problem stringlengths 10 7.44k | answer stringlengths 1 270 | difficulty stringclasses 8
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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 ... | 4 | 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: `Eric`, `Arnold`, `Peter`
- The people keep unique animals: `horse`, `bird... | Holly | 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`, `Alice`, `Arnold`, `Peter`, `Eric`
- Each person has a unique level... | high school | 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... | 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: `Alice`, `Eric`, `Peter`, `Bob`, `Arnold`
- People have unique favorite sp... | Peter | 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: `Arnold`, `Peter`, `Eric`, `Alice`, `Bob`
- Each person has a unique hobby... | 4 | 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`, `Arnold`, `Peter`
- The people keep unique animals: `cat`... | stew | 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: `Arnold`, `Eric`, `Peter`
- Each person has a unique level of education: `... | milk | 2/8 |
4. The continuation of the height $B H$ of triangle $A B C$ intersects the circumscribed circle around it at point $D$ (points $B$ and $D$ lie on opposite sides of line $A C$). The degree measures of arcs $A D$ and $C D$, not containing point $B$, are $120^{\circ}$ and $90^{\circ}$, respectively. Determine in what rati... | 1:\sqrt{3} | 0/8 |
Example 2. Given that $D, F$ are points on the sides $A B$, $A C$ of $\triangle A B C$ respectively, and $A D: D B=C F: F A=2: 3$. Connect $D F$ to intersect the extension of side $B C$ at point $E$. Then, $E F: F D=$ $\qquad$
(3rd Zu Chongzhi Cup Junior High School Mathematics Invitational Competition) | 2: 1 | 2/8 |
1. In a computer game, a turtle moves across a grid on the computer screen, which contains 5 columns and 7 rows. Initially, it is located at the bottom-left corner of the screen - on the cell with coordinates $(0,0)$. If the program instructs the turtle to move off the screen, it reappears on the opposite side - for ex... | (2,6) | 1/8 |
2. In the room, there are knights who always tell the truth, and liars who always lie. 10 of them said: "In this room, there are more knights than liars." 15 said: "In this room, there are more liars than knights." The remaining 25 said: "In this room, there are an equal number of liars and knights." How many liars cou... | 25;35 | 0/8 |
4. Adam and Bohouš participated in a tournament played in a round-robin system, where each player was supposed to play one match per day. However, Adam and Bohouš were the only ones who did not complete the tournament due to illness. Bohouš withdrew five days earlier than Adam. In total, 350 matches were played. How ma... | 15 | 0/8 |
10.208. Perpendiculars are drawn from the vertex of the acute angle of a rhombus to the lines containing the sides of the rhombus to which this vertex does not belong. The length of each perpendicular is 3 cm, and the distance between their bases is $3 \sqrt{3}$ cm. Calculate the lengths of the diagonals of the rhombus... | 6 | 5/8 |
2. The eight-digit number $\square 2 \square 0 \square 2 \square 2$ (the digits in $\square$ can repeat) is a multiple of 72. There are $\qquad$ such eight-digit numbers. | 200 | 4/8 |
2. (12 points) In a family, there are four children of different ages. Their total age is 31 years. Four years ago, the total age of all the children in the family was 16 years, 7 years ago it was 8 years, and 11 years ago it was 1 year. How old are the children at present? (Age is always expressed as a whole number of... | 3 | 0/8 |
| $A$ | $B$ | $C$ | $D$ | $E$ |
| :--- | :--- | :--- | :--- | :--- |
| $E$ | $D$ | $C$ | $B$ | $A$ |
| $F$ | $F$ | $F$ | $F$ | $F$ |$+$
A fragment of a conversation on the beach:
Feri: Don't rush me now, I need to write digits here instead of the letters (Fig. 1), so that the addition is correct, the same digit for t... | 32 | 0/8 |
8. Find all positive integers $a$ such that for any positive integer $n \geqslant 5$, we have $\left(2^{n}-n^{2}\right) \mid\left(a^{n}-n^{a}\right)$. | 2 | 0/8 |
On colors some cells of an $8 \times 8$ chessboard in red. How many cells can be colored at most if we want there to be no red tromino? How many cells can be colored at least if we want every tromino to have at least one red cell? | 32 | 0/8 |
Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. It is given that there exist points $X$ and $Y$ on the circumference of $\omega$ such that $\angle BXC=\angle BYC=90^\circ$. Suppose further that $X$, $I$, and $Y$ are collinear. If $AB=80$ and $AC=97$, compute the length of $BC$. | 59 | 1/8 |
Let $n$ and $k$ be two integers which are greater than $1$. Let $a_1,a_2,\ldots,a_n,c_1,c_2,\ldots,c_m$ be non-negative real numbers such that
i) $a_1\ge a_2\ge\ldots\ge a_n$ and $a_1+a_2+\ldots+a_n=1$;
ii) For any integer $m\in\{1,2,\ldots,n\}$, we have that $c_1+c_2+\ldots+c_m\le m^k$.
Find the maximum of $c_1a_1^k+c... | 1 | 4/8 |
[ Arithmetic. Mental calculation, etc.]
$[\quad$ Invariants $]$
The rabbits are sawing the log again, but now both ends of the log are secured. Ten middle pieces fell, while the two end pieces remained secured. How many cuts did the rabbits make? | 11 | 3/8 |
36. Mathematician Gauss invented the floor function $[x]$ when studying integer problems, denoting by $[x]$ the greatest integer not exceeding $x$.
Question: When the value of the natural number $n$ is taken as $1, 2, 3, \ldots, 2019$, how many different possible values does $\left[\frac{n}{2}\right]+\left[\frac{n}{3}\... | 1347 | 4/8 |
8. On the edge $AS$ of the tetrahedron $S-ABC$, mark points $M, N$ such that $AM=MN=NS$. If the areas of $\triangle ABC$, $\triangle MBC$, and $\triangle SBC$ are $1$, $2$, and $\sqrt{37}$, respectively, find the area of $\triangle NBC$. | 4 | 4/8 |
[ Concerning the sphere ] $[$ Regular tetrahedron $]$
Four spheres of radius 1 touch each other pairwise. Find the radius of the sphere that touches all four spheres. | \sqrt{6} - 2 | 0/8 |
\( ABCD \) is a cyclic quadrilateral. A perpendicular to \( BA \) erected at point \( A \) intersects line \( CD \) at point \( M \); a perpendicular to \( DA \) erected at point \( A \) intersects line \( BC \) at point \( N \). Prove that \( MN \) passes through the center of the circle. | MN \text{ passes through the center of the circle} | 4/8 |
2. In $\square A B C D$, $\angle B<90^{\circ}, A B<B C$. From point $D$ draw tangents to the circumcircle $\Gamma$ of $\triangle A B C$, the points of tangency are $E$ and $F$. It is known that $\angle E D A=\angle F D C$. Find $\angle A B C$ | 60^\circ | 4/8 |
Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Suppose that $\angle BAC =60^\circ$ and $O$ lies on $\gamma$. If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, compute $100a+b$.
[i]Proposed by Kaan Dokmeci[/i] | 408 | 3/8 |
3. In triangle $A B C$, point $M$ is the midpoint of $A C$, moreover, $B C=2 A C / 3$ and $\angle B M C=2 \angle A B M$. Find the ratio $A M / A B$. | \dfrac{3\sqrt{5}}{10} | 5/8 |
11. Given a regular quadrilateral pyramid $V-A B C D$ with all edges equal to $a$, the midpoints of the lateral edges $V B$ and $V D$ are $H$ and $K$, respectively. If the plane passing through points $A$, $H$, and $K$ intersects the lateral edge $V C$ at $L$, then the area of quadrilateral $A H L K$ is $\qquad$ | \dfrac{a^2 \sqrt{5}}{6} | 0/8 |
## Task B-1.1.
Write the expression $\left[27^{-2 m+3} \cdot\left(\frac{1}{9}\right)^{3-2 m}\right]^{-2}: 81^{1+m}-6 \cdot 81^{-3}$, where $m$ is an integer, in the form of a power with a positive exponent. | \left(\frac{1}{3}\right)^{11} | 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`, `Alice`, `Eric`, `Peter`
- They all have a unique favorite flowe... | oneplus 9 | 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`, `Arnold`, `Eric`, `Bob`, `Peter`
- Everyone has a favorite smooth... | grilled cheese | 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`, `Peter`, `Alice`
- People have unique heights: `very sho... | tall | 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`, `Arnold`, `Eric`, `Alice`, `Bob`
- Each person has a unique level... | Samantha | 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`, `Arnold`, `Peter`
- Each person prefers a unique type of vacation:... | tesla model 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: `Arnold`, `Alice`, `Bob`, `Eric`, `Peter`
- Each person has a favorite col... | iphone 13 | 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`, `Eric`, `Peter`, `Alice`, `Bob`, `Arnold`
- Everyone has somethin... | green | 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... | 1 | 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`, `Arnold`, `Eric`, `Peter`
- Each person has a unique level of edu... | 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: `Alice`, `Eric`, `Bob`, `Arnold`, `Peter`, `Carol`
- People have unique fa... | 6 | 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`, `Alice`, `Arnold`, `Bob`, `Peter`, `Carol`
- 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: `Alice`, `Bob`, `Arnold`, `Eric`, `Peter`
- People have unique favorite bo... | mystery | 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`, `Bob`, `Arnold`, `Carol`, `Eric`, `Alice`
- Everyone has somethin... | master | 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: `Arnold`, `Eric`, `Peter`, `Bob`, `Alice`
- Everyone has a favorite smooth... | 1 | 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`, `Arnold`, `Peter`
- People own unique car models: `ford f150`, `te... | mountain | 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: `Eric`, `Peter`, `Arnold`, `Bob`, `Carol`, `Alice`
- The people keep uniqu... | 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`, `Arnold`, `Alice`, `Peter`
- The mothers' names in different house... | Kailyn | 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`, `Arnold`, `Bob`, `Alice`, `Carol`
- The people keep uniqu... | 4 | 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 has a uni... | sept | 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`, `Alice`, `Peter`, `Eric`
- Each person has a unique birthday mon... | sept | 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: `Arnold`, `Eric`, `Peter`
- People have unique favorite music genres: `pop... | 1 | 5/8 |
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`
- Each person prefers a unique type of vacation:... | 1 | 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`, `Eric`, `Peter`
- People have unique favorite book genres: `scie... | Peter | 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`
- Each person has a unique hobby: `cook... | roses | 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`, `Peter`, `Arnold`, `Eric`, `Alice`, `Carol`
- People have unique ha... | high school | 0/8 |
There are 2 houses, numbered 1 to 2 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`
- Each person lives in a unique style of house: `victoria... | very short | 2/8 |
Let \( ABC \) be an acute triangle and \( O \) be its circumcenter. Let \( D \) be the midpoint of \( [AB] \). The circumcircle of \( \triangle ADO \) meets \( [AC] \) at \( A \) and \( E \). If \( |AE|=7 \), \( |DE|=8 \), and \( m(\widehat{AOD}) = 45^\circ \), what is the area of \( \triangle ABC \) in the form \( x \... | 56 | 2/8 |
Let $(1+\sqrt{2})^{2012}=a+b\sqrt{2}$, where $a$ and $b$ are integers. Find the greatest common divisor of $b$ and $81$. | 3 | 3/8 |
Let \(M\) be the intersection of the diagonals \(AC\) and \(BD\) of cyclic quadrilateral \(ABCD\). If \(|AB|=5\), \(|CD|=3\), and \(m(\widehat{AMB}) = 60^\circ\), find the circumradius of the quadrilateral. If the circumradius is expressed in the form \(\frac{a\sqrt{b}}{c}\), where \(a\), \(b\), and \(c\) are integers,... | 13 | 4/8 |
Two obvious approximations to the length of the perimeter of the ellipse with semi-axes $a$ and $b$ are $\pi(a+b)$ and $2 \pi(a b)^{1/2}$. Which one comes nearer the truth when the ratio $b / a$ is very close to 1? Write your answer in the form of $\pi(x+y)$, where $x$ and $y$ are integers. Find the value of $x + y$. | 2 | 2/8 |
The solution of the equation \(7^{x+7} = 8^x\) can be expressed in the form \(x = \log_b 7^7\). The original answer is in the format \(\frac{k}{m}\). Please find the value of \(k + m\). | 15 | 2/8 |
Let \(M\) be the intersection of diagonals of the convex quadrilateral \(ABCD\), where \(m(\widehat{AMB})=60^\circ\). Let the points \(O_1\), \(O_2\), \(O_3\), \(O_4\) be the circumcenters of the triangles \(ABM\), \(BCM\), \(CDM\), \(DAM\), respectively. The original answer is in the form \(\frac{k}{m}\), where k and ... | 5 | 1/8 |
The Hawks scored a certain number of points, and the total points scored by both teams together is 82. If the difference between the points scored by the Eagles and the Hawks is 18, and the spectator claimed that the Hawks scored 40 points, then determine the actual number of points the Hawks scored. | 32 | 5/8 |
Consider the set of all four-digit rising numbers using the digits 1 through 7. Find the digit that the 35th number in the list from smallest to largest does not contain. | 3 | 0/8 |
In an isosceles triangle, one of the angles measures $60^\circ$. Determine the sum of the three possible values of another angle $y^\circ$ in the triangle. | 180^\circ | 4/8 |
Consider a modified finite sequence of four-digit integers where the tens, hundreds, and units digits of each term are, respectively, the thousands, hundreds, and tens digits of the next term, and the tens, hundreds, and units digits of the last term are, respectively, the thousands, hundreds, and tens digits of the fi... | 101 | 3/8 |
Jonas sets his watch correctly at 8:00 AM and notices that his watch reads 9:48 AM at the actual time of 10:00 AM. Assuming his watch loses time at a constant rate, calculate the actual time when his watch will first read 5:00 PM. | 6:00 PM | 0/8 |
Square $ABCD$ has its vertex $A$ at the origin of the coordinate system, and side length $AB = 2$. Vertex $E$ of isosceles triangle $\triangle ABE$, where $AE=BE$, is inside the square. A circle is inscribed in $\triangle ABE$ tangent to side $AB$ at point $G$, exactly at the midpoint of $AB$. Find the area of $\triang... | 1 | 3/8 |
Jack drove 150 miles in 2.5 hours. His average speed during the first hour was 50 mph. After a 15-minute stop, he resumed travel for another hour at an average speed of 55 mph. Calculate his average speed, in mph, during the last 30 minutes. | 90 | 3/8 |
The interior of a quadrilateral is bounded by the graphs of $(x+by)^2 = 9b^2$ and $(bx-y)^2 = 4b^2$, where $b$ is a positive real number. Determine the area of this region in terms of $b$, valid for all $b > 0$. | \frac{24b^2}{1 + b^2} | 5/8 |
In our number system, the base is ten. If the base were changed to seven, count the twenty-fifth number in the new base. | 34_7 | 0/8 |
Twenty-five percent of the audience listened to the entire 90-minute talk, and fifteen percent did not pay attention at all. Of the remainder, 40% caught half of the talk, and the rest heard only one fourth of it. Calculate the average time in minutes the talk was heard by the audience members. | 41.4 | 5/8 |
Calculate the sum $E(1)+E(2)+E(3)+\cdots+E(500)$, where $E(n)$ denotes the sum of the even digits of $n$. For example, $E(5681) = 6 + 8 = 14$.
A) 2000
B) 2200
C) 2400
D) 2500
E) 2600 | 2600 | 0/8 |
Given circle $O$, point $C$ is on the opposite side of diameter $\overline{AB}$ from point $A$, and point $D$ is on the same side as point $A$. Given $\angle AOC = 40^{\circ}$, and $\angle DOB = 60^{\circ}$, calculate the ratio of the area of the smaller sector $COD$ to the area of the circle. | \frac{4}{9} | 1/8 |
7. (4 points) With the number written on the board, one of the following operations is allowed:
1) If there is a digit in the original number that is not equal to 9 and has two neighboring digits greater than 0, you can increase this digit by 1, and decrease the neighboring digits by 1.
2) Subtract 1 from any non-zero ... | 3 | 0/8 |
Given the figure, $\odot O$ is the circumcircle of $\triangle A B C$. The circle $\odot J$ is inscribed in $\odot O$ and tangential to $A B$ and $A C$ at points $D$ and $E$, respectively. The line segment $F G$ is tangent to $\odot O$ at point $A$ and satisfies $A F = A G = A D$. The circumcircle of $\triangle A F B$ i... | \text{The circumcircle of } \triangle ASG \text{ is tangent to } \odot J | 4/8 |
$\underline{\text { Tolkpy A.K. }}$
A circle is divided into seven arcs such that the sum of any two adjacent arcs does not exceed $103^{\circ}$.
Name the largest number $A$ such that in any such division, each of the seven arcs contains at least $A^{\circ}$. | 51 | 5/8 |
As shown in Figure 2, given a square $ABCD$, extend $BC$ and $DC$ to $M$ and $N$ respectively, such that $S_{\triangle QMN} = S_{\text{square } ABCD}$. Determine the degree measure of $\angle MAN$. | 45 | 2/8 |
Choose positive integers \( b_{1}, b_{2}, \ldots \) satisfying
\[
1=\frac{b_{1}}{1^{2}}>\frac{b_{2}}{2^{2}}>\frac{b_{3}}{3^{2}}>\frac{b_{4}}{4^{2}}>\cdots
\]
and let \( r \) denote the largest real number satisfying \( \frac{b_{n}}{n^{2}} \geq r \) for all positive integers \( n \). What are the possible values of \( r... | [0, \frac{1}{2}] | 0/8 |
II. (40 points) Find all positive integers $m, n$ such that $\frac{m^{3}+n^{3}-m^{2} n^{2}}{(m+n)^{2}}$ is a non-negative integer. | (2, 2) | 2/8 |
Kazitsyna T.V.
In triangle $ABC$, points $X$ and $Y$ are taken on sides $AC$ and $BC$ such that $\angle ABX = \angle YAC$, $\angle AYB = \angle BXC$, and $XC = YB$. Find the angles of triangle $ABC$. | 60^\circ | 4/8 |
15. Given the parabola $y=a x^{2}$ passes through the point $P(-1,1)$, a line $l$ with a positive slope is drawn through the point $Q\left(-\frac{1}{2}, 0\right)$ intersecting the parabola at points $M, N$ (point $M$ is between $Q$ and $N$). A line parallel to the $x$-axis is drawn through point $M$, intersecting $O P$... | S_1 > 3S_2 | 4/8 |
Define $ n!!$ to be $ n(n\minus{}2)(n\minus{}4)\ldots3\cdot1$ for $ n$ odd and $ n(n\minus{}2)(n\minus{}4)\ldots4\cdot2$ for $ n$ even. When $ \displaystyle \sum_{i\equal{}1}^{2009} \frac{(2i\minus{}1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $ 2^ab$ with $ b$ odd. Find $ \displaystyle... | 401 | 1/8 |
# Task 2. (3 points)
In a certain company, the $20 \%$ most useful employees perform $80 \%$ of the work. What is the smallest percentage of work that the $40 \%$ most useful employees can perform?
We will consider an employee more useful if they perform more work. | 85 | 1/8 |
41. $y=\lg (x-2)$.
The above text is translated into English as follows, retaining the original text's line breaks and format:
41. $y=\lg (x-2)$. | (2, \infty) | 1/8 |
5. [5 points] Given the numbers $\log _{\sqrt{29-x}}\left(\frac{x}{7}+7\right), \log _{(x+1)^{2}}(29-x), \log _{\sqrt{\frac{x}{7}+7}}(-x-1)$. For which $x$ are two of these numbers equal, and the third one greater than them by 1? | -7 | 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`, `Bob`, `Carol`, `Eric`, `Peter`, `Arnold`
- Each person has a uni... | pop | 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`, `Eric`, `Peter`
- They all have a uniqu... | 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: `Arnold`, `Alice`, `Eric`, `Bob`, `Peter`
- Each person lives in a unique ... | huawei p50 | 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`, `Arnold`, `Peter`
- The people keep unique animals: `horse`, `bird... | red | 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: `Alice`, `Arnold`, `Bob`, `Eric`, `Peter`
- Each person has a unique hobby... | knitting | 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`
- Each person lives in a unique ... | 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: `Peter`, `Eric`, `Arnold`, `Bob`, `Alice`
- Each person has a unique favor... | modern | 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`
- Everyone has something unique ... | 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`, `Peter`, `Eric`, `Alice`, `Arnold`
- Everyone has a favorite smooth... | Peter | 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`, `Arnold`, `Eric`, `Peter`
- Each mother is accompanied by their c... | 3 | 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: `Eric`, `Arnold`, `Peter`
- Everyone has something unique for lunch: `spag... | Peter | 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`, `Peter`, `Alice`, `Eric`
- Each person has a unique favor... | 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: `Arnold`, `Bob`, `Alice`, `Eric`, `Peter`
- People have unique heights: `v... | very short | 2/8 |
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.