problem stringlengths 10 7.44k | answer stringlengths 1 270 | difficulty stringclasses 8
values |
|---|---|---|
10. All positive integer solution pairs $(x, y)$ of the equation $x^{2}+y^{2}=208(x-y)$ are $\qquad$ . | (48, 32) | 0/8 |
8. As shown in Figure $11, \angle A O B=$ $30^{\circ}, \angle A O B$ contains a fixed point $P$, and $O P=10, O A$ has a point $Q, O B$ has a fixed point $R$. If the perimeter of $\triangle P Q R$ is minimized, find its minimum value. | 10 | 2/8 |
9.1. Find the smallest natural number $n$ for which the decimal representation of $n$ together with $n^{2}$ uses all the digits from 1 to 9 exactly once. | 567 | 0/8 |
A function $f:\mathbb{N} \to \mathbb{N}$ is given. If $a,b$ are coprime, then $f(ab)=f(a)f(b)$. Also, if $m,k$ are primes (not necessarily different), then $$f(m+k-3)=f(m)+f(k)-f(3).$$ Find all possible values of $f(11)$. | 1 | 3/8 |
Problem 4. The number 1 is written on the blackboard. After that a sequence of numbers is created as follows: at each step each number $a$ on the blackboard is replaced by the numbers $a-1$ and $a+1$; if the number 0 occurs, it is erased immediately; if a number occurs more than once, all its occurrences are left on t... | \dbinom{n}{\lfloor n/2 \rfloor} | 0/8 |
10. Let $n$ be a positive integer. If in the $2 n+1$ consecutive natural numbers including 2009, the sum of the squares of the first $n+1$ numbers is equal to the sum of the squares of the last $n$ numbers, then the value of $n$ is $\qquad$. | 31 | 3/8 |
## Task 2
Subtract the number 5 from 9! Write down the equation for this! | 4 | 0/8 |
In a school, $2n$ subjects are studied. All students have grades of "four" and "five," and no two students have the same grades. It is not possible to say that one of any two students is studying better than the other. Prove that the number of students in the school does not exceed $\frac{(2n)}{n!n!}$! | \dfrac{(2n)!}{n! \cdot n!} | 0/8 |
Petya is playing a shooting game. If he scores less than 1000 points, the computer will add $20 \%$ of his score. If he scores from 1000 to 2000 points, the computer will add $20 \%$ of the first thousand points and $30 \%$ of the remaining points. If Petya scores more than 2000 points, the computer will add $20 \%$ of... | 470 | 5/8 |
2. In $\triangle A B C$, $\angle B=2 \angle C, A D$ is the angle bisector of $\angle A$, $m A B=n B D(n>m>0)$. Then $\cos C=$ $\qquad$ . | \dfrac{m + n}{2n} | 3/8 |
## Task 3.
On the side $\overline{A B}$ of the cyclic quadrilateral $A B C D$, there is a point $X$ with the property that the diagonal $\overline{B D}$ bisects the length of $\overline{C X}$, and the diagonal $\overline{A C}$ bisects the length of $\overline{D X}$.
What is the smallest possible ratio $|A B|:|C D|$ i... | 2 | 2/8 |
We folded a rectangular piece of paper along one of its diagonals. After the fold, the four vertices ended up at the four vertices of a trapezoid, three sides of which are of equal length. What is the length of the shorter side of the original rectangle if the longer side is 12 cm? | 4\sqrt{3} | 1/8 |
In a right triangle \(ABC\) with right angle at \(A\), the altitude \(AH\) is drawn. On the extension of the hypotenuse \(BC\) beyond point \(C\), there is a point \(X\) such that
\[ HX = \frac{BH + CX}{3}. \]
Prove that \( 2 \angle ABC = \angle AXC \). | 2 \angle ABC = \angle AXC | 1/8 |
Perimeter of triangle $ABC$ is $1$. Circle $\omega$ touches side $BC$, continuation of side $AB$ at $P$ and continuation of side $AC$ in $Q$. Line through midpoints $AB$ and $AC$ intersects circumcircle of $APQ$ at $X$ and $Y$.
Find length of $XY$. | \dfrac{1}{2} | 1/8 |
3. There are 12 students in the class. Each of them was asked how many friends they have in the class. Each number \( n \) was either mentioned exactly \( n \) times or not at all. What is the maximum value that the sum of all the numbers mentioned can take? | 90 | 1/8 |
10. Given that $\lambda$ is a positive real number, the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=\frac{1}{3}, a_{n+1}=\frac{2 a_{n}^{2}}{4 a_{n}^{2}-2 a_{n}+1}, n=1,2,3, \cdots$. If for any $m \in \mathbf{N}^{*}$, we have $\sum_{k=1}^{m} a_{k}<\lambda$, find the minimum value of $\lambda$. | 1 | 3/8 |
Given a positive integer \( N \), let \( a_1, a_2, \ldots, a_n \) be \( n \) distinct positive integers less than \( N \). Denote the least common multiple of these \( n \) numbers as \( M \) and the greatest common divisor as \( m \). Given that \( M \mid m^2 \), prove that \( n^3 < 4 N \). | n^3 < 4N | 2/8 |
Test: Given that $a$ is a natural number, a quadratic trinomial with integer coefficients and $a$ as the leading coefficient has two distinct positive roots less than 1. Find the minimum value of $a$.
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The above text translated into English, preserving the original text's line breaks and format. | 5 | 5/8 |
The lines of the first given table are all arithmetic progressions of the same ratio, and the columns of this table are all arithmetic progressions of the same ratio. In the second given table, the same formation rule was used, but someone erased some numbers, leaving only three. What is the number that was in the posi... | 102 | 0/8 |
There are three printing machines. The first machine, with a card containing the numbers $a$ and $b$, outputs a card with the numbers $a+1$ and $b+1$; the second machine, with a card containing even numbers $a$ and $b$, outputs a card with the numbers $a / 2$ and $b / 2$; the third machine, with a pair of cards contain... | \text{No} | 3/8 |
2. Given a positive integer $n$, define $f(0, j)=f(i, 0)=0, f(1,1)=n$ and
$$
f(i, j)=\left\lfloor\frac{f(i-1, j)}{2}\right\rfloor+\left\lfloor\frac{f(i, j-1)}{2}\right\rfloor
$$
for all positive integers $i$ i $j,(i, j) \neq(1,1)$. How many ordered pairs of positive integers $(i, j)$ are there for which $f(i, j)$ is ... | n | 0/8 |
2. Given $x \in \mathbf{R}, f(x)=x^{2}+12 x+30$. Then the solution set of the equation $f(f(f(f(f(x)))))=0$ is $\qquad$ . | \{-6 + 6^{1/32}, -6 - 6^{1/32}\} | 0/8 |
6. Given $[x]$ represents the greatest integer not exceeding the real number $x$. If $P=\sum_{i=1}^{2010}\left[\frac{6^{i}}{7}\right]$, then the remainder when $P$ is divided by 35 is | 20 | 3/8 |
5. There is a method of setting a six-digit password: the first 2 digits are the last 2 digits of the birth year, the middle 2 digits are the birth month, and the last 2 digits are the birth date. For example, if born on April 23, 1998, the six-digit password is set as 980423. The number of possible passwords set in th... | 36525 | 1/8 |
2. Let $\alpha, \beta$ be the roots of the quadratic equation $x^{2}+(p-2) x+1$ $=0$ and. $[1+\alpha(p+\alpha)-\beta][1+\beta(p+\beta)$ $-\alpha]=-\frac{7}{2}$, find the value of $p$. | \dfrac{9}{2} | 0/8 |
# Problem 7.5 (7 points)
On a plane, 6 lines are drawn and several points are marked. It turned out that on each line exactly 3 points are marked. What is the minimum number of points that could have been marked? | 7 | 3/8 |
1. Solve the equation $x^{\log _{2}\left(0.25 x^{3}\right)}=512 x^{4}$. | \dfrac{1}{2} | 1/8 |
In a given group of numbers, one number is equal to the average of all, the largest number is 7 greater than the average, the smallest is 7 less than the average, and most of the numbers in the group have below-average values.
What is the smallest number of numbers that can be in the group?
(K. Pazourek)
Hint. What ... | 7 | 3/8 |
5.9 The International Mathematical Olympiad Problem Selection Committee has 34 countries participating, with each country being represented by a team leader and a deputy leader. Before the meeting, the participants shake hands with each other, but the team leader and deputy leader from the same country do not shake han... | 33 | 5/8 |
3. Petrov lists the odd numbers: $1,3,5, \ldots, 2013$, while Vasechkin lists the even numbers: $2,4, \ldots, 2012$. Each of them calculated the sum of all digits of all their numbers and told the excellent student Masha. Masha subtracted Vasechkin's result from Petrov's result. What did she get? | 1007 | 1/8 |
Let $ ABCD$ be a convex quadrilateral. We have that $ \angle BAC\equal{}3\angle CAD$, $ AB\equal{}CD$, $ \angle ACD\equal{}\angle CBD$. Find angle $ \angle ACD$ | 30^\circ | 2/8 |
Viruses are capable of reproducing inside bacteria, essentially the bacteria's organism reproduces them. When a virus enters a bacterium, after a certain time, the bacterium dies and the resulting viruses are released.
Suppose a virus enters a colony of $n$ bacteria and immediately enters one of the bacteria; after on... | n | 3/8 |
4. As shown in the upper right figure, the radii of semicircles $A$ and $B$ are equal, they are tangent to each other, and both are internally tangent to the semicircle $O$ with radius 1. $\odot O_{1}$ is tangent to both of them, and $\odot O_{2}$ is tangent to $\odot O_{1}$, semicircle $O$, and semicircle $B$. Then th... | \dfrac{1}{6} | 5/8 |
12. In a game, two players take turns to "eat squares" from a $5 \times 7$ grid chessboard. To "eat a square," a player selects an uneaten square and moves the piece to that square, then all the squares in the quadrant formed (along the left edge of the square upwards, and along the bottom edge of the square to the rig... | 792 | 3/8 |
1B. Find the angle $\alpha$ of triangle $ABC$, if for its angles $\alpha, \beta, \gamma$ the equality $\sin \alpha=\frac{\sin \beta+\sin \gamma}{\cos \beta+\cos \gamma}$ holds. | \dfrac{\pi}{2} | 5/8 |
There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes: Nationality, Music-Genre, Hobby, Pet, Sport, Transport.
The attributes have the following possible values:
- Nationality: american, australian, malaysian, nigerian, pakistani, spanish
- Music-... | pop | 0/8 |
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Nationality, Music-Genre, Pet, Transport.
The attributes have the following possible values:
- Nationality: argentine, brazilian, chinese, japanese, spanish
- Music-Genre: d&b, folk, hip-hop, ... | japanese | 0/8 |
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Job, Food, Beverage, Movie-Genre, Music-Genre, Nationality.
The attributes have the following possible values:
- Job: accountant, firefighter, journalist, manager
- Food: asparagus, cucumber, ... | manager | 0/8 |
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Food, Music-Genre, Sport, Transport.
The attributes have the following possible values:
- Food: avocado, cherry, lime, pineapple, zucchini
- Music-Genre: classical, dubstep, gospel, pop, techn... | pineapple | 1/8 |
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Food, Movie-Genre, Pet, Transport.
The attributes have the following possible values:
- Food: artichoke, avocado, grapefruit, pear
- Movie-Genre: disaster, family, musical, mystery
- Pet: chin... | musical | 0/8 |
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Nationality, Movie-Genre, Food, Transport.
The attributes have the following possible values:
- Nationality: brazilian, british, german, indonesian, turkish
- Movie-Genre: crime, documentary, ... | thriller | 0/8 |
There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes: Job, Hobby, Movie-Genre, Music-Genre, Transport.
The attributes have the following possible values:
- Job: accountant, doctor, musician, paramedic, teacher, videographer
- Hobby: drawing, read... | musician | 0/8 |
There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes: Beverage, Hobby, Music-Genre, Pet.
The attributes have the following possible values:
- Beverage: coffee, hot-chocolate, juice, mirinda, soy-milk, water
- Hobby: baking, card-games, drawing, f... | juice | 0/8 |
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Nationality, Hobby, Sport, Transport.
The attributes have the following possible values:
- Nationality: chinese, colombian, french, japanese, pakistani
- Hobby: board-games, card-games, dancin... | french | 0/8 |
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Nationality, Hobby, Food, Pet.
The attributes have the following possible values:
- Nationality: australian, french, japanese, mexican, pakistani
- Hobby: chess, dancing, gardening, singing, v... | pakistani | 0/8 |
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Nationality, Hobby, Movie-Genre, Music-Genre, Food, Transport.
The attributes have the following possible values:
- Nationality: argentine, chinese, egyptian, indonesian
- Hobby: fishing, gard... | romance | 0/8 |
There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes: Beverage, Food, Movie-Genre, Transport.
The attributes have the following possible values:
- Beverage: coffee, cola, juice, milk, soy-milk, water
- Food: banana, cauliflower, grapefruit, papay... | radish | 0/8 |
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Beverage, Food, Movie-Genre, Music-Genre, Sport.
The attributes have the following possible values:
- Beverage: cola, fanta, hot-chocolate, soy-milk, tea
- Food: apricot, cranberry, mango, pap... | fanta | 0/8 |
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Food, Movie-Genre, Sport, Transport.
The attributes have the following possible values:
- Food: artichoke, blueberry, grapes, mango
- Movie-Genre: animation, drama, epic, family
- Sport: badmi... | soccer | 0/8 |
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Job, Food, Hobby, Beverage, Movie-Genre, Sport.
The attributes have the following possible values:
- Job: freelancer, manager, nurse, project-manager
- Food: artichoke, cabbage, corn, zucchini... | project-manager | 0/8 |
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Job, Hobby, Beverage, Movie-Genre, Music-Genre, Sport.
The attributes have the following possible values:
- Job: paramedic, pilot, scientist, teacher
- Hobby: chess, rock-climbing, skydiving, ... | hot-chocolate | 0/8 |
There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes: Nationality, Food, Movie-Genre, Beverage, Transport.
The attributes have the following possible values:
- Nationality: german, italian, japanese, mexican, spanish, thai
- Food: cauliflower, cr... | tea | 0/8 |
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Job, Hobby, Movie-Genre, Pet.
The attributes have the following possible values:
- Job: accountant, bartender, coach, police-officer, scientist
- Hobby: dancing, reading, rock-climbing, sudoku... | musical | 1/8 |
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Nationality, Food, Hobby, Beverage, Pet, Transport.
The attributes have the following possible values:
- Nationality: french, japanese, malaysian, polish
- Food: asparagus, lemon, mango, potat... | polish | 0/8 |
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Beverage, Food, Hobby, Music-Genre, Sport.
The attributes have the following possible values:
- Beverage: almond-milk, cola, iced-tea, lemonade, sprite
- Food: cabbage, garlic, orange, plum, z... | lemonade | 0/8 |
There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes: Job, Hobby, Music-Genre, Pet.
The attributes have the following possible values:
- Job: architect, entrepreneur, journalist, manager, nurse, project-manager
- Hobby: baking, camping, chess, ma... | project-manager | 0/8 |
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Beverage, Hobby, Music-Genre, Pet, Sport.
The attributes have the following possible values:
- Beverage: iced-tea, milk, mirinda, tea
- Hobby: baking, board-games, fishing, singing
- Music-Gen... | tea | 0/8 |
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Job, Food, Beverage, Pet, Sport.
The attributes have the following possible values:
- Job: analyst, chef, fisherman, police-officer
- Food: kiwi, nectarine, orange, peas
- Beverage: almond-mil... | police-officer | 0/8 |
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Beverage, Food, Movie-Genre, Music-Genre, Pet, Sport.
The attributes have the following possible values:
- Beverage: coffee, fanta, hot-chocolate, mirinda, sprite
- Food: apple, garlic, kiwi, ... | hot-chocolate | 0/8 |
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Job, Food, Beverage, Movie-Genre, Sport.
The attributes have the following possible values:
- Job: bartender, dancer, firefighter, manager, security-guard
- Food: avocado, kale, lettuce, spina... | mirinda | 0/8 |
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Job, Hobby, Food, Movie-Genre, Pet, Sport.
The attributes have the following possible values:
- Job: engineer, librarian, manager, paramedic, photographer
- Hobby: cooking, filmmaking, reading... | reading | 0/8 |
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Nationality, Hobby, Beverage, Transport.
The attributes have the following possible values:
- Nationality: egyptian, pakistani, polish, russian
- Hobby: board-games, reading, rock-climbing, wr... | pakistani | 0/8 |
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Job, Hobby, Movie-Genre, Music-Genre, Transport.
The attributes have the following possible values:
- Job: accountant, architect, journalist, photographer
- Hobby: cooking, photography, puzzle... | jazz | 0/8 |
There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes: Hobby, Music-Genre, Pet, Sport, Transport.
The attributes have the following possible values:
- Hobby: baking, filmmaking, photography, reading, skydiving, video-games
- Music-Genre: ambient, ... | trance | 0/8 |
There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes: Job, Hobby, Movie-Genre, Music-Genre, Transport.
The attributes have the following possible values:
- Job: bartender, chef, coach, journalist, lawyer, librarian
- Hobby: board-games, camping, ... | indie | 0/8 |
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Job, Food, Movie-Genre, Sport.
The attributes have the following possible values:
- Job: bartender, dancer, dressmaker, musician
- Food: carrot, cranberry, pepper, radish
- Movie-Genre: advent... | pepper | 0/8 |
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Job, Beverage, Movie-Genre, Nationality, Sport, Transport.
The attributes have the following possible values:
- Job: entrepreneur, lawyer, social-worker, software-developer
- Beverage: almond-... | mystery | 0/8 |
Example 9. Given the dihedral angle $C-A B=D=120^{\circ}$, $\angle C B A=60^{\circ}, \angle D A B=30^{\circ}, A B=\sqrt{37} \mathrm{~cm}$ (Figure 17). Find the distance $d$ between the skew lines $B C, A D$. | 3 | 5/8 |
Find the number of functions $f(x)$ from $\{1, 2, 3, 4, 5\}$ to $\{1, 2, 3, 4, 5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1, 2, 3, 4, 5\}$ .
Please give the answer directly without any intermediate steps. | 756 | 0/8 |
2. At the parade of the royal musketeers, seeing that they could not be arranged in rows of 11 musketeers, D'Artagnan decided to arrange the musketeers in rows of 10 musketeers, but it turned out that there was one empty place in the last row. Then he tried to arrange them in rows of 9, 8, 7, 6, 5, 4, 3, 2 musketeers, ... | 5039 | 5/8 |
6. As shown in the figure, given that the area of the equilateral $\triangle A B C$ is $S, D$ is the midpoint of $A B, D E \perp B C, E F \perp A C, F G \perp$ $A B$. Then the area of quadrilateral $D E F G$ is ${ }^{\circ}$ . $\qquad$ | \dfrac{51}{128} S | 5/8 |
# Problem 2. (2 points)
In a positive non-constant geometric progression, the arithmetic mean of the third, fourth, and eighth terms is equal to some term of this progression. What is the minimum possible number of this term? | 4 | 5/8 |
5. Daisies grew along the path. Between every two daisies, a cornflower grew, and then between each cornflower and daisy - a dandelion. It turned out that now 101 flowers grow along the path. How many daisies grow along the path? | 26 | 5/8 |
## Task 2 - 220812
For a 22-digit number $z$, the following properties are required:
$z$ has the unit digit 7; if this last digit is removed and placed in front of the remaining 21 digits, the result is the same as the multiplication $7 \cdot z$.
Prove that there is exactly one such number $z$! Determine this number... | 1014492753623188405797 | 1/8 |
10. The edge length of the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ is 3, and points $P, Q, R$ are on the edges $A A_{1}, B B_{1}, C_{1} D_{1}$ respectively, with $A_{1} P=1, B_{1} Q=2, C_{1} R=1$. Let the dihedral angle (acute) between plane $P Q R$ and plane $A B C D$ be $\alpha$, find $\alpha$. | \arctan \frac{\sqrt{34}}{9} | 2/8 |
Given \( n \) points on a plane, where no three points are collinear, and some of these points are connected by line segments, let \( N_{k} \) denote the number of complete graphs of \( k \) points. Prove that:
$$
\frac{N_{k+1}}{N_{k}} \geqslant \frac{1}{k^{2}-1}\left(\frac{k^{2} N_{k}}{N_{k+1}}-n\right),
$$
for \( k... | \frac{N_{k+1}}{N_{k}} \geqslant \frac{1}{k^{2}-1}\left(\frac{k^{2} N_{k}}{N_{k+1}}-n\right) | 3/8 |
In a circle, let $AB$ and $BC$ be chords , with $AB =\sqrt3, BC =3\sqrt3, \angle ABC =60^o$. Find the length of the circle chord that divides angle $ \angle ABC$ in half. | 4 | 3/8 |
Task 2. Mom baked a cake for Anya's birthday, which weighs an integer number of grams. Before decorating it, Mom weighed the cake on digital scales that round the weight to the nearest ten grams (if the weight ends in 5, the scales round it down). The result was 1440 g. When Mom decorated the cake with identical candle... | 4 | 3/8 |
4. Find the number of finite increasing sequences of natural numbers $a_{1}, a_{2}, \ldots, a_{k}$, of all possible lengths $k$, for which $a_{1}=1, a_{i} \mid a_{i+1}$ for $i=1,2, \ldots, k-1$, and $a_{k}=969969$. | 4683 | 4/8 |
6. Let the complex number $z$ satisfy $|z|=1$, such that the equation $z x^{2}+2 \bar{z} x+2=0$ has real roots, then the sum of such complex numbers $z$ is $\qquad$ . | -\dfrac{3}{2} | 5/8 |
Circles \( S_1 \) and \( S_2 \) intersect at points \( A_1 \) and \( A_4 \), circles \( S_2 \) and \( S_3 \) intersect at points \( A_2 \) and \( A_5 \), circles \( S_3 \) and \( S_1 \) intersect at points \( A_3 \) and \( A_6 \). A broken line \( M_1 M_2 M_3 M_4 M_5 M_6 M_7 \) has the property that each segment \( M_k... | M_1 \text{ and } M_7 \text{ coincide} | 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: `Arnold`, `Alice`, `Peter`, `Eric`
- People have unique favorite music gen... | pop | 4/8 |
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`, `Peter`, `Eric`
- Each person prefers a unique type of vacation:... | grilled cheese | 1/8 |
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`, `Eric`, `Peter`
- Everyone has something unique ... | 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: `Arnold`, `Eric`, `Peter`, `Carol`, `Bob`, `Alice`
- Everyone has a favori... | 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`, `Peter`, `Arnold`, `Eric`
- People own unique car models: `honda ... | honda civic | 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: `Carol`, `Bob`, `Alice`, `Peter`, `Eric`, `Arnold`
- Each person has a uni... | 5 | 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`, `Arnold`, `Peter`
- People have unique hair colors... | roses | 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: `Carol`, `Bob`, `Alice`, `Eric`, `Arnold`, `Peter`
- Each person has a uni... | yellow | 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... | mystery | 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... | 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: `Arnold`, `Eric`, `Alice`, `Peter`
- People own unique car models: `honda ... | white | 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: `Alice`, `Arnold`, `Peter`, `Carol`, `Bob`, `Eric`
- Each person prefers a... | prince | 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: `Peter`, `Arnold`, `Eric`, `Bob`, `Carol`, `Alice`
- Each person prefers a... | 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: `Eric`, `Bob`, `Carol`, `Arnold`, `Peter`, `Alice`
- Everyone has a unique... | water | 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: `Eric`, `Arnold`, `Peter`, `Bob`, `Alice`, `Carol`
- People have unique fa... | 5 | 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: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`, `Carol`
- People have unique fa... | 3 | 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: `Arnold`, `Carol`, `Peter`, `Eric`, `Bob`, `Alice`
- Each person lives in ... | mediterranean | 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`, `Eric`, `Arnold`
- Each person has a unique birthday month: `jan`... | 3 | 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: `Eric`, `Peter`, `Bob`, `Alice`, `Arnold`
- People have unique heights: `v... | 5 | 0/8 |
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.