task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | 7.189. $\log _{x} m \cdot \log _{\sqrt{m}} \frac{m}{\sqrt{2 m-x}}=1$. | m | 38 | 1 |
math | ## Task A-3.1.
Determine all pairs of real numbers $(x, y)$ such that $x, y \in\left[0, \frac{\pi}{2}\right]$ for which
$$
\frac{2 \sin ^{2} x+2}{\sin x+1}=3+\cos (x+y)
$$ | (x,y)=(\frac{\pi}{2},\frac{\pi}{2}) | 75 | 17 |
math | Consider equilateral triangle $ABC$ with side length $1$. Suppose that a point $P$ in the plane of the triangle satisfies \[2AP=3BP=3CP=\kappa\] for some constant $\kappa$. Compute the sum of all possible values of $\kappa$.
[i]2018 CCA Math Bonanza Lightning Round #3.4[/i] | \frac{18\sqrt{3}}{5} | 82 | 13 |
math | 8. Let the area of $\triangle A B C$ be $1, D$ be a point on side $B C$, and $\frac{B D}{D C}=\frac{1}{2}$. If a point $E$ is taken on side $A C$ such that the area of quadrilateral $A B D E$ is $\frac{4}{5}$, then the value of $\frac{A E}{E C}$ is $\qquad$. | \frac{7}{3} | 99 | 7 |
math | ## Task B-2.4.
Determine all natural numbers $x$ that satisfy the system of inequalities
$$
x-\frac{6}{x} \geq 1, \quad \frac{1}{x-6} \leq 0
$$ | x\in{3,4,5} | 57 | 10 |
math | Let $u$ and $v$ be integers satisfying $0<v<u.$ Let $A=(u,v),$ let $B$ be the reflection of $A$ across the line $y=x,$ let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the y-axis. The area of pentagon $ABCDE$ is 451. Find $u+v.$ | 21 | 109 | 2 |
math | 191. Perform the operations: a) $(5+3 i)(5-3 i) ;$ b) $(2+5 i)(2-$ $-5 i)$; c) $(1+i)(1-i)$. | 34,29,2 | 49 | 7 |
math | 7. Let $\alpha, \beta$ be a pair of conjugate complex numbers. If $|\alpha-\beta|=2 \sqrt{3}$ and $\frac{\alpha}{\beta^{2}}$ is a real number, then $|\alpha|=$ $\qquad$ . | 2 | 59 | 1 |
math | 5. Given real numbers $x, y$ satisfy
$$
x^{2}+3 y^{2}-12 y+12=0 \text {. }
$$
then the value of $y^{x}$ is $\qquad$ | 1 | 52 | 1 |
math | 2. If the equation with respect to $x$
$$
\sqrt{1+a-x}-a+\sqrt{x}=0
$$
has real solutions, then the range of positive real number $a$ is $\qquad$ | \left[\frac{1+\sqrt{5}}{2}, 1+\sqrt{3}\right] | 48 | 23 |
math | Example 5. Investigate the convergence of the infinite product
$$
\prod_{k=1}^{\infty}\left(1-\frac{1}{k+1}\right)=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right) \ldots\left(1-\frac{1}{k+1}\right) \ldots
$$ | 0 | 89 | 1 |
math | 7. (10 points) In a math competition, there are three problems: $A, B, C$. Among the 39 participants, each person answered at least one question correctly. Among those who answered $A$ correctly, those who only answered $A$ are 5 more than those who answered other questions as well; among those who did not answer $A$ correctly, those who answered $B$ correctly are twice as many as those who answered $C$ correctly; it is also known that the number of people who only answered $A$ correctly is equal to the sum of those who only answered $B$ correctly and those who only answered $C$ correctly. Therefore, the maximum number of people who answered $A$ correctly is $\qquad$. | 23 | 158 | 2 |
math | 19. Let $a_{k}=\frac{k+1}{(k-1)!+k!+(k+1)!}$, then $a_{2}+a_{3}+\cdots+a_{1991}+\frac{1}{1992!}=$ | \frac{1}{2} | 63 | 7 |
math | 1. In the set of real numbers, solve the equation
$$
\log _{1997}\left(\sqrt{1+x^{2}}+x\right)=\log _{1996}\left(\sqrt{1+x^{2}}-x\right)
$$ | 0 | 62 | 1 |
math | 11. Let $f(x)=x^{2}+6 x+c$ for all real numbers $x$, where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly 3 distinct real roots? | \frac{11-\sqrt{13}}{2} | 55 | 14 |
math | 1A. a) Let $f$ be a real function defined on $\mathbb{R}$. Find the real function $g(x)$, if $f(x)=3x+2$ and $f\left(x^{2}+x g(x)\right)=3 x^{2}+6 x+5$, for every $x \in \mathbb{R} \backslash\{0\}$.
b) Let $f(x)=x+2$, for every $x \in \mathbb{R}$. Find the function $g(x)$ such that $f(g(f(x)))=5 x-1$. | (x)=2+\frac{1}{x},\,()=5t-13 | 134 | 19 |
math | Our school's ball last year allocated 10% of its net income to the acquisition of specialized clubs, and the remaining portion exactly covered the rental fee for the sports field. This year, we cannot issue more tickets, and the rental fee remains unchanged, so the share for the clubs could only be increased by raising the ticket price. By what percentage would the ticket price need to be increased to make the share 20%? | 12.5 | 88 | 4 |
math | 3. The equation $x^{2}+a x+3=0$ has two distinct roots $x_{1}$ and $x_{2}$; in this case,
$$
x_{1}^{3}-\frac{99}{2 x_{2}^{2}}=x_{2}^{3}-\frac{99}{2 x_{1}^{2}}
$$
Find all possible values of $a$. | -6 | 92 | 2 |
math | In the decimal expansion of $\sqrt{2}=1.4142\dots$, Isabelle finds a sequence of $k$ successive zeroes where $k$ is a positive integer.
Show that the first zero of this sequence can occur no earlier than at the $k$-th position after the decimal point. | k | 65 | 2 |
math | 3. Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0), F$ is the right focus of the ellipse, and $AB$ is a chord passing through the center $O$. Then the maximum value of the area of $\triangle ABF$ is $\qquad$. | b \sqrt{a^{2}-b^{2}} | 77 | 12 |
math | 4. (1990 National High School League Question) Find the maximum value of the function $f(x)=\sqrt{x^{4}-3 x^{2}-6 x+13}-\sqrt{x^{4}-x^{2}+1}$. | \sqrt{10} | 55 | 6 |
math | [ Algebra and arithmetic (miscellaneous).] $[$ Sequences (miscellaneous) ]
Find the value of the expression $1!* 3-2!* 4+3!* 5-4!* 6+\ldots-2000!* 2002+2001!$. | 1 | 65 | 1 |
math | Example 2.3.5 Derangement Problem: Find the number of permutations $\left\{x_{1}, x_{2}, \cdots, x_{n}\right\}$ that satisfy $\forall i$, $x_{i} \neq i,(i=1,2, \cdots, n)$, denoted as $D_{n}$. | D_{n}=n!(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+(-1)^{n}\frac{1}{n!}) | 77 | 48 |
math | 6. If a student buys 5 notebooks, 20 kn will remain, but if he buys 8 notebooks, 16 kn will be missing. How many kuna did the student have? | 80\mathrm{kn} | 42 | 7 |
math | 15. How many sequences of $0 \mathrm{~s}$ and $1 \mathrm{~s}$ are there of length 10 such that there are no three 0s or 1 s consecutively anywhere in the sequence? | 178 | 52 | 3 |
math | 6. Let $a, b, c$ be nonnegative rational numbers. If $a(b+c)=36, b(a+c)=50$ and $c(a+b)=56$, what is $a b c$? | 105 | 49 | 3 |
math | Find the natural integers $n$ such that the fraction
$$
\frac{21 n+4}{14 n+1}
$$
is irreducible.
The second theorem is due to Carl Friedrich Gauss.
Theorem 2.2.18 (Gauss's Theorem).
Let $a, b$, and $c$ be three natural integers such that $a$ divides $b \times c$. If $a$ and $b$ are coprime, then $a$ divides $c$.
Proof. Since $a \wedge b=1$, Bézout's theorem indicates that there exist two integers $m$ and $n$ such that $a \times m+b \times n=1$. Let $k$ be an integer such that $b \times c=a \times k$. It then appears that
$$
c=c \times(a \times m+b \times n)=a \times c \times m+b \times c \times n=a \times c \times m+a \times k \times n
$$
is indeed a multiple of $a$. | n\not\equiv1(\bmod5) | 232 | 11 |
math | 17. $\log _{x} \sqrt{5}+\log _{x}(5 x)-2.25=\left(\log _{x} \sqrt{5}\right)^{2}$. | x_{1}=\sqrt[5]{5},x_{2}=5 | 46 | 16 |
math | Find all such integers $x$ that $x \equiv 3(\bmod 7), x^{2} \equiv 44\left(\bmod 7^{2}\right), x^{3} \equiv 111\left(\bmod 7^{3}\right)$. | x\equiv17(\bmod343) | 64 | 12 |
math | Two disks of radius 1 are drawn so that each disk's circumference passes through the center of the other disk. What is the circumference of the region in which they overlap? | \frac{4\pi}{3} | 35 | 9 |
math | ## Subject 3.
Consider the triangle $ABC$ where $m(\widehat{ABC})=30^{\circ}$ and $m(\widehat{ACB})=15^{\circ}$. Point $M$ is the midpoint of side $[BC]$. Determine the measure of the angle $\widehat{AMB}$.
| Details of solution | Associated grading |
| :--- | :---: |
| Consider the ray (Cx such that ray (CA is the angle bisector of $\widehat{BCx}$. Let $\{F\}=AB \cap (Cx$. Triangle $FBC$ is isosceles with base $[BC]$, so $FM \perp BC$ and $m(\widehat{BFM})=m(\widehat{CMF})=m(\widehat{BFx})=60^{\circ}$. | $\mathbf{3 p}$ |
| For triangle $CMF$, ray (CA is the internal angle bisector, and ray (FA is the external angle bisector. We deduce that ray (MA is also the external angle bisector for triangle $CMF$. | $\mathbf{3 p}$ |
| It follows that $m(\widehat{AMB})=45^{\circ}$. | $\mathbf{1 p}$ | | 45 | 281 | 2 |
math | Example 12. Solve the equation
$$
\sqrt{2 x+5}+\sqrt{x-1}=8
$$ | 10 | 28 | 2 |
math | 4. A two-digit number was increased by 3, and it turned out that the sum is divisible by 3. When 7 was added to this same two-digit number, the resulting sum was divisible by 7. If 4 is subtracted from this two-digit number, the resulting difference is divisible by four. Find this two-digit number. | 84 | 72 | 2 |
math | ## Task A-1.4.
In the plane, one hundred circles with the same center and radii $1,2, \ldots, 100$ are drawn. The smallest circle is colored red, and each of the 99 circular rings bounded by two circles is colored either red or green such that adjacent areas are of different colors.
Determine the total area of the green-colored regions. | 5050\pi | 85 | 6 |
math | Example 1. Find $\int \frac{\operatorname{sh} x d x}{\sqrt{1+\operatorname{ch}^{2} x}}$
Solution It is not hard to notice the possibility of bringing $\operatorname{ch} x$ under the differential sign and applying the tabular formula 13:
$$
\int \frac{\operatorname{sh} x d x}{\sqrt{1+\operatorname{ch}^{2} x}}=\int \frac{d(\operatorname{ch} x)}{\sqrt{1+\operatorname{ch}^{2} x}}=\ln \left(\operatorname{ch} x+\sqrt{1+\operatorname{ch}^{2} x}\right)+C
$$
$$
\text { Example } 2 . \text { Find } \int \frac{d x}{\operatorname{ch} x} \text {. }
$$ | 2\operatorname{arctg}e^{x}+C | 196 | 15 |
math | Oldjuk meg az $1+\cos 3 x=2 \cos 2 x$ egyenletet.
| x_{1}=2k\pi,\quadx_{2}=\frac{\pi}{6}+k\pi,\quadx_{3}=\frac{5\pi}{6}+k\pi | 25 | 44 |
math | Given a regular triangular pyramid $S A B C$. Point $S$ is the apex of the pyramid, $S A=2 \sqrt{3}, B C=3, B M$ is the median of the base of the pyramid, $A R$ is the height of triangle $A S B$. Find the length of segment $M R$.
Answer
## $\frac{9}{4}$
Given a unit cube $A B C D A 1 B 1 C 1 D 1$. Find the radius of the sphere passing through points $A, B, C 1$ and the midpoint of edge $B 1 C 1$. | \frac{\sqrt{14}}{4} | 138 | 11 |
math | Consider a fair coin and a fair 6-sided die. The die begins with the number 1 face up. A [i]step[/i] starts with a toss of the coin: if the coin comes out heads, we roll the die; otherwise (if the coin comes out tails), we do nothing else in this step. After 5 such steps, what is the probability that the number 1 is face up on the die? | \frac{37}{192} | 93 | 10 |
math | A trapezium $ ABCD$, in which $ AB$ is parallel to $ CD$, is inscribed in a circle with centre $ O$. Suppose the diagonals $ AC$ and $ BD$ of the trapezium intersect at $ M$, and $ OM \equal{} 2$.
[b](a)[/b] If $ \angle AMB$ is $ 60^\circ ,$ find, with proof, the difference between the lengths of the parallel sides.
[b](b)[/b] If $ \angle AMD$ is $ 60^\circ ,$ find, with proof, the difference between the lengths of the parallel sides.
[b][Weightage 17/100][/b] | 2\sqrt{3} | 150 | 8 |
math | Problem 1. Find all pairs $(x, y)$ of integers that satisfy the equation
$$
x^{2} y+y^{2}=x^{3} \text {. }
$$
(Daniel Paleka) | (0,0),(-4,-8) | 45 | 10 |
math | [ Motion problems ] [ Arithmetic. Mental calculation, etc. ]
If Anya walks to school and takes the bus back, she spends a total of 1.5 hours on the road. If she takes the bus both ways, the entire journey takes her 30 minutes. How much time will Anya spend on the road if she walks to and from school? | 2.5 | 75 | 3 |
math | 8.2. a) Agent 007 wants to encrypt his number using two natural numbers $m$ and $n$ such that $0.07=\frac{1}{m}+\frac{1}{n}$. Can he do this? b) Can his colleague, Agent 013, encrypt his number in a similar way? | )\frac{1}{20}+\frac{1}{50};b)\frac{1}{8}+\frac{1}{200} | 74 | 33 |
math | 1. If two prime numbers $p, q$ satisfy $3 p^{2}+5 q=517$, then $p+q=$ $\qquad$ | 103 \text{ or } 15 | 36 | 11 |
math | Daniel, Clarence, and Matthew split a \$20.20 dinner bill so that Daniel pays half of what Clarence pays. If Daniel pays \$6.06, what is the ratio of Clarence's pay to Matthew's pay?
[i]Proposed by Henry Ren[/i] | 6:1 | 58 | 3 |
math | In a $4 \times 4$ chessboard composed of 16 small squares, 8 of the small squares are to be colored black, such that each row and each column has exactly 2 black squares. There are $\qquad$ different ways to do this. | 90 | 57 | 2 |
math | 7. Variant 1.
Numbers $x$ and $y$ satisfy the equation $\frac{x}{x+y}+\frac{y}{2(x-y)}=1$. Find all possible values of the expression $\frac{5 x+y}{x-2 y}$, and in the answer, write their sum. | 21 | 65 | 2 |
math | Let $k$ be a fixed positive integer. The $n$th derivative of $\tfrac{1}{x^k-1}$ has the form $\tfrac{P_n(x)}{(x^k-1)^{n+1}}$, where $P_n(x)$ is a polynomial. Find $P_n(1)$. | (-1)^n n! k^n | 71 | 9 |
math | Sure, here is the translated text:
```
Solve the following system of equations:
$$
\begin{aligned}
& 3^{x+2}=\sqrt[y+1]{9^{8-y}} \\
& 2^{x-2}=\sqrt[y+1]{8^{3-y}}
\end{aligned}
$$
``` | 5,1 | 72 | 3 |
math | 3. (10 points) Each lead type is engraved with a digit. If printing twelve pages of a book, the lead types for page numbers used are the following 15: $1,2,3,4,5,6,7,8,9,1,0,1,1,1,2$.
Now, a new book is to be printed, and 2011 lead types for page numbers are taken from the warehouse, with some left over after typesetting. Then, this book can have at most $\qquad$ pages. The minimum number of leftover lead types is $\qquad$. | 706 | 131 | 3 |
math | 37th Putnam 1976 Problem B5 Find ∑ 0 n (-1) i nCi ( x - i ) n , where nCi is the binomial coefficient. Solution | n! | 42 | 2 |
math | Task 2 - 260712 In the material issuance of a company, the keys of twelve padlocks have been mixed up due to a mishap.
Since only one of the twelve keys fits each padlock and only one padlock fits each key, which cannot be distinguished from each other externally, it must be determined which key belongs to which padlock.
Apprentice Bernd, who was assigned this task, thought: "Now I have to try twelve keys on twelve padlocks, so if I'm unlucky, I have to perform $12 \cdot 12=144$ tests."
However, his friend Uwe said that much fewer tests would suffice.
Determine the smallest number of tests with which one can certainly (i.e., even in the worst case) find the matching key for each padlock! | 66 | 173 | 2 |
math | ## Task B-2.3.
Given is the equation $x^{2}-p x+q=0$, where $p$ and $q$ are positive real numbers. If the difference between the solutions of the equation is 1, and the sum of the solutions is 2, calculate $p$ and $q$. | p=2,q=\frac{3}{4} | 68 | 11 |
math | 10.223. The perimeter of a right-angled triangle is 60 cm. Find its sides if the height drawn to the hypotenuse is 12 cm. | 15,20,25 | 39 | 8 |
math | 26. $5^{\lg x}-3^{\lg x-1}=3^{\lg x+1}-5^{\lg x-1}$. | 100 | 35 | 3 |
math | Example 27 ([38.5]) Find all positive integer pairs $\{a, b\}$, satisfying the equation
$$a^{b^{2}}=b^{a}.$$ | \{1,1\}, \{16,2\}, \{27,3\} | 40 | 23 |
math | Problem 5. Inside triangle ABC, two points are given. The distances from one of them to the lines AB, BC, and AC are 1, 3, and 15 cm, respectively, and from the other - 4, 5, and 11 cm. Find the radius of the circle inscribed in triangle ABC. | 7 | 72 | 1 |
math | 5. We know: $9=3 \times 3, 16=4 \times 4$, here, $9, 16$ are called "perfect squares". Among the first 300 natural numbers, if we remove all the "perfect squares", what is the sum of the remaining natural numbers? | 43365 | 68 | 5 |
math | 10.149. A circle with radius $R$ is divided into two segments by a chord equal to the side of the inscribed square. Determine the area of the smaller of these segments. | \frac{R^{2}(\pi-2)}{4} | 42 | 15 |
math | 3. Find the sum of the first 10 elements that are found both in the arithmetic progression $\{4,7,10,13, \ldots\}$ and in the geometric progression $\{10,20,40,80, \ldots\} \cdot(10$ points $)$ | 3495250 | 70 | 7 |
math | 【Question 6】
Among all the factors of $11!(11!=11 \times 10 \times \cdots \times 1)$, the largest factor that can be expressed as $6k+1$ (where $k$ is a natural number) is $\qquad$. | 385 | 65 | 3 |
math | 6.17 The sum of the first three terms of a geometric progression is 21, and the sum of their squares is 189. Find the first term and the common ratio of this progression. | b_{1}=3,q=2 | 44 | 8 |
math | We randomly place points $A, B, C$, and $D$ on the circumference of a circle, independently of each other. What is the probability that the chords $AB$ and $CD$ intersect? | \frac{1}{3} | 43 | 7 |
math | 3. If the complex number $z$ satisfies $|z|=2$, then the maximum value of $\frac{\left|z^{2}-z+1\right|}{|2 z-1-\sqrt{3} \mathrm{i}|}$ is . $\qquad$ | \frac{3}{2} | 58 | 7 |
math | 1.22 Divide a set containing 12 elements into 6 subsets, so that each subset contains exactly two elements. How many ways are there to do this? | 10395 | 35 | 5 |
math | 11. (20 points) Let $[x]$ denote the greatest integer not exceeding the real number $x$. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=1, a_{n+1}=1+\frac{1}{a_{n}}+\ln a_{n} \text {. }
$$
Let $S_{n}=\left[a_{1}\right]+\left[a_{2}\right]+\cdots+\left[a_{n}\right]$. Find $S_{n}$. | 2n-1 | 116 | 4 |
math | Example 20. From an urn containing 3 blue and 2 red balls, balls are drawn sequentially according to a random selection scheme without replacement. Find the probability $P_{k}$ that a red ball will appear for the first time on the $k$-th trial ( $k=1,2,3,4$ ). | p_{1}=0.4,p_{2}=0.3,p_{3}=0.2,p_{4}=0.1 | 70 | 28 |
math | Concerning the Homothety of Tangent Circles helps to solve the problem, $\quad]$
In a right triangle $ABC$, angle $C$ is a right angle, and side $CA=4$. A point $D$ is taken on the leg $BC$, such that $CD=1$. A circle with radius $\frac{\sqrt{5}}{2}$ passes through points $C$ and $D$ and is tangent at point $C$ to the circumcircle of triangle $ABC$. Find the area of triangle $ABC$. | 4 | 113 | 1 |
math | Let's say a positive integer is simple if it has only the digits 1 or 2 (or both). How many numbers less than 1 million are simple? | 126 | 34 | 3 |
math | 2. (5 points) The number of different divisors of 60 (excluding 1) is | 11 | 22 | 2 |
math | A wheel is rolled without slipping through $15$ laps on a circular race course with radius $7$. The wheel is perfectly circular and has radius $5$. After the three laps, how many revolutions around its axis has the wheel been turned through? | 21 | 51 | 2 |
math | 1. The fraction $\frac{1}{5}$ is written as an infinite binary fraction. How many ones are there among the first 2022 digits after the decimal point in such a representation? (12 points) | 1010 | 47 | 4 |
math | Find the largest $k$ such that for every positive integer $n$ we can find at least $k$ numbers in the set $\{n+1, n+2, ... , n+16\}$ which are coprime with $n(n+17)$. | k = 1 | 59 | 5 |
math | 11.43 Solve the equation $x+\lg \left(1+4^{x}\right)=\lg 50$. | 1 | 29 | 1 |
math | Problem 2. $n$ mushroom pickers went to the forest and brought a total of 338 mushrooms (it is possible that some of them did not bring any mushrooms home). Boy Petya, upon learning this, said: "Some two of them must have brought the same number of mushrooms!" For what smallest $n$ will Petya definitely be right? Don't forget to justify your answer. | 27 | 86 | 2 |
math | 6. Pete came up with all the numbers that can be formed using the digits 2, 0, 1, 8 (each digit can be used no more than once). Find their sum. | 78331 | 42 | 5 |
math | Let $a, b, x, y$ be positive real numbers such that $a+b=1$. Prove that $\frac{1}{\frac{a}{x}+\frac{b}{y}}\leq ax+by$
and find when equality holds. | \frac{1}{\frac{a}{x} + \frac{b}{y}} \leq ax + by | 57 | 27 |
math | Example 5. Solve the equation $\sqrt{x^{2}+6 x+36}$
$$
=\sqrt{x^{2}-14 x+76}+8 \text{. }
$$ | x=10 | 44 | 4 |
math | 51 Given $\alpha, \beta>0, x, y, z \in \mathbf{R}^{+}, x y z=2004$. Find the maximum value of $u$, where, $u=$
$$\sum_{\mathrm{oc}} \frac{1}{2004^{\alpha+\beta}+x^{\alpha}\left(y^{2 \alpha+3 \beta}+z^{2 \alpha+3 \beta}\right)} .$$ | 2004^{-(\alpha+\beta)} | 106 | 11 |
math | Example 1 Let $m, n$ be positive integers, find the minimum value of $\left|12^{m}-5^{m}\right|$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 7 | 58 | 1 |
math | 4.018. The denominator of the geometric progression is $1 / 3$, the fourth term of this progression is $1 / 54$, and the sum of all its terms is 121/162. Find the number of terms in the progression. | 5 | 59 | 1 |
math | Below are five facts about the ages of five students, Adyant, Bernice, Cici, Dara, and Ellis.
- Adyant is older than Bernice.
- Dara is the youngest.
- Bernice is older than Ellis.
- Bernice is younger than Cici.
- Cici is not the oldest.
Determine which of the five students is the third oldest. | Bernice | 81 | 2 |
math | 1. If real numbers $x, y$ satisfy $x^{2}-2 x y+5 y^{2}=4$, then the range of $x^{2}+y^{2}$ is $\qquad$ . | [3-\sqrt{5},3+\sqrt{5}] | 47 | 13 |
math | Find $x+y+z$ when $$a_1x+a_2y+a_3z= a$$$$b_1x+b_2y+b_3z=b$$$$c_1x+c_2y+c_3z=c$$ Given that $$a_1\left(b_2c_3-b_3c_2\right)-a_2\left(b_1c_3-b_3c_1\right)+a_3\left(b_1c_2-b_2c_1\right)=9$$$$a\left(b_2c_3-b_3c_2\right)-a_2\left(bc_3-b_3c\right)+a_3\left(bc_2-b_2c\right)=17$$$$a_1\left(bc_3-b_3c\right)-a\left(b_1c_3-b_3c_1\right)+a_3\left(b_1c-bc_1\right)=-8$$$$a_1\left(b_2c-bc_2\right)-a_2\left(b_1c-bc_1\right)+a\left(b_1c_2-b_2c_1\right)=7.$$
[i]2017 CCA Math Bonanza Lightning Round #5.1[/i] | \frac{16}{9} | 292 | 8 |
math | A quadrilateral $ABCD$ has a right angle at $\angle ABC$ and satisfies $AB = 12$, $BC = 9$, $CD = 20$, and $DA = 25$. Determine $BD^2$.
.
| 769 | 54 | 3 |
math | 7. Given a sequence of numbers: $1,3,3,3,5,3,7,3,9,3, \cdots . .$. , the sum of the first 100 terms of this sequence is $\qquad$ _. | 2650 | 55 | 4 |
math | Let's consider the inequality $ a^3\plus{}b^3\plus{}c^3<k(a\plus{}b\plus{}c)(ab\plus{}bc\plus{}ca)$ where $ a,b,c$ are the sides of a triangle and $ k$ a real number.
[b]a)[/b] Prove the inequality for $ k\equal{}1$.
[b]b) [/b]Find the smallest value of $ k$ such that the inequality holds for all triangles. | k = 1 | 106 | 5 |
math | 3. For any real numbers $x, y$, the inequality
$$
|x-1|+|x-3|+|x-5| \geqslant k(2-|y-9|)
$$
always holds. Then the maximum value of the real number $k$ is $\qquad$ | 2 | 67 | 1 |
math | $\begin{array}{l}\text { B. } \frac{\sqrt{1^{4}+2^{4}+1}}{1^{2}+2^{2}-1}+\frac{\sqrt{2^{4}+3^{4}+1}}{2^{2}+3^{2}-1}+\cdots+\frac{\sqrt{99^{4}+100^{4}+1}}{99^{2}+100^{2}-1} \\ =\end{array}$ | \frac{9999 \sqrt{2}}{200} | 115 | 17 |
math | Seven boys and three girls are playing basketball. I how many different ways can they make two teams of five players so that both teams have at least one girl? | 105 | 32 | 3 |
math | Danka had a paper flower with ten petals. On each petal, there was exactly one digit, and no digit was repeated on any other petal. Danka tore off two petals so that the sum of the numbers on the remaining petals was a multiple of nine. Then she tore off another two petals so that the sum of the numbers on the remaining petals was a multiple of eight. Finally, she tore off two more petals so that the sum of the numbers on the remaining petals was a multiple of ten.
Find the three sums that could have remained after each tearing. Determine all such triplets of sums.
(E. Novotná) | (36,32,20),(36,32,30),(36,24,20),(36,24,10) | 131 | 37 |
math | What is the area of a quadrilateral whose sides are: $a=52 \mathrm{~m}, b=56 \mathrm{~m}, c=33 \mathrm{~m}, d=39 \mathrm{~m}$ and one of its angles $\alpha=112^{\circ} 37^{\prime} 12^{\prime \prime} ?$ | 1774 | 86 | 4 |
math | Example 5 Let $n(\geqslant 3)$ be a positive integer, and $M$ be an $n$-element set. Find the maximum positive integer $k$ such that:
there exists a family $\psi$ of $k$ three-element subsets of $M$ such that the intersection of any two elements of $\psi$ (note: elements of $\psi$ are three-element sets) is non-empty. | C_{n-1}^{2} | 90 | 9 |
math | 9. Solve the equation $12 \sin x-5 \cos x=13$.
$$
12 \sin x-5 \cos x=13
$$ | x-\operatorname{arctg}\frac{5}{12}=\frac{\pi}{2}+2\pik | 38 | 28 |
math | Aliens from Lumix have one head and four legs, while those from Obscra have two heads and only one leg. If 60 aliens attend a joint Lumix and Obscra interworld conference, and there are 129 legs present, how many heads are there? | 97 | 60 | 2 |
math | [ Arithmetic. Mental calculation, etc. ] $[\quad$ Case enumeration $\quad]$
109 apples are distributed into bags. In some bags, there are $x$ apples, and in others, there are three apples.
Find all possible values of $x$, if the total number of bags is 20. | 10or52 | 67 | 5 |
math | 22. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that for all real $x$ satisfy the relations
$$
\underbrace{f(f(f \ldots f}_{13}(x) \ldots))=-x, \quad \underbrace{f(f(f \ldots f}_{8}(x) \ldots))=x
$$ | f(x)=-x | 86 | 5 |
math | Find the number of positive integers $n$ for which
(i) $n \leq 1991$;
(ii) 6 is a factor of $(n^2 + 3n +2)$. | 1328 | 47 | 4 |
math | Find the smallest exact square with last digit not $0$, such that after deleting its last two digits we shall obtain another exact square. | 121 | 27 | 3 |
math | |Column 15 Given the sets $A=\left\{x \mid x^{2}-x-2<0\right\}, B=\left\{x \mid x^{2}-a x-a^{2}<0\right.$, $a \in \mathbf{R}\}$. If $A \subseteq B$, find the range of real numbers for $a$. | (-\infty,-1-\sqrt{5})\cup(\frac{1+\sqrt{5}}{2},+\infty) | 82 | 29 |
math | 11.095. A sphere is circumscribed around a regular triangular prism, the height of which is twice the side of the base. How does its volume relate to the volume of the prism? | \frac{64\pi}{27} | 43 | 11 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.