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 | Example 6 For any non-empty subset $X$ of the set $M=\{1,2, \cdots, 1000\}$, let $\alpha_{X}$ denote the sum of the maximum and minimum numbers in $X$. Find the arithmetic mean of all such $\alpha_{X}$.
(1991, National High School Mathematics Joint Competition) | 1001 | 79 | 4 |
math | Let $a, b \in \mathbf{R}^{*}$, find the minimum value of $y=\frac{a}{\sin ^{3 / 2} \theta}+\frac{b}{\cos ^{3 / 2} \theta}\left(\theta \in\left(0, \frac{\pi}{2}\right)\right)$. | (^{\frac{4}{7}}+b^{\frac{4}{7}})^{\frac{7}{4}} | 80 | 26 |
math | 1. Given $S=1^{2}-2^{2}+3^{2}-4^{2}+\cdots-100^{2}+$ $101^{2}$. Then the remainder when $S$ is divided by 103 is | 1 | 57 | 1 |
math | 1. There are weights of $11 \mathrm{~g}$ and $17 \mathrm{~g}$ available in sufficient quantity. To weigh an object of mass $3 \mathrm{~g}$ on a balance, at least $\qquad$ such weights are needed. | 13 | 59 | 2 |
math | # Problem 5. Maximum 20 points
The commander of a tank battalion, in celebration of being awarded a new military rank, decided to invite soldiers to a tank festival, where the main delicacy is buckwheat porridge. The commander discovered that if the soldiers are lined up by height, there is a certain pattern in the ch... | 150 | 477 | 3 |
math | Find all functions $f:[0,1] \rightarrow[0,1]$ such that for all $0 \leqslant x \leqslant 1, f(2 x-f(x))=x$.
| f(x)=x | 48 | 4 |
math | 3. In the number $2 * 0 * 1 * 6 * 0 *$, each of the 5 asterisks needs to be replaced with any of the digits $0,1,2,3,4,5,6,7,8$ (digits can repeat) so that the resulting 10-digit number is divisible by 18. In how many ways can this be done? | 3645 | 86 | 4 |
math | $(NET 3)$ Let $x_1, x_2, x_3, x_4,$ and $x_5$ be positive integers satisfying
\[x_1 +x_2 +x_3 +x_4 +x_5 = 1000,\]
\[x_1 -x_2 +x_3 -x_4 +x_5 > 0,\]
\[x_1 +x_2 -x_3 +x_4 -x_5 > 0,\]
\[-x_1 +x_2 +x_3 -x_4 +x_5 > 0,\]
\[x_1 -x_2 +x_3 +x_4 -x_5 > 0,\]
\[-x_1 +x_2 -x_3 +x_4 +x_5 > 0\]
$(a)$ Find the maximum of $(x_1 + x_3)^{... | (a + c)^{b + d} = 499^{499} | 244 | 21 |
math | In a pile you have 100 stones. A partition of the pile in $ k$ piles is [i]good[/i] if:
1) the small piles have different numbers of stones;
2) for any partition of one of the small piles in 2 smaller piles, among the $ k \plus{} 1$ piles you get 2 with the same number of stones (any pile has at least 1 stone).
... | k_{\text{max}} = 13 | 107 | 12 |
math | 7. (2002 National High School Competition Question) The range of negative values of $a$ for which the inequality $\sin ^{2} x+a \cdot \cos x+a^{2}>1+\cos x$ holds for all $x \in \mathbf{R}$ is $\qquad$ . | \leqslant-2 | 67 | 7 |
math | Example 1 Solve the system of equations
$$
\left\{\begin{array}{l}
x+a y+a^{2} z+a^{3}=0 \\
x+b y+b^{2} z+b^{3}=0 \\
x+c y+c^{2} z+c^{3}=0
\end{array}\right.
$$ | x=-a b c, y=a b+b c+c a, z=-(a+b+c) | 71 | 21 |
math | Find a number $\mathrm{N}$ with five digits, all different and none zero, which equals the sum of all distinct three digit numbers whose digits are all different and are all digits of $\mathrm{N}$. | 35964 | 44 | 5 |
math | 31. a) Find all integers that start with the digit 6 and decrease by 25 times when this digit is erased.
b) Prove that there do not exist integers that decrease by 35 times when the first digit is erased. | 6250\ldots0(n=0,1,2,\ldots) | 52 | 19 |
math | Let the sequence $u_n$ be defined by $u_0=0$ and $u_{2n}=u_n$, $u_{2n+1}=1-u_n$ for each $n\in\mathbb N_0$.
(a) Calculate $u_{1990}$.
(b) Find the number of indices $n\le1990$ for which $u_n=0$.
(c) Let $p$ be a natural number and $N=(2^p-1)^2$. Find $u_N$. | 0 | 117 | 1 |
math | Five, let $A=\{1,2,3, \cdots, 17\}$. For any function $f: A \rightarrow A$, denote
$$
f^{[1]}(x)=f(x), f^{[k+1]}(x)=f\left(f^{[k]}(x)\right)
$$
$(k \in \mathbb{N})$. Find the natural number $M$ such that:
$$
\begin{array}{l}
\text { (1) When } m<M, 1 \leqslant i \leqslant 16, \text { we have } \\
f^{[m]}(i+1)-f^{[m]}(i... | 8 | 356 | 1 |
math | 14 Let $a_{1}, a_{2}, a_{3} \in \mathbf{R}^{+}$, find
$$\frac{a_{1} a_{2}}{\left(a_{2}+a_{3}\right)\left(a_{3}+a_{1}\right)}+\frac{a_{2} a_{3}}{\left(a_{3}+a_{1}\right)\left(a_{1}+a_{2}\right)}+\frac{a_{3} a_{1}}{\left(a_{1}+a_{2}\right)\left(a_{2}+a_{3}\right)}$$
the minimum value. | \frac{3}{4} | 145 | 7 |
math | 3. Let the sides of the cyclic quadrilateral $ABCD$ be $AB=3, BC=4, CD=5, DA=6$, then the area of quadrilateral $ABCD$ is $\qquad$ . | 6\sqrt{10} | 48 | 7 |
math | 3. Determine all two-digit numbers that are equal to twice the product of their digits. | 36 | 18 | 2 |
math | Example 8. The probability density function of a random variable $X$ is given by
$$
p(x)=a x^{2} e^{-k x} \quad(k>0, \quad 0 \leq x<+\infty)
$$
Find the value of the coefficient $a$. Find the distribution function $F(x)$ of the variable $X$. | =\frac{k^3}{2},\quadF(x)=1-\frac{k^2x^2+2kx+2}{2}e^{-kx} | 78 | 36 |
math | We wrote to ten of our friends and put the letters into the addressed envelopes randomly. What is the probability that exactly 5 letters end up with the person they were intended for? | \frac{11}{3600} | 36 | 11 |
math | ## Task A-1.6.
Borna wants to color each of the numbers $2,3, \ldots, 32$ with one of $k$ colors $(k \in \mathbb{N})$ such that no number is a multiple of another number of the same color. Determine the smallest natural number $k$ for which Borna can achieve this. | 5 | 79 | 1 |
math | 10.4. How many solutions in integers $x, y$ does the equation $|3 x+2 y|+|2 x+y|=100$ have? | 400 | 38 | 3 |
math | Example 4 The 10 complex roots of the equation $x^{10}+(13 x-1)^{10}=0$ are $r_{1}, \overline{r_{1}}, r_{2}, \overline{r_{2}}, r_{3}, \overline{r_{3}}, r_{4}$, $\overline{r_{4}}, r_{5}, \overline{r_{5}}$. Find the value of the algebraic expression $\frac{1}{r_{1} \overline{r_{1}}}+\frac{1}{r_{2} \overline{r_{2}}}+\cdot... | 850 | 158 | 3 |
math | 2. (5 points) At the World Meteorologists Conference, each participant in turn announced the average monthly temperature in their hometown. At this moment, all the others recorded the product of the temperatures in their and the speaker's cities. In total, 92 positive and 40 negative numbers were recorded. What is the ... | 2 | 78 | 1 |
math | 11. In a non-isosceles $\triangle ABC$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively, and it satisfies $(2c - b) \cos C = (2b - c) \cos B$.
(1) Find the size of angle $A$;
(2) If $a = 4$, find the range of the area of $\triangle ABC$. | 60,(0,4\sqrt{3}) | 93 | 11 |
math | 10. (7 points) In the multiplication problem below, each box must be filled with a digit; each Chinese character represents a digit, different characters represent different digits, and the same character represents the same digit. Therefore, the final product of this multiplication problem is . $\qquad$
将上面的文本翻译成英文，请... | 39672 | 81 | 5 |
math | 10.1. What two digits need to be appended to the right of the number 2013 so that the resulting six-digit number is divisible by 101? Find all possible solutions. | 94 | 43 | 2 |
math | Let's determine all integer pairs \((x, y)\) for which the following equation holds:
$$
1+2^{x}+2^{2 x+1}=y^{2}
$$ | (0;2),(0;-2),(4;23),(4;-23) | 41 | 19 |
math | 5. Howard chooses $n$ different numbers from the list $2,3,4,5,6,7,8,9,10,11$, so that no two of his choices add up to a square.
What is the largest possible value of $n$ ? | 7 | 59 | 1 |
math | 4. Given an isosceles triangle $\triangle A B C$ with side lengths $a$, $b$, and $c$ all being integers, and satisfying $a+b c+b+c a=24$. Then the number of such triangles is $\qquad$. | 3 | 56 | 1 |
math | 5. In a regular tetrahedron $ABCD$, $E$ and $F$ are on edges $AB$ and $AC$ respectively, satisfying $BE=3$, $EF=4$, and $EF$ is parallel to plane $BCD$. Then the area of $\triangle DEF$ is $\qquad$. | 2\sqrt{33} | 68 | 7 |
math | The product of two of the four roots of the quartic equation $x^4 - 18x^3 + kx^2+200x-1984=0$ is $-32$. Determine the value of $k$. | 86 | 55 | 2 |
math | Find all integers $n$ such that $\frac{70 n+200}{n^{2}+1}$ is an integer.
Initial text: 150 | n=0, \pm 1, \pm 2, \pm 3, -67 | 37 | 22 |
math | Example 10 The sequence $a_{1}, a_{2}, a_{3}, \cdots, a_{2 n}, a_{2 n+1}$ forms an arithmetic sequence, and the sum of the terms with odd indices is 60, while the sum of the terms with even indices is 45. Then the number of terms $n=$ $\qquad$ | 3 | 80 | 1 |
math | $$
\begin{array}{l}
A=\{x|| x-2 \mid<a\}, \\
B=\left\{x \mid x^{2}-2 x-3<0\right\} .
\end{array}
$$
If $B \subseteq A$, then the range of real number $a$ is $\qquad$ | a \geqslant 3 | 74 | 8 |
math | 4. If the equation $\sqrt{a x^{2}+a x+2}=a x+2$ has exactly one real root, then the range of the real number $a$ is | {-8}\cup[1,+\infty) | 41 | 11 |
math | 10. In quadrilateral $A B C D$, $A B=B C=C D=$ $26, A D=30 \sqrt{3}, A C$ intersects $B D$ at point $O, \angle A O B=$ $60^{\circ}$. Then $S_{\text {quadrilateral } A B C D}=$ $\qquad$ | 506\sqrt{3} | 82 | 8 |
math | Example 6 Let $n \equiv 1(\bmod 4)$ and $n>1, P=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ be any permutation of $\{1,2,3, \cdots, n\}$, and $k_{p}$ denote the largest index $k$ such that the following inequality holds,
$$
a_{1}+a_{2}+\cdots+a_{k}<a_{k+1}+a_{k+2}+\cdots+a_{n} \text {. }
$$
Find the sum of $k_{p}$ for... | \frac{1}{2}(n-1)(n!) | 147 | 13 |
math | $56$ lines are drawn on a plane such that no three of them are concurrent. If the lines intersect at exactly $594$ points, what is the maximum number of them that could have the same slope? | 44 | 47 | 2 |
math | For a certain task, $B$ needs 6 more days than $A$, and $C$ needs 3 more days than $B$. If $A$ works for 3 days and $B$ works for 4 days together, they accomplish as much as $C$ does in 9 days. How many days would each take to complete the task alone? | 18,24,27 | 76 | 8 |
math | Calculate the sum of $n$ addends
$$7 + 77 + 777 +...+ 7... 7.$$ | 7 \left( \frac{10^{n+1} - 9n - 10}{81} \right) | 30 | 29 |
math | Determine all prime numbers $p, q$ and $r$ such that $p+q^{2}=r^{4}$.
(Karl Czakler)
Answer. The only solution is $p=7, q=3, r=2$. | p=7,q=3,r=2 | 54 | 9 |
math | (7) $\cos \frac{\pi}{15}-\cos \frac{2 \pi}{15}-\cos \frac{4 \pi}{15}+\cos \frac{7 \pi}{15}=$ $\qquad$ | -\frac{1}{2} | 54 | 7 |
math | # 8.2. Condition:
On an island, there are two tribes: knights, who always tell the truth, and liars, who always lie. Four islanders lined up, each 1 meter apart from each other.
- The leftmost in the row said: "My fellow tribesman in this row stands 2 meters away from me."
- The rightmost in the row said: "My fellow ... | 1 | 168 | 1 |
math | Anumber of schools took part in a tennis tournament. No two players from the same school played against each other. Every two players from different schools played exactly one match against each other. A match between two boys or between two girls was called a [i]single[/i] and that between a boy and a girl was called ... | 3 | 126 | 3 |
math | 12.222. The base of the pyramid is a rhombus, one of the angles of which is $\alpha$. The lateral faces are equally inclined to the plane of the base. A plane is drawn through the midpoints of two adjacent sides of the base and the vertex of the pyramid, forming an angle $\beta$ with the plane of the base. The area of ... | 2\sqrt{\frac{2S\cos\beta}{\sin\alpha}} | 95 | 18 |
math | Circle $\omega_1$ and $\omega_2$ have centers $(0,6)$ and $(20,0)$, respectively. Both circles have radius $30$, and intersect at two points $X$ and $Y$. The line through $X$ and $Y$ can be written in the form $y = mx+b$. Compute $100m+b$.
[i]Proposed by Evan Chen[/i] | 303 | 93 | 3 |
math | 4. Let $x, y, z$ be non-negative numbers such that $x^{2}+y^{2}+z^{2}+x+2 y+3 z=\frac{13}{4}$. Find the minimum value of $x+y+z$.
(1 mark)設 $x 、 y 、 z$ 為非負數, 使得 $x^{2}+y^{2}+z^{2}+x+2 y+3 z=\frac{13}{4}$ 。求 $x+y+z$ 的最小值。 | \frac{-3+\sqrt{22}}{2} | 124 | 13 |
math | 6. A number is the product of 5 twos, 3 threes, 2 fives, and 1 seven. This number, of course, has many two-digit divisors. Among these two-digit divisors, what is the largest?
保留源文本的换行和格式，翻译结果如下：
6. A number is the product of 5 twos, 3 threes, 2 fives, and 1 seven. This number, of course, has many two-digit divisors.... | 96 | 119 | 2 |
math | In a country with $n$ cities, all direct airlines are two-way. There are $r>2014$ routes between pairs of different cities that include no more than one intermediate stop (the direction of each route matters). Find the least possible $n$ and the least possible $r$ for that value of $n$. | 2016 | 69 | 4 |
math | $2 \cdot 48$ Try to find the remainder when $10^{10}+10^{10^{2}}+10^{10^{3}}+\cdots+10^{10^{10}}$ is divided by 7.
(5th Moscow Mathematical Olympiad, 1939) | 5 | 73 | 1 |
math | Find all real numbers $x, y, z$ satisfying:
$$
\left\{\begin{array}{l}
(x+1) y z=12 \\
(y+1) z x=4 \\
(z+1) x y=4
\end{array}\right.
$$
Elementary Symmetric Polynomials In this section, we are interested in the links between the coefficients of a polynomial and its roots.
## Viète's Formulas
Proposition (Viète's Fo... | (2,-2,-2)(\frac{1}{3},3,3) | 272 | 18 |
math | Example 1. Find the maximum and minimum values of the function $z=f(x, y)=2 x^{2}-2 y^{2}$ in the circle $x^{2}+y^{2} \leq 9$. | z_{\text{max}}=18,\;z_{\text{}}=-18 | 49 | 21 |
math | $$
\text { Three. (25 points) In } \triangle A B C \text {, } A B=8+2 \sqrt{6} \text {, }
$$
$B C=7+2 \sqrt{6}, C A=3+2 \sqrt{6}, O, I$ are the circumcenter and incenter of $\triangle A B C$, respectively. Is the length of segment $O I$ a rational number or an irrational number? Make a judgment and explain your reasoni... | 2.5 | 110 | 3 |
math | G11 (3-6, Romania) Given a plane $E$ and three non-collinear points $A, B, C$ on the same side of $E$, and the plane through $A, B, C$ is not parallel to plane $E$. Take any three points $A^{\prime}, B^{\prime}, C^{\prime}$ on plane $E$. Points $L, M, N$ are the midpoints of segments $A A^{\prime}, B B^{\prime}, C C^{\... | \frac{1}{6}(a_{3}+b_{3}+c_{3}) | 179 | 21 |
math | Example 3 Given $x+2 y+3 z+4 u+5 v=30$, find the minimum value of $w$ $=x^{2}+2 y^{2}+3 z^{2}+4 u^{2}+5 v^{2}$ (Example 6 from reference [1]). | 60 | 69 | 2 |
math |
NT1 Find all the pairs positive integers $(x, y)$ such that
$$
\frac{1}{x}+\frac{1}{y}+\frac{1}{[x, y]}+\frac{1}{(x, y)}=\frac{1}{2}
$$
where $(x, y)$ is the greatest common divisor of $x, y$ and $[x, y]$ is the least common multiple of $x, y$.
| (8,8),(9,24),(24,9),(5,20),(20,5),(12,15),(15,12),(8,12),(12,8),(6,12),(12,6) | 98 | 57 |
math | Example 3 Let $n$ be a positive integer,
$$
\begin{aligned}
S= & \{(x, y, z) \mid x, y, z \in\{0,1, \cdots, n\}, \\
& x+y+z>0\}
\end{aligned}
$$
is a set of $(n+1)^{3}-1$ points in three-dimensional space. Try to find the minimum number of planes whose union contains $S$ but does not contain $(0,0,0)$. ${ }^{[4]}$ | 3n | 121 | 2 |
math | 10. An integer $n$ allows the polynomial $f(x)=3 x^{3}-n x-n-2$ to be expressed as the product of two non-constant polynomials with integer coefficients, then the sum of all possible values of $n$ is $\qquad$ . | 192 | 60 | 3 |
math | Find the number of ordered pairs $(a, b)$ of positive integers that are solutions of the following equation: \[a^2 + b^2 = ab(a+b).\] | 1 | 37 | 1 |
math | 19. $\tan \frac{A}{2} \tan \frac{B}{2} \tan \frac{C}{2}=\frac{r}{p}$. | \frac{r}{p} | 38 | 7 |
math | Y62 ** Find the smallest natural number $n$, such that $n!$ ends with exactly 1987 zeros. | 7960 | 28 | 4 |
math | 3. A merchant bought several bags of salt in Tver and sold them in Moscow with a profit of 100 rubles. With all the money earned, he again bought salt in Tver (at the Tver price) and sold it in Moscow (at the Moscow price). This time the profit was 120 rubles. How much money did he spend on the first purchase? | 500 | 82 | 3 |
math | We define the weight $W$ of a positive integer as follows: $W(1) = 0$, $W(2) = 1$, $W(p) = 1 + W(p + 1)$ for every odd prime $p$, $W(c) = 1 + W(d)$ for every composite $c$, where $d$ is the greatest proper factor of $c$. Compute the greatest possible weight of a positive integer less than 100. | 12 | 100 | 2 |
math | Angela, Bill, and Charles each independently and randomly choose a subset of $\{ 1,2,3,4,5,6,7,8 \}$ that consists of consecutive integers (two people can select the same subset). The expected number of elements in the intersection of the three chosen sets is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positi... | 421 | 113 | 3 |
math | Consider coins with positive real denominations not exceeding 1. Find the smallest $C>0$ such that the following holds: if we have any $100$ such coins with total value $50$, then we can always split them into two stacks of $50$ coins each such that the absolute difference between the total values of the two stacks is ... | \frac{50}{51} | 90 | 9 |
math | 99. When dividing a five-digit number, consisting of identical digits, by a four-digit number, also consisting of identical digits, the quotient was 16 and some remainder. After discarding one digit from both the dividend and the divisor, the quotient remained unchanged, but the remainder decreased by 2000. Find these ... | 555553333 | 70 | 9 |
math | 99 Set $A=\left\{z \mid z^{18}=1\right\}, B=\left\{w \mid w^{48}=1\right\}$, are both sets of complex unit roots of 1. $C=$ $\{z w \mid z \in A, w \in B\}$ is also a set of complex unit roots of 1, then the set $C$ contains $\qquad$ elements. | 144 | 97 | 3 |
math | 6. The polynomial $p(x)=x^{2}-3 x+1$ has zeros $r$ and $s$ and a quadratic polynomial $q(x)$ has leading coefficient 1 and zeros $r^{3}$ and $s^{3}$. Find $q(1)$. | -16 | 60 | 3 |
math | 1. For all real numbers $x, y$, if the function $f$ satisfies:
$$
f(x y)=f(x) \cdot f(y)
$$
and $f(0) \neq 0$, then $f(1998)=$ $\qquad$ . | 1 | 62 | 1 |
math | 162. $7^{x}-3 \cdot 7^{x-1}+7^{x+1}=371$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
162. $7^{x}-3 \cdot 7^{x-1}+7^{x+1}=371$. | 2 | 85 | 1 |
math | 9. Let $\triangle A B C$ have internal angles $\angle A, \angle B, \angle C$ with opposite sides $a, b, c$ respectively, and $\angle A - \angle C = \frac{\pi}{2}, a, b, c$ form an arithmetic sequence. Then the value of $\cos B$ is . $\qquad$ | \frac{3}{4} | 77 | 7 |
math | 2. (Easy/Average) Find the remainder when $14^{100}$ is divided by 45 . | 31 | 26 | 2 |
math | 4. Two students, A and B, play chess. Winning a game earns 2 points, drawing a game earns 1 point each, and losing a game earns 0 points. They play three consecutive games, and the one with more points wins. What is the probability that A wins? | \frac{10}{27} | 60 | 9 |
math | 3. The greatest common divisor of two natural numbers $a$ and $b$ is $d$. Determine the greatest common divisor of the numbers $5a + 3b$ and $13a + 8b$. | d | 47 | 1 |
math | Example 13. Map the unit circle $|z|<1$ onto the unit circle $|\boldsymbol{w}|<1$. | e^{i\alpha}\frac{z-z_{0}}{1-z\bar{z}_{0}} | 30 | 23 |
math | 20. In $\triangle A B C$, $3 \sin A+4 \cos B=6, 4 \sin B+3 \cos A=1$, find the degree measure of angle $C$.
In $\triangle A B C$, $3 \sin A+4 \cos B=6, 4 \sin B+3 \cos A=1$, find the degree measure of angle $C$. | \frac{\pi}{6} | 88 | 7 |
math | 4. The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=2, a_{n+1}=\frac{2(n+2)}{n+1} a_{n}\left(n \in \mathbf{N}^{*}\right)$, then $\frac{a_{2014}}{a_{1}+a_{2}+\cdots+a_{2013}}$ | \frac{2015}{2013} | 94 | 13 |
math | 4.4 How many positive integer factors does the number 20! have?
(Mexico Mathematical Olympiad, 1990) | 41040 | 30 | 5 |
math | $3 \cdot 27$ When the natural number $n \geqslant 2$ is the smallest, find integers $a_{1}, a_{2}, \cdots, a_{n}$, such that the following equation $a_{1}+a_{2}+\cdots+a_{n}=a_{1} \cdot a_{2} \cdot \cdots a_{n}=1990$ holds. | 5 | 94 | 1 |
math | Let $n,k$ be positive integers so that $n \ge k$.Find the maximum number of binary sequances of length $n$ so that fixing any arbitary $k$ bits they do not produce all binary sequances of length $k$.For exmple if $k=1$ we can only have one sequance otherwise they will differ in at least one bit which means that bit pro... | \sum_{i=0}^{k-1} \binom{n}{i} | 92 | 20 |
math | 1. Find all integers $a$ for which the modulus of the number $\left|a^{2}-3 a-6\right|$ is a prime number. Answer. $-1 ; 4$. | -1,4 | 43 | 4 |
math | 2 [Arithmetic. Mental arithmetic, etc.]
There are nuts in the boxes. In the first box, there are 6 kg fewer nuts than in the other two together. And in the second box, there are 10 kg fewer nuts than in the other two together. How many nuts are in the third box? | 8 | 66 | 1 |
math | 6. Find the integer solutions of the equation $x \sqrt{14-3 y^{2}}-y \sqrt{21-3 x^{2}}=7 \sqrt{2}$. | 1,-2 | 43 | 3 |
math | 7. Nestašni Darko, upon arriving at the amusement park, forgot about his savings. He spent 4 kuna on two scoops of ice cream, and then spent half of the remaining money on a ride in the bumper cars. Then he bought a chocolate banana for 2 kuna, and spent half of the remaining money on a ride on the carousel. After that... | 80 | 165 | 2 |
math | Andover has a special weather forecast this week. On Monday, there is a $\frac{1}{2}$ chance of rain. On Tuesday, there is a $\frac{1}{3}$ chance of rain. This pattern continues all the way to Sunday, when there is a $\frac{1}{8}$ chance of rain. The probability that it doesn't rain in Andover all week can be expressed... | 9 | 121 | 3 |
math | 1. The front tire of a motorcycle wears out after $25000 \mathrm{~km}$, and the rear tire after $15000 \mathrm{~km}$. After how many kilometers should the tires be swapped so that both tires wear out simultaneously? After how many kilometers should the motorcyclist buy new tires? | 18750\mathrm{~} | 72 | 10 |
math | Task B-4.7. Determine all natural numbers $n$ such that the value of the following expression
$$
\left(\frac{1+i \operatorname{tan} \frac{\pi}{38 n}}{1-i \operatorname{tan} \frac{\pi}{38 n}}\right)^{2014}
$$
is a real number. | n\in{1,2,53,106} | 81 | 15 |
math | The numbers $\frac{1}{1}, \frac{1}{2}, ... , \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$, $y$, erases them and then writes down $x + y + xy$. He continues to do this until only one number is left on the blackboard. What is this number? | 2010 | 87 | 4 |
math | Given an $n \times n \times n$ grid of unit cubes, a cube is [i]good[/i] if it is a sub-cube of the grid and has side length at least two. If a good cube contains another good cube and their faces do not intersect, the first good cube is said to [i]properly[/i] contain the second. What is the size of the largest possib... | (n-1)^3 + (n-2)^3 | 105 | 14 |
math | 2. (8 points) Zombies are coming, and Fire Dragon Grass is responsible for defense:
First-stage Fire Dragon Grass always spits red flames;
Second-stage Fire Dragon Grass spits red flames half the time and blue flames the other half;
Third-stage Fire Dragon Grass always spits blue flames.
If the attack power of the seco... | 3 | 107 | 1 |
math | 5. (10 points) There is a strange computer with a button. If the number on the computer is a multiple of 3, pressing the button will divide it by 3; if the number is not a multiple of 3, pressing the button will multiply it by 6. Xiao Ming pressed the button 6 times in a row without looking at the number on the screen,... | 27 | 110 | 2 |
math | We write down the positive integers from 1 in sequence up to some fixed $n$. Below them, we write the same numbers, but in reverse order. We then form the absolute value of the difference of the numbers that are one below the other, and add up these values. What will be the sum thus obtained? | \frac{n^2}{2} | 65 | 8 |
math | 1. (2 points) Boy Vasya wrote down the non-zero coefficients of the polynomial $P(x)$ of the seventh degree in his notebook. Then he calculated the derivative of the resulting polynomial and wrote down its non-zero coefficients, and so on, until he obtained a constant, which he also wrote down.
What is the smallest nu... | 7 | 114 | 1 |
math | 3B. Determine the angle $\alpha$ if it is known that $\alpha \in\left[0, \frac{\pi}{2}\right]$ and
$$
\sin \alpha=\frac{1-\sqrt{2}}{\sqrt{6-3 \sqrt{2}}-\sqrt{2+\sqrt{2}}}
$$ | \frac{7\pi}{24} | 71 | 10 |
math | ## Task B-2.6.
Solve the equation $i z^{2}+2 \bar{z}=0$ in the set of complex numbers. Calculate the quotient of the sum of the cubes and the product of all solutions that are different from zero. | 3 | 55 | 1 |
math | \section*{Problem 3B - 251043B}
Given are real numbers \(a_{1}, a_{2}, a_{3}, a_{4}\).
Determine for each possible case of these \(a_{1}, \ldots, a_{4}\) all triples \(\left(b_{1}, b_{2}, b_{3}\right)\) of real numbers (or prove, if applicable, that no such triples exist) for which the system of equations
\[
\begin{... | (b_{1},b_{2},b_{3})=(0,0,0) | 237 | 19 |
math | Example 2 Let $x, y \in \mathbf{R}, M=\max || x+y |$, $|x-y|,|1-x|,|1-y|\}$. Try to find the minimum value of $M$. | \frac{2}{3} | 51 | 7 |
math | Lyla and Isabelle run on a circular track both starting at point $P$. Lyla runs at a constant speed in the clockwise direction. Isabelle also runs in the clockwise direction at a constant speed $25 \%$ faster than Lyla. Lyla starts running first and Isabelle starts running when Lyla has completed one third of one lap. ... | 17 | 96 | 2 |
math | The integers $a$, $b$, $c$ and $d$ are such that $a$ and $b$ are relatively prime, $d\leq 2022$ and $a+b+c+d = ac + bd = 0$. Determine the largest possible value of $d$, | 2016 | 63 | 4 |
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