task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | For a positive integer $n$, let $\Omega(n)$ denote the number of prime factors of $n$, counting multiplicity. Let $f_1(n)$ and $f_3(n)$ denote the sum of positive divisors $d|n$ where $\Omega(d)\equiv 1\pmod 4$ and $\Omega(d)\equiv 3\pmod 4$, respectively. For example, $f_1(72) = 72 + 2 + 3 = 77$ and $f_3(72) = 8+12+18... | \frac{1}{10}(6^{2021} - 3^{2021} - 2^{2021} - 1) | 158 | 37 |
math | ## Task 32/66
An electric heater contains two resistors $R_{1}$ and $R_{2}$, which can be switched in four stages $A, B, C$, and $D$.
In stage $A$, $R_{1}$ and $R_{2}$ are connected in parallel, in stages $B$ and $C$ one of the two resistors is in operation, and in stage $D$ $R_{1}$ and $R_{2}$ are connected in serie... | R_{1}\approx13\Omega,R_{2}\approx21.1\Omega,N_{B}\approx3710\mathrm{~W},N_{C}\approx2290\mathrm{~W},N_{D}\approx1420\mathrm{~W} | 175 | 65 |
math | Let $n,k$ be positive integers such that $n\geq2k>3$ and $A= \{1,2,...,n\}.$ Find all $n$ and $k$ such that the number of $k$-element subsets of $A$ is $2n-k$ times bigger than the number of $2$-element subsets of $A.$ | n = 27 | 81 | 6 |
math | 275. Maximum Number. Let a set of distinct complex numbers $z_{i}, i=1,2, \ldots, n$, be given, satisfying the inequality
$$
\min _{i \neq j}\left|z_{i}-z_{j}\right| \geqslant \max _{i}\left|z_{i}\right|
$$[^16]
Find the maximum possible $n$ and for this $n$ all sets satisfying the condition of the problem. | 7 | 108 | 1 |
math | Solve the following equation
$$
\sqrt[3]{\sqrt{5}+x}+\sqrt[3]{\sqrt{5}-x}=\sqrt{5}
$$ | \2 | 39 | 2 |
math | Example 3. A pocket contains 7 white balls and 3 black balls of the same size. If two balls are drawn at random, what is the probability of getting one white ball and one black ball? (Exercise 6, Question 6) | \frac{7}{15} | 52 | 8 |
math | Let $A$ and $B$ be sets such that there are exactly $144$ sets which are subsets of either $A$ or $B$. Determine the number of elements $A \cup B$ has. | 8 | 46 | 1 |
math | 4. Let the function $f(x)$ have the domain $D=(-\infty, 0) \bigcup(0,+\infty)$, and for any $x \in D$, it holds that $f(x)=\frac{f(1) \cdot x^{2}+f(2) \cdot x-1}{x}$, then the sum of all zeros of $f(x)$ is $\qquad$ . | -4 | 93 | 2 |
math | Golenischeva-Kumuzova T.I.
Yura has a calculator that allows multiplying a number by 3, adding 3 to a number, or (if the number is divisible by 3) dividing the number by 3. How can Yura use this calculator to get from the number 1 to the number 11? | 11 | 71 | 2 |
math | 11. From 30 people with distinct ages, select two groups, the first group consisting of 12 people and the second group consisting of 15 people, such that the oldest person in the first group is younger than the youngest person in the second group. How many different ways are there to do this? | 4060 | 66 | 4 |
math | 4. Solve the equation $2^{x}+3^{y}+3=10^{z}$ in natural numbers. | (2,1,1),(4,4,2) | 27 | 13 |
math | What digit can the positive integer $n \geq 3$ end with, if $n+n^{2}+\ldots+n^{2 n-3}-4$ is a prime number? | 5 | 41 | 1 |
math | Compute $\sqrt{(31)(30)(29)(28)+1}$. | 869 | 19 | 3 |
math | $10 \cdot 86$ Find all integers $n>3$, such that there exist $n$ points $A_{1}, A_{2}, \cdots$, $A_{n}$ and real numbers $r_{1}, r_{2}, \cdots, r_{n}$ in the plane, satisfying the following two conditions:
(1) No three points among $A_{1}, A_{2}, \cdots, A_{n}$ are collinear;
(2) For each triplet $\left\{A_{i}, A_{j}, ... | 4 | 182 | 1 |
math | 16.2.39 ** Find the number of positive integers such that in base $n$ their digits are all different, and except for the leftmost digit, each digit differs from some digit to its left by $\pm 1$.
untranslated text remains unchanged. | 2^{n+1}-2n-2 | 56 | 10 |
math | Nick is a runner, and his goal is to complete four laps around a circuit at an average speed of 10 mph. If he completes the first three laps at a constant speed of only 9 mph, what speed does he need to maintain in miles per hour on the fourth lap to achieve his goal? | 15 \text{ mph} | 64 | 7 |
math | 2. The equation about $x$
$$
x^{2}-2 a x+a^{2}-4 a=0
$$
has an imaginary root with a modulus of 3. Then the value of the real number $a$ is $\qquad$ | 2-\sqrt{13} | 54 | 7 |
math | A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$.
[i]Proposed by Aaron Lin[/i] | 2842 | 73 | 4 |
math | We calculated the square root of every four-digit number, and if we did not get an integer, we rounded it to the nearest integer. Did we round up or down more often? | 24 | 37 | 2 |
math | 685. $y=\sin ^{2}(2 x-1)$, i.e. $y=u^{2}$, where $u=\sin (2 x-1)$. | 2\sin2(2x-1) | 40 | 10 |
math | Example 4 Let the inequality
$$
x^{2}-(a+1) x-a^{2}>0
$$
hold for all $a \in(1,2)$. Find the range of $x$. | x \geqslant 4 \text{ or } x \leqslant -1 | 47 | 21 |
math | (2) (20 points) Let $P$ be a moving point on the line $y=x-2$, and draw the tangent lines from point $P$ to the parabola $y=\frac{1}{2} x^{2}$, with the points of tangency being $A$ and $B$.
(1) Prove that the line $AB$ passes through a fixed point;
(2) Find the minimum value of the area $S$ of $\triangle PAB$, and the... | S_{}=3\sqrt{3},atP(1,-1) | 120 | 17 |
math | In the right parallelopiped $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$, with $AB=12\sqrt{3}$ cm and $AA^{\prime}=18$ cm, we consider the points $P\in AA^{\prime}$ and $N\in A^{\prime}B^{\prime}$ such that $A^{\prime}N=3B^{\prime}N$. Determine the length of the line segment $AP$ such that for any position of the poi... | \frac{27}{2} | 136 | 8 |
math | A natural number $k$ is said $n$-squared if by colouring the squares of a $2n \times k$ chessboard, in any manner, with $n$ different colours, we can find $4$ separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in functio... | 2n^2 - n + 1 | 99 | 11 |
math | 12. In the Cartesian coordinate system $x 0 y$, given two points $M(-1,2)$ and $N(1,4)$, point $P$ moves on the $x$-axis. When $\angle M P N$ takes the maximum value, the x-coordinate of point $P$ is $\qquad$ $-$.
| 1 | 74 | 1 |
math | Three. (20 points) Given the parabola $C: y=\frac{1}{2} x^{2}$, $A_{1}\left(x_{1}, 0\right)$ and $A_{2}\left(x_{2}, 0\right)$ are two points on the $x$-axis $\left(x_{1}+x_{2} \neq 0, x_{1} x_{2} \neq 0\right)$. Perpendicular lines to the $x$-axis are drawn through points $A_{1}$ and $A_{2}$, intersecting the parabola ... | \frac{1}{3} | 276 | 7 |
math | 356. A simple cryptarithm. Each letter in the following cryptarithm stands for a definite decimal digit:
$$
3(B I D F O R)=4(F O R B I D)
$$
Restore the original record. | 3(571428)=4(428571) | 50 | 18 |
math | 10.1. Find all numbers $a$ and $b$ for which the equality $|a x+b y|+|b x+a y|=|x|+|y|$ holds for all values of the variables $x$ and $\cdot y$. | =\1,b=0or=0,b=\1 | 55 | 11 |
math | Find all polynomials $P(x)$ with integer coefficients, such that for all positive integers $m, n$, $$m+n \mid P^{(m)}(n)-P^{(n)}(m).$$
[i]Proposed by Navid Safaei, Iran[/i] | P(x) \equiv c | 61 | 7 |
math | Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that
$$
f(f(x)+x f(y))=x+f(x) y
$$
where $\mathbb{Q}$ is the set of rational numbers. | f(x)=x | 56 | 4 |
math | 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 \text{ or } 11 | 87 | 9 |
math | Five, let the sequence of positive numbers $a_{0}, a_{1}, a_{2}, \cdots, a_{n}, \cdots$ satisfy
$$
\sqrt{a_{n} a_{n-2}}-\sqrt{a_{n-1} a_{n-2}}=2 a_{n-1} \quad(n \geqslant 2) .
$$
and $a_{0}=a_{1}=1$. Find the general term formula for $\left\{a_{n}\right\}$. | a_{n}={\begin{pmatrix}1&(n=0),\\\prod_{k=1}^{n}(2^{k}-1)^{2}&(n\inN)0\end{pmatrix}.} | 117 | 50 |
math | 4. We will call a non-empty set of distinct natural numbers from 1 to 13 good if the sum of all the numbers in it is even. How many good sets are there in total? | 2^{12}-1 | 42 | 6 |
math | Task 3. The average number of airplanes arriving at the airport per minute is 4. Find the probabilities that in 3 minutes a) two airplanes will arrive, b) fewer than two, c) no fewer than two airplanes will arrive. | )0.0005;b)0.0001;)0.9999 | 50 | 22 |
math | Find all integers $x, y \geq 1$ such that $x^{3}-y^{3}=x y+61$.
## Second Part
To solve Diophantine equations, we often use congruences by considering the equation modulo $N$. But which modulo $N$ should we choose?
- If there are $p$-th powers with $p$ prime, try $N=p^{2} p^{3}$, etc.
Indeed, if $a$ is not divisibl... | (6,5) | 944 | 5 |
math | 11.6. The sequence of numbers $a_{1}, a_{2}, \ldots, a_{2022}$ is such that $a_{n}-a_{k} \geqslant$ $\geqslant n^{3}-k^{3}$ for any $n$ and $k$ such that $1 \leqslant n \leqslant 2022$ and $1 \leqslant k \leqslant$ $\leqslant 2022$. Moreover, $a_{1011}=0$. What values can $a_{2022}$ take?
(N. Agakhanov) | 2022^{3}-1011^{3} | 148 | 14 |
math | # Task 4. (12 points)
A shooting tournament involves several series of 10 shots each. In one series, Ivan scored 82 points, as a result of which the average number of points he scored per series increased from 75 to 76 points. How many points does Ivan need to score in the next series of shots to make the average numb... | 84 | 89 | 2 |
math | 18. Master Li made 8 identical rabbit lanterns in three days, making at least 1 per day. Master Li has ( ) different ways to do this. | 21 | 35 | 2 |
math | 2、Function $f(x)=\sqrt{5 x+15}+\sqrt{12-x}$ has the range of | [\sqrt{15},3\sqrt{10}] | 27 | 13 |
math | Task B-1.6. If $\frac{z}{x+y}=2$ and $\frac{y}{x+z}=3$, what is the value of $\frac{z}{y+z}$? | \frac{8}{17} | 43 | 8 |
math | 1. Find the positive integer tuple $(a, b, c)$, such that $a^{2}+b+3=\left(b^{2}-c^{2}\right)^{2}$. | (2,2,1) | 42 | 7 |
math | 11.149. The base of a right parallelepiped is a parallelogram, one of the angles of which is $30^{\circ}$. The area of the base is 4 dm $^{2}$. The areas of the lateral faces of the parallelepiped are 6 and 12 dm ${ }^{2}$. Find the volume of the parallelepiped. | 12 | 84 | 2 |
math | Example 3 The equation $z^{6}+z^{3}+1=0$ has a complex root, and on the complex plane, the argument of this root is between $90^{\circ}$ and $180^{\circ}$. Find the degree measure of $\theta$.
(2nd American Invitational Mathematics Examination) | 160 | 73 | 3 |
math | 15. For natural numbers $n$ greater than 0, define an operation “ $G$ " as follows:
(1) When $n$ is odd, $G(n)=3 n+1$;
(2) When $n$ is even, $G(n)$ equals $n$ continuously divided by 2 until the quotient is odd;
The $k$-th “ $G$ ” operation is denoted as $G^{k}$, for example, $G^{1}(5)=3 \times 5+1=16, G^{2}(5)=G^{1}(1... | 63,34,4 | 238 | 7 |
math | ## Task 31/81
We are looking for all Pythagorean number triples $(a ; b ; c)$ for which $(c ; a b ; 4 a-b)$ is also a Pythagorean number triple. | (4;3;5) | 48 | 7 |
math | 3. In $\triangle A B C$, $A B=2, A C=1, B C=\sqrt{7}$, $O$ is the circumcenter of $\triangle A B C$, and $\overrightarrow{A O}=\lambda \overrightarrow{A B}+\mu \overrightarrow{A C}$. Then $\lambda+\mu=$ $\qquad$ . | \frac{13}{6} | 80 | 8 |
math | \section*{Problem 6 - 311046}
Let \(q\) be the larger of the two solutions to the equation \(x^{2}-4 x+1=0\).
Determine the last digit (units place) in the decimal digit representation of the number \(\left[q^{1992}\right]\).
Hint: If \(z\) is a real number, the integer \(g\) such that \(g \leq z < g+1\) is denoted ... | 1 | 113 | 1 |
math | 6. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=6, a_{2}=20, a_{n}\left(a_{n}-8\right)=a_{n-1} a_{n+1}-12(n \geqslant 2) \text {. }
$$
Let $\{x\}=x-[x]$, where $[x]$ denotes the greatest integer not exceeding the real number $x$. Then $\lim _{n \rightarrow \infty}\left\{\sqrt{a_{n}}\right\}=$ $\qquad$ | \frac{1}{2} | 133 | 7 |
math | Example 4 Given $\cdots 1 \leqslant a \leqslant 1$, the inequality $\left(\frac{1}{2}\right)^{x^{2}+a x}<\left(\frac{1}{2}\right)^{2 x+a-1}$ always holds. Find the range of values for $x$. | x>2 \text{ or } x<0 | 75 | 11 |
math | Find all pairs of positive integers $m, n \ge 3$ for which there exist infinitely many positive integers $a$ such that \[\frac{a^{m}+a-1}{a^{n}+a^{2}-1}\] is itself an integer. | (m, n) = (5, 3) | 58 | 13 |
math | 2. The sum of the terms of an infinite geometric series is 2 and the sum of the squares of the terms of this series is 6 . Find the sum of the cubes of the terms of this series. | \frac{96}{7} | 44 | 8 |
math | Let $p_1,p_2,p_3,p_4$ be four distinct primes, and let $1=d_1<d_2<\ldots<d_{16}=n$ be the divisors of $n=p_1p_2p_3p_4$. Determine all $n<2001$ with the property that
$d_9-d_8=22$. | n = 1995 | 86 | 8 |
math | 4. For the set $S=\{1,2,3, \cdots, 2009\}$, a 12-element subset $T=\left\{a_{1}, a_{2}, \cdots, a_{12}\right\}$ is such that the absolute difference between any two elements is not 1. The number of such 12-element subsets $T$ is $\qquad$ | C_{1998}^{12} | 91 | 11 |
math | Positive $x,y,z$ satisfy a system: $\begin{cases} x^2 + xy + y^2/3= 25 \\
y^2/ 3 + z^2 = 9 \\
z^2 + zx + x^2 = 16 \end{cases}$
Find the value of expression $xy + 2yz + 3zx$. | 24\sqrt{3} | 83 | 7 |
math | Given is an acute angled triangle $ ABC$. Determine all points $ P$ inside the triangle with
\[1\leq\frac{\angle APB}{\angle ACB},\frac{\angle BPC}{\angle BAC},\frac{\angle CPA}{\angle CBA}\leq2\] | O | 65 | 2 |
math | 11.2. On the board, there are 4 numbers. Vasya multiplied the first of these numbers by $\sin \alpha$, the second - by $\cos \alpha$, the third - by $\operatorname{tg} \alpha$, and the fourth - by $\operatorname{ctg} \alpha$ (for some angle $\alpha$) and obtained a set of the same 4 numbers (possibly in a different ord... | 3 | 109 | 1 |
math | 2. Laura has 2010 lamps and 2010 switches in front of her, with different switches controlling different lamps. She wants to find the correspondence between the switches and the lamps. For this, Charlie operates the switches. Each time Charlie presses some switches, and the number of lamps that light up is the same as ... | 11 | 151 | 2 |
math | 9. (This question is worth 16 points) The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=3$, and for any positive integers $m, n$, it holds that $a_{m+n}=a_{m}+a_{n}+2 m n$
(1) Find the general term formula for $\left\{a_{n}\right\}$;
(2) If there exists a real number $c$ such that $\sum_{i=1}^{k} \frac{1}{a_{i}}<c$ ... | \in[\frac{3}{4},+\infty) | 139 | 13 |
math | Task 1. For which values of the variables $x$ and $y$, the difference between the expressions $\frac{2 x+15}{8}$ and $1 \frac{1}{3} \cdot(y-1)$ will be 3 times smaller than the expression $2 \cdot(5-2 y)$, and the expression $\frac{x+5 \frac{3}{4}}{2}$ will be 0.125 greater than $3 y?$ | \frac{1}{2},1 | 101 | 8 |
math | Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. Points $D$ and $E$ are chosen on $AC$ and $AB$, respectively, such that quadrilateral $BCDE$ is cyclic and when the triangle is folded along segment $DE$, point $A$ lies on side $BC$. If the length of $DE$ can be expressed as $\tfrac{m}{n}$ for relatively p... | 6509 | 130 | 4 |
math | 7 Let $f(x)=\sin ^{4} x-\sin x \cos x+\cos ^{4} x$, then the range of $f(x)$ is $\qquad$ . | 0\leqslantf(x)\leqslant\frac{9}{8} | 41 | 20 |
math | Find the smallest natural number $ k$ for which there exists natural numbers $ m$ and $ n$ such that $ 1324 \plus{} 279m \plus{} 5^n$ is $ k$-th power of some natural number. | 3 | 56 | 1 |
math | All integers from 1 to 100 are written in a string in an unknown order. With one question about any 50 numbers, you can find out the order of these 50 numbers relative to each other. What is the minimum number of questions needed to definitely find out the order of all 100 numbers?
# | 5 | 70 | 1 |
math | A magician with an assistant is going to perform the following trick. A spectator writes a sequence of $N$ digits on a board. The magician's assistant covers two adjacent digits with a black circle. Then the magician enters. His task is to guess both covered digits (and the order in which they are located). For what sm... | 101 | 85 | 3 |
math | Márcia is in a store buying a recorder that she has wanted for a long time. When the cashier registers the price, she exclaims: "That can't be right, you entered the number backwards, you switched the order of two digits, I remember it cost less than 50 reais last week!" The cashier replies: I'm sorry, but all of our i... | 54 | 104 | 2 |
math | Example 3 Given that $x, y, z$ are non-negative real numbers, not all zero. Find
$$
u=\frac{\sqrt{x^{2}+y^{2}+x y}+\sqrt{y^{2}+z^{2}+y z}+\sqrt{z^{2}+x^{2}+z x}}{x+y+z}
$$
the minimum value. | \sqrt{3} | 87 | 5 |
math | 221. Differentiate the function $y=\left(x^{2}+2\right)(2 x+1)$. | 6x^2+2x+4 | 27 | 9 |
math | ## SUBJECT 4
Let the sequence of real numbers $\left(x_{n}\right)_{n \geq 1}$, given by $x_{1}=\alpha$, with $\alpha \in \mathrm{R}^{*}, \alpha \neq 2$ and $x_{n+1}=\frac{x_{n}^{2}+2 x_{n}+2}{x_{n}^{2}+1},(\forall) n \geq 1$. Calculate:
a) $\lim _{n \rightarrow \infty} x_{n}$,
b) $\lim _{n \rightarrow \infty}\left(x_{... | 2 | 244 | 1 |
math | 9. (1 mark) Let $x_{1}, y_{1}, x_{2}, y_{2}$ be real numbers satisfying the equations $x_{1}^{2}+5 x_{2}^{2}=10$, $x_{2} y_{1}-x_{1} y_{2}=5$ and $x_{1} y_{1}+5 x_{2} y_{2}=\sqrt{105}$. Find the value of $y_{1}^{2}+5 y_{2}^{2}$.
(1 mark) Let $x_{1}, y_{1}, x_{2}, y_{2}$ be real numbers satisfying the equations $x_{1}^{... | 23 | 236 | 2 |
math | 8. Let $0<\alpha \leqslant \beta \leqslant \gamma$, and $\alpha+\beta+\gamma=\pi$, then the range of $\min \left\{\frac{\sin \beta}{\sin \alpha}, \frac{\sin \gamma}{\sin \beta}\right\}$ is | [1,\frac{1+\sqrt{5}}{2}) | 70 | 14 |
math | Find the number of ordered pairs $(a, b)$ of positive integers such that $a$ and $b$ both divide $20^{19}$, but $a b$ does not. | 444600 | 41 | 6 |
math | 9.4. Along the shore of a circular lake with a perimeter of 1 km, two salmons are swimming - one at a constant speed of $500 \mathrm{m} /$ min clockwise, the other at a constant speed of 750 m/min counterclockwise. Along the edge of the shore, a bear is running, always moving along the shore at a speed of 200 m/min in ... | 7 | 216 | 1 |
math | 1. Given the function $f(x)$ satisfies $f(1)=2$, and
$$
f(x+1)=\frac{1+f(x)}{1-f(x)}
$$
for any $x$ in its domain. Then $f(2016)=$
$\qquad$ | \frac{1}{3} | 64 | 7 |
math | Example 1. Let $a x+b y=c\left(a, b, c \in R^{+}, \boldsymbol{x}\right.$ , $\left.y \in \overline{R^{-}}\right)$, find the extremum of $f(x, y)=m \sqrt{x}+n \sqrt{y}(m$, $n>0)$. | f_{\text{max}}(x, y)=\sqrt{\frac{c}{a b}\left(a n^{2}+b m^{2}\right)}, \quad f_{\text{min}}(x, y)=\sqrt{c} k | 79 | 56 |
math | 3. Given real numbers $a, b, c$ satisfy
$$
a+b+c=11 \text { and } \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}=\frac{13}{17} \text {. }
$$
Then the value of $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}$ is $\qquad$ . | \frac{92}{17} | 100 | 9 |
math | 7. The 3 roots of the equation $x^{3}-\sqrt{3} x^{2}-(2 \sqrt{3}+1) x+3+\sqrt{3}=0$ are $\qquad$ . | 1+\sqrt{3}, \frac{-1 \pm \sqrt{1+4 \sqrt{3}}}{2} | 49 | 26 |
math | 1. Determine the real number $a$ such that the polynomials $x^{2}+a x+1$ and $x^{2}+x+a$ have at least one common root.
(3rd Canadian Olympiad) | =1or=-2 | 49 | 5 |
math | Example 8. When $\mathrm{n}=1,2,3, \cdots, 1988$, find the sum $\mathrm{s}$ of the lengths of the segments cut off by the x-axis for all functions
$$
y=n(n+1) x^{2}-(2 n+1) x+1
$$ | \frac{1988}{1989} | 71 | 13 |
math | 20. How many subsets of the set $\{1,2,3, \ldots, 9\}$ do not contain consecutive odd integers? | 208 | 32 | 3 |
math | 14. A rectangular prism with integer edge lengths is painted red on all its surfaces, and then it is cut into small cubes with edge lengths of 1. Among them, there are 40 small cubes with two red faces, and 66 small cubes with one red face. What is the volume of this rectangular prism? | 150 | 68 | 3 |
math | Which is the smallest positive integer that, when divided by 5, leaves a remainder of 2, when divided by 7, leaves a remainder of 3, and when divided by 9, leaves a remainder of 4? | 157 | 48 | 3 |
math | Example 1. Find a point on the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$ such that the product of its distances to the two foci is 16. | \left(\frac{15}{4}, \frac{3 \sqrt{7}}{4}\right),\left(\frac{15}{4},-\frac{3 \sqrt{7}}{4}\right),\left(-\frac{15}{4}, \frac{3 \sqrt{7}}{4}\right),\left(-\frac{15}{4},-\frac{3 \sqrt{7}}{4}\right) | 49 | 98 |
math | 23 Among all triangles with a given perimeter, find the triangle with the largest inradius.
Among all triangles with a given perimeter, find the triangle with the largest inradius. | r \leqslant \frac{p}{\sqrt{27}} | 36 | 17 |
math | 1. Calculate the sum $S=S_{1}+S_{2}$, where $S_{1}$ and $S_{2}$ are given by
$$
S_{1}=\frac{1}{\log _{\operatorname{tg} 1^{\circ}} 2}+\frac{2}{\log _{\operatorname{tg} 2^{\circ}} 2^{2}}+\ldots+\frac{44}{\log _{\operatorname{tg} 44^{\circ}} 2^{44}} \text { and } S_{2}=\frac{46}{\log _{\operatorname{tg} 46^{\circ}} 2^{4... | 0 | 215 | 1 |
math | Given are positive reals $x_1, x_2,..., x_n$ such that $\sum\frac {1}{1+x_i^2}=1$. Find the minimal value of the expression $\frac{\sum x_i}{\sum \frac{1}{x_i}}$ and find when it is achieved. | n-1 | 68 | 3 |
math | 1.1. In a cross-country skiing race, Petya and Vasya started simultaneously. Vasya ran the entire race at a constant speed of 12 km/h. Petya ran the first half of the distance at a speed of 9 km/h and fell behind Vasya. What should Petya's speed be on the second half of the distance to catch up with Vasya and finish at... | 18 | 103 | 2 |
math | ## 253. Matheknobelei $6 / 86$
Statistische Angaben aus dem Jahre 1981 besagen, dass zu dieser Zeit 11 Prozent der Weltbevölkerung in Afrika lebten. Dieser Kontinent nimmt 20 Prozent des Festlandes der Erde ein. In Europa dagegen lebten 15,5 Prozent aller Menschen auf 7,1 Prozent des Festlandes. Wie viele Mal war Euro... | 3.97 | 124 | 4 |
math | $1 \cdot 21$ Team A has $2 m$ people, Team B has $3 m$ people. Now, $14-m$ people are drawn from Team A, and $5 m-11$ people are drawn from Team B to participate in the game. How many people are there in Team A and Team B respectively? How many different ways are there to select the participants for the game? | 150 | 87 | 3 |
math | 8. Given that one edge of a tetrahedron is 6, and the other edges are all 5, then the radius of the circumscribed sphere of this tetrahedron is $\qquad$ | \frac{20\sqrt{39}}{39} | 45 | 15 |
math | 11.1. Petya wrote ten natural numbers on the board, none of which are equal. It is known that among these ten numbers, three can be chosen that are divisible by 5. It is also known that among the ten numbers written, four can be chosen that are divisible by 4. Can the sum of all the numbers written on the board be less... | 71 | 91 | 2 |
math | 1. Let $a, b$ be two positive integers, their least common multiple is 232848, then the number of such ordered pairs of positive integers $(a, b)$ is $\qquad$
$\qquad$ groups. | 945 | 52 | 3 |
math | ## Problem Statement
Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.
$$
A(-3 ; -7 ; -5), B(0 ; -1 ; -2), C(2 ; 3 ; 0)
$$ | 1 | 61 | 1 |
math | 14. A wooden block floating in a cylindrical vessel with a base area of $\mathrm{S}=25 \mathrm{~cm}^2$, partially filled with water, had a small stone placed on it. As a result, the block remained afloat, and the water level in the vessel rose by $h_{1}=1.5 \mathrm{~cm}$. Then the stone was removed from the block and d... | 3\frac{\mathrm{r}}{\mathrm{}^{3}} | 143 | 14 |
math | 1. Let $\triangle A B C$ be a right triangle with right angle at $B$. Let the points $D, E$, and $F$ be on $A B, B C$, and $C A$, respectively, such that $\triangle D E F$ is an equilateral triangle and $E C=F C$. If $D B=5 \sqrt{3}, B E=3$, and $\sin \angle A C B=4 \sqrt{3} / 7$, find the perimeter of $\triangle A D F... | 35\sqrt{3}+63+2\sqrt{21} | 113 | 18 |
math | 3. (5 points) This year, Lingling is 8 years old, and her grandmother is 60 years old. In \qquad years, her grandmother's age will be 5 times Lingling's age. | 5 | 47 | 1 |
math | 3. Let $\mathrm{P}_{1}, \mathrm{P}_{2}, \ldots, \mathrm{P}_{41}$ be 41 distinct points on the segment $\mathrm{BC}$ of a triangle $\mathrm{ABC}$, where $\mathrm{AB}=\mathrm{AC}$ $=7$. Evaluate the sum $\sum_{i=1}^{41}\left(\mathrm{AP}_{i}^{2}+\mathrm{P}_{i} \mathrm{~B} \cdot \mathrm{P}_{i} \mathrm{C}\right)$. | 2009 | 122 | 4 |
math | Malcolm writes a positive integer on a piece of paper. Malcolm doubles this integer and subtracts 1, writing this second result on the same piece of paper. Malcolm then doubles the second integer and adds 1, writing this third integer on the paper. If all of the numbers Malcolm writes down are prime, determine all poss... | 2 \text{ and } 3 | 72 | 8 |
math | 4. (17th Nordic Mathematical Contest) Find all integer triples $(x, y, z)$ such that
$$
x^{3}+y^{3}+z^{3}-3 x y z=2003 \text {. }
$$ | (668,668,667),(668,667,668),(667,668,668) | 54 | 37 |
math | 9.027. $(x+1)(3-x)(x-2)^{2}>0$. | x\in(-1;2)\cup(2;3) | 23 | 14 |
math |
4. For a positive integer $n$ the number $P(n)$ is the product of the positive divisors of $n$. For example, $P(20)=8000$, as the positive divisors of 20 are $1,2,4,5,10$ and 20 , whose product is $1 \cdot 2 \cdot 4 \cdot 5 \cdot 10 \cdot 20=8000$.
(a) Find all positive integers $n$ satisfying $P(n)=15 n$.
(b) Show ... | 15 | 147 | 2 |
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