(x)}{x}$. If
\[\sum_{r\in\mathcal{R}}\frac{1}{r^2}=\frac{a^b-c}{d}\]
for positive integers $a,b,c,d$, where $b$ is as large as possible and $c$ and $d$ are relatively prime, find $a+b+c+d$. | 4060 | 0/8 |
[
Two tangents drawn from one point
Auxiliary similar triangles Properties and characteristics of isosceles triangles.
Inscribed and circumscribed circles
Pythagorean theorem (direct and inverse)
In an isosceles triangle \(ABC (AB = BC)\), a circle is inscribed. A line parallel to side \(BC\) and tangent to the ci... | \dfrac{3}{2} | 3/8 |
II. (25 points) As shown in Figure 4, the incircle $\odot I$ of $\triangle ABC$ touches $AB$ and $AC$ at points $D$ and $E$, respectively. Extend $DI$ to $M$ and $EI$ to $N$ such that $IM = IN = 9IE$. Points $B$, $I$, $C$, $M$, and $N$ are concyclic. The ratio of the perimeter of $\triangle ABC$ to side $BC$ is the sim... | 2009 | 1/8 |
Kalinin A.
Anya and Borya, whose speeds are constant but not necessarily the same, simultaneously set out from villages A and B towards each other. If Anya had set out 30 minutes earlier, they would have met 2 km closer to village B. If Borya had set out 30 minutes earlier, the meeting would have taken place closer to... | 2 | 4/8 |
Find prime numbers $p , q , r$ such that $p+q^2+r^3=200$. Give all the possibilities.
Remember that the number $1$ is not prime. | (23, 13, 2) | 1/8 |
Example 7 Given $x \geqslant 0, x^{2}+(y-2)^{2}=1$. Find
$$
W=\frac{3 x^{2}+2 \sqrt{3} x y+5 y^{2}}{x^{2}+y^{2}}
$$
the maximum and minimum values. | 6 | 0/8 |
Problem 4. Point $O$ is the center of the circumcircle of triangle $ABC$ with sides $AB=8$ and $AC=5$. Find the length of side $BC$ if the length of the vector $\overrightarrow{OA}+3 \overrightarrow{OB}-4 \overrightarrow{OC}$ is 10. | 4 | 5/8 |
3. In rectangle $A B C D$, $A B=4, A D=3$, fold the rectangle along diagonal $A C$ so that the projection of point $D$ on plane $A B C$, point $E$, falls on line $A B$. At this moment, the dihedral angle $B-A C-D$ is $\qquad$ | \arccos \frac{9}{16} | 4/8 |
3. Alice and Bob are independently trying to figure out a secret password to Cathy's bitcoin wallet. Both of them have already figured out that:
- it is a 4-digit number whose first digit is 5;
- it is a multiple of 9 ;
- The larger number is more likely to be a password than a smaller number.
Moreover, Alice figured o... | 5949 | 1/8 |
Consider the sequence of polynomials $P_0(x) = 2$ , $P_1(x) = x$ and $P_n(x) = xP_{n-1}(x) - P_{n-2}(x)$ for $n \geq 2$ . Let $x_n$ be the greatest zero of $P_n$ in the the interval $|x| \leq 2$ . Show that $$ \lim \limits_{n \to \infty}n^2\left(4-2\pi +n^2\int \limits_{x_n}^2P_n(x)dx\right)=2\pi - 4-\frac... | 2\pi - 4 - \frac{\pi^3}{12} | 1/8 |
Let $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and $S = \{(a,b) \in X \times X : x^2 + ax + b \text{ and } x^3 + bx + a \text{ have at least a common real zero.}\}$.
How many elements are there in $S$? | 24 | 5/8 |
A parliament consists of $n$ senators. These senators are organized into 10 parties and 10 committees. Each senator belongs to exactly one party and one committee.
Determine the smallest possible value of $n$ such that it is possible to label the parties and the committees with numbers from 1 to 10, ensuring that the... | 101 | 1/8 |
Let $p(x) = x^4 - 5773x^3 - 46464x^2 - 5773x + 46$. Determine the sum of $\arctan$ of its real roots. | 0 | 2/8 |
Consider an infinite sequence $x_1, x_2, \dots$ of positive integers such that, for every integer $n \geq 1$:
- If $x_n$ is even, then $x_{n+1} = \frac{x_n}{2}$.
- If $x_n$ is odd, then $x_{n+1} = \frac{x_n - 1}{2} + 2^{k-1}$, where $2^{k-1} \leq x_n < 2^k$.
Determine the smallest possible value of $x_1$ for which $... | 1183 | 1/8 |
Suppose we have $10$ balls and $10$ colors. For each ball, we independently color it with one of the $10$ colors, then group the balls together by color at the end. If $S$ is the expected value of the square of the number of distinct colors used on the balls, find the sum of the digits of $S$ written as a decimal. | 55 | 4/8 |
Cat and Claire are having a conversation about Cat’s favorite number. Cat says, "My favorite number is a two-digit perfect square!"
Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I’d know for certain what it is?"... | 36 | 4/8 |
A function $f$ defined on the positive integers (and taking positive integer values) is given by:
$$\begin{align*}
f(1) &= 1, \\
f(3) &= 3, \\
f(2 \cdot n) &= f(n), \\
f(4 \cdot n + 1) &= 2 \cdot f(2 \cdot n + 1) - f(n), \\
f(4 \cdot n + 3) &= 3 \cdot f(2 \cdot n + 1) - 2 \cdot f(n),
\end{align*}$$
for all positive i... | 92 | 0/8 |
Let $ABCD$ be a convex quadrilateral with positive integer side lengths. Given that $\angle A = \angle B = 120^{\circ}$, $|AD - BC| = 42$, and $CD = 98$, find the maximum possible value of $AB$. | 69 | 1/8 |
We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A, B, C)$ of distinct chosen points such that $A$ and $B$ have the same $y$-coordinate, and $B$ and $C$ have the same $x$-coordinate. Find the greatest value that $N$ can attain considering all possible ways to choose the points. | 8100 | 4/8 |
At the Lexington High School, each student is given a unique five-character ID consisting of uppercase letters. Compute the number of possible IDs that contain the string "LMT". | 2028 | 4/8 |
A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$-sided die, with faces labeled $0, 1, 2, \ldots, 2018$, and with each face equally likely to come up. Every second, the mouse rolls the die. If the die lands on $k$, and the mouse has at least $k$ slices of cheese remaining, then ... | 2019 | 4/8 |
Suppose tetrahedron $PABC$ has volume $420$ and satisfies $AB = 13$, $BC = 14$, and $CA = 15$. The minimum possible surface area of $PABC$ can be written as $m+n\sqrt{k}$, where $m$, $n$, and $k$ are positive integers, and $k$ is not divisible by the square of any prime. Compute $m+n+k$. | 346 | 4/8 |
Let $\triangle ABC$ satisfy $AB = 17$, $AC = \frac{70}{3}$, and $BC = 19$. Let $I$ be the incenter of $\triangle ABC$ and $E$ be the excenter of $\triangle ABC$ opposite $A$. (Note: this means that the circle tangent to ray $AB$ beyond $B$, ray $AC$ beyond $C$, and side $BC$ is centered at $E$.) Suppose the circle with... | 22 | 0/8 |
Given triangle $ABC$. Let $A_1B_1$, $A_2B_2$, $\ldots$, $A_{2008}B_{2008}$ be $2008$ lines parallel to $AB$ which divide triangle $ABC$ into $2009$ equal areas. Calculate the value of $$ \left\lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + \ldots + \frac{A_1B_1}{2A_{2008}B_{2008}} \right\rfloor$$ | 43 | 5/8 |
Caltech's 900 students are evenly spaced along the circumference of a circle. How many equilateral triangles can be formed with at least two Caltech students as vertices? | 808500 | 1/8 |
An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies the recurrence
\[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \]for every positive integer $n$. Given that $a_1 = a_3 = 1$ and $a_{98} = a_{99}$, compute $a_1 + a_2 + \dots + a_{100}$. | 3 | 1/8 |
Euclid, Pythagoras, Ptolemy, and Hypatia are playing a game where they all have to think of a number, and then cube that number 20 times. Hypatia doesn't want to cube large numbers, so she chooses the number 1. Euclid thinks the same thing and also chooses the number 1. However, Pythagoras and Ptolemy don't think ahead... | 2 | 5/8 |
Let $k$ be the smallest positive integer such that the binomial coefficient $\binom{10^9}{k}$ is less than the binomial coefficient $\binom{10^9 + 1}{k - 1}$. Let $a$ be the first (from the left) digit of $k$ and let $b$ be the second (from the left) digit of $k$. What is the value of $10a + b$? | 38 | 3/8 |
In $\vartriangle ABC, \angle A = 52^o$ and $\angle B = 57^o$. One circle passes through the points $B, C$, and the incenter of $\vartriangle ABC$, and a second circle passes through the points $A, C$, and the circumcenter of $\vartriangle ABC$. Find the degree measure of the acute angle at which the two circles interse... | 59 | 1/8 |
Let $P$ be the set of positive integers that are prime numbers. Find the number of subsets of $P$ that have the property that the sum of their elements is $34$ such as $\{3, 31\}$. | 9 | 2/8 |
Six different small books and three different large books sit on a shelf. Three children may each take either two small books or one large book. Find the number of ways the three children can select their books. | 1176 | 4/8 |
A storage depot is a pyramid with height $30$ and a square base with side length $40$. Determine how many cubical $3\times 3\times 3$ boxes can be stored in this depot if the boxes are always packed so that each of their edges is parallel to either an edge of the base or the altitude of the pyramid. | 471 | 2/8 |
Fiona had a solid rectangular block of cheese that measured $6$ centimeters from left to right, $5$ centimeters from front to back, and $4$ centimeters from top to bottom. Fiona took a sharp knife and sliced off a $1$ centimeter thick slice from the left side of the block and a $1$ centimeter slice from the right side ... | 340 | 2/8 |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Eric`
- People have unique heights: `tall`, `... | 4 | 4/8 |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Arnold`, `Carol`, `Peter`, `Eric`, `Bob`
- Each person has a uni... | Peter | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Arnold`, `Bob`, `Peter`, `Alice`
- Each person has a unique birth... | jan | 0/8 |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Bob`, `Arnold`, `Eric`, `Peter`, `Carol`, `Alice`
- Each person has a uni... | woodworking | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Bob`, `Arnold`, `Alice`, `Peter`, `Eric`
- Each person has a unique hobby... | tennis | 0/8 |
There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Peter`, `Arnold`
- Everyone has a favorite smoothie: `cherry`, `w... | horse | 1/8 |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Arnold`, `Peter`, `Alice`
- People have unique favorite book genr... | pop | 2/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Arnold`, `Alice`, `Bob`, `Peter`
- Each person lives in a unique ... | 3 | 2/8 |
There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Arnold`, `Eric`, `Peter`
- People have unique favorite book genres: `scie... | 3 | 2/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Peter`, `Bob`
- People use unique phone models... | 4 | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Peter`, `Alice`, `Bob`, `Arnold`
- The people are of nationalitie... | Peter | 0/8 |
Given the positive integers \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \). Any set of four numbers selected and summed results in the set \(\{44, 45, 46, 47\}\). Determine these five numbers. | 10, 11, 11, 12, 13 | 2/8 |
Let \( a, b, c \in \left[\frac{1}{2}, 1\right] \). Define \( s = \frac{a+b}{1+c} + \frac{b+c}{1+a} + \frac{c+a}{1+b} \). What is the range of possible values for \( s \)? | [2, 3] | 4/8 |
We start with the numbers \( a, b, c, d \). We then replace them with \( a' = a - b \), \( b' = b - c \), \( c' = c - d \), \( d' = d - a \). We carry out this process 1996 times. Is it possible to end up with numbers \( A, B, C, D \) such that \( |BC - AD| \), \( |AC - BD| \), \( |AB - CD| \) are all primes? | \text{No} | 4/8 |
Problem 4. Find the number of all natural numbers $n, 4 \leq n \leq$ 1023, such that their binary representations do not contain three consecutive equal digits.
Emil Kolev | 228 | 2/8 |
3. As shown in Figure 4, it is known that quadrilateral $ABCD$ is inscribed in $\odot O$, $AB$ is the diameter, $AD=DC$, and $BA$, $CD$ are extended to intersect at point $E$, $BF \perp EC$ intersects the extension of $EC$ at point $F$. If $AE=AO, BC=6$, then the length of $CF$ is $\qquad$. | \dfrac{3\sqrt{2}}{2} | 3/8 |
Subsets and Splits
Filtered Answers A-D
Retrieves 100 rows where the answer is a single letter from A to D, providing basic filtering of the dataset.