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 | 5. In triangle $A B C, \angle A=45^{\circ}$ and $M$ is the midpoint of $\overline{B C}$. $\overline{A M}$ intersects the circumcircle of $A B C$ for the second time at $D$, and $A M=2 M D$. Find $\cos \angle A O D$, where $O$ is the circumcenter of $A B C$. | -\frac{1}{8} | 92 | 7 |
math | 5th APMO 1993 Problem 4 Find all positive integers n for which x n + (x+2) n + (2-x) n = 0 has an integral solution. Solution | 1 | 44 | 1 |
math | 11.3. The fractional part of a positive number, its integer part, and the number itself form an increasing geometric progression. Find all such numbers. | \frac{\sqrt{5}+1}{2} | 32 | 12 |
math | 13.94 For which natural numbers $n$ and $k$, are the binomial coefficients
$$
C_{n}^{k-1}, C_{n}^{k}, C_{n}^{k+1}
$$
in arithmetic progression? | n=u^2-2,k=C_{u}^{2}-1ork=C_{u+1}^{2}-1,u\geqslant3 | 55 | 33 |
math | 8,9
Angle $A$ at the vertex of isosceles triangle $A B C$ is $100^{\circ}$. On ray $A B$, segment $A M$ is laid off, equal to the base $B C$. Find angle $B C M$. | 10 | 62 | 2 |
math | Let $\alpha$ denote $\cos^{-1}(\tfrac 23)$. The recursive sequence $a_0,a_1,a_2,\ldots$ satisfies $a_0 = 1$ and, for all positive integers $n$, $$a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}.$$ Suppose that the series $$\sum_{k=0}^\infty\dfrac{a_k}{2^k}$$ can be expressed uniquely as $\tfrac{p\sqrt q}r$, where $p$ and $r$ are coprime positive integers and $q$ is not divisible by the square of any prime. Find the value of $p+q+r$. | 23 | 181 | 2 |
math | ## Task B-2.2.
The function $f(x)=x^{2}+p x+q$ takes negative values only for $x \in\langle-3,14\rangle$. How many integer values from the set $[-100,-10]$ can the function $f$ take? | 63 | 67 | 2 |
math | Solve the following equation:
$$
\sqrt{4^{x}+\frac{17}{64}}-\sqrt{2 \cdot 4^{x}-\frac{7}{64}}=\sqrt{4^{x}-\frac{1}{16}}
$$ | -\frac{3}{2} | 59 | 7 |
math | 921. Find the mass of a hemisphere if the surface density at each of its points is numerically equal to the distance of this point from the radius perpendicular to the base of the hemisphere. | \frac{\pi^{2}R^{3}}{2} | 40 | 14 |
math | 1. Let $i_{1}, i_{2}, \cdots, i_{10}$ be a permutation of $1,2, \cdots, 10$. Define $S=\left|i_{1}-i_{2}\right|+\left|i_{3}-i_{4}\right|+\cdots+\left|i_{9}-i_{10}\right|$. Find all possible values of $S$.
[2] | 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25 | 94 | 39 |
math | Find all pairs of numbers $(n, k)$ such that in the decimal system:
$n^{n}$ is written with $k$ digits,
$k^{k}$ is written with $n$ digits. | (1,1),(8,8),(9,9) | 41 | 13 |
math | Let the $A$ be the set of all nonenagative integers.
It is given function such that $f:\mathbb{A}\rightarrow\mathbb{A}$ with $f(1) = 1$ and for every element $n$ od set $A$ following holds:
[b]1)[/b] $3 f(n) \cdot f(2n+1) = f(2n) \cdot (1+3 \cdot f(n))$;
[b]2)[/b] $f(2n) < 6f(n)$,
Find all solutions of $f(k)+f(l) = 293$, $k<l$. | (5, 47), (7, 45), (13, 39), (15, 37) | 144 | 31 |
math | # 8. Variant 1
Petya wrote down all positive numbers that divide some natural number $N$. It turned out that the sum of the two largest written numbers is 3333. Find all such $N$. If there are several numbers, write their sum in the answer. | 2222 | 62 | 4 |
math | 4-124 m, n are two different positive integers, find the common complex roots of the equations
$$
\text { and } \quad \begin{aligned}
x^{m+1}-x^{n}+1=0 \\
x^{n+1}-x^{m}+1=0
\end{aligned}
$$ | \frac{1}{2}\\frac{\sqrt{3}}{2}i | 73 | 17 |
math | 4. (10 points) A school has two classes each in the third and fourth grades. Class 3-1 has 4 more students than Class 3-2, Class 4-1 has 5 fewer students than Class 4-2, and the third grade has 17 fewer students than the fourth grade. Therefore, Class 3-1 has fewer students than Class 4-2 by ____. | 9 | 88 | 1 |
math | Let $ABCD$ be a square, and let $M$ be the midpoint of side $BC$. Points $P$ and $Q$ lie on segment $AM$ such that $\angle BPD=\angle BQD=135^\circ$. Given that $AP<AQ$, compute $\tfrac{AQ}{AP}$. | \sqrt{5} | 71 | 5 |
math | ## Task A-3.4.
How many ordered pairs of natural numbers $(a, b)$ satisfy
$$
\log _{2023-2(a+b)} b=\frac{1}{3 \log _{b} a} ?
$$ | 5 | 54 | 1 |
math | Zadam Heng bets Saniel Dun that he can win in a free throw contest. Zadam shoots until he has made $5$ shots. He wins if this takes him $9$ or fewer attempts. The probability that Zadam makes any given attempt is $\frac{1}{2}$. What is the probability that Zadam Heng wins the bet?
[i]2018 CCA Math Bonanza Individual Round #4[/i] | \frac{1}{2} | 92 | 7 |
math | Let $x$ and $y$ be two non-zero numbers such that $x^{2} + x y + y^{2} = 0$ (where $x$ and $y$ are complex numbers, but that's not too important). Find the value of
$$
\left(\frac{x}{x+y}\right)^{2013} + \left(\frac{y}{x+y}\right)^{2013}
$$ | -2 | 96 | 2 |
math | Task 1. (5 points) Solve the equation $x^{9}-2022 x^{3}-\sqrt{2021}=0$.
# | {-\sqrt[6]{2021};\sqrt[3]{\frac{\sqrt{2021}\45}{2}}} | 36 | 31 |
math | Example 14. Find the positive integer solutions of the equation $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$. | \left\{\begin{array}{l}x=2,2,3 \\ y=3,4,3 \\ z=6,4,3\end{array}\right.} | 36 | 41 |
math | 4. [5 points] Find the number of triples of natural numbers $(a ; b ; c)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
\operatorname{GCD}(a ; b ; c)=22 \\
\operatorname{LCM}(a ; b ; c)=2^{16} \cdot 11^{19}
\end{array}\right.
$$ | 9720 | 91 | 4 |
math | 18.3.19 $\star \star$ Find all positive integer triples $(a, b, c)$ that satisfy $a^{2}+b^{2}+c^{2}=2005$ and $a \leqslant b \leqslant c$. | (23,24,30),(12,30,31),(9,30,32),(4,30,33),(15,22,36),(9,18,40),(4,15,42) | 62 | 60 |
math |
4. Determine all pairs of positive real numbers $(a, b)$ with $a>b$ that satisfy the following equations:
$$
a \sqrt{a}+b \sqrt{b}=134 \quad \text { and } \quad a \sqrt{b}+b \sqrt{a}=126 .
$$
| (\frac{81}{4},\frac{49}{4}) | 72 | 16 |
math | Example 11 Find the least common multiple of $8127, 11352, 21672$ and 27090. | 3575880 | 38 | 7 |
math | 24. Let $P$ be a regular 2006-gon. If an end of a diagonal of $P$ divides the boundary of $P$ into two parts, each containing an odd number of sides of $P$, then the diagonal is called a "good diagonal". It is stipulated that each side of $P$ is also a "good diagonal". Given that 2003 non-intersecting diagonals inside $P$ divide $P$ into several triangles. How many isosceles triangles with two "good diagonals" can there be at most in such a division?
（47th IMO Problem） | 1003 | 134 | 4 |
math | 4. Does there exist a convex polygon with 2015 diagonals?
# | 65 | 18 | 2 |
math | 9. Let $N$ be the smallest positive integer such that $N / 15$ is a perfect square, $N / 10$ is a perfect cube, and $N / 6$ is a perfect fifth power. Find the number of positive divisors of $N / 30$. | 8400 | 65 | 4 |
math | 3. On the extensions of sides $\boldsymbol{A B}, \boldsymbol{B C}, \boldsymbol{C D}$ and $\boldsymbol{A}$ of the convex quadrilateral $\boldsymbol{A} \boldsymbol{B C D}$, points $\boldsymbol{B}_{1}, \boldsymbol{C}_{1}, \boldsymbol{D}_{1}$ and $\boldsymbol{A}_{1}$ are taken such that $\boldsymbol{B} \boldsymbol{B}_{1}=\boldsymbol{A B}, \boldsymbol{C} \boldsymbol{C}_{1}=\boldsymbol{B C}, \boldsymbol{D D}_{1}=\boldsymbol{C D}$ and $\boldsymbol{B} \boldsymbol{B}_{1}=\boldsymbol{A B}$ and $\boldsymbol{A} \boldsymbol{A}_{1}=\boldsymbol{A}$. How many times smaller is the area of quadrilateral $\boldsymbol{A} \boldsymbol{B} \boldsymbol{C D}$ compared to the area of quadrilateral $\boldsymbol{A}_{1} \boldsymbol{B}_{1} C_{1} \boldsymbol{D}_{1}$. (10 points) | 5 | 265 | 1 |
math | 3. The product of two different natural numbers is 15 times greater than their sum. What values can the difference between the larger and smaller number take? | 224,72,40,16 | 32 | 12 |
math | A sphere with a surface area of $195.8 \mathrm{dm}^{2}$ is floating on $+4 \mathrm{C}$ water, submerged to a depth of $1.2 \mathrm{dm}$. What is the area of the submerged part of the surface, and what is the weight of the sphere? What is the central angle of the spherical sector corresponding to the submerged surface? | 9147^{\}18^{\\},29.75\mathrm{}^{2},16.04\mathrm{~} | 85 | 33 |
math | 1. Given $\operatorname{ctg} \theta=\sqrt[3]{7}\left(0^{\circ}<\theta<90^{\circ}\right)$. Then, $\frac{\sin ^{2} \theta+\sin \theta \cos \theta+2 \cos ^{2} \theta}{\sin ^{2} \theta+\sin \theta \cos \theta+\cos ^{2} \theta}=$ $\qquad$ . | \frac{13-\sqrt[3]{49}}{6} | 100 | 16 |
math | 1. When one of two integers was increased 1996 times, and the other was reduced 96 times, their sum did not change. What can their quotient be? | 2016 | 38 | 4 |
math | For any natural number $n$, we denote $\mathbf{S}(n)$ as the sum of the digits of $n$. Calculate $\mathbf{S}^{5}\left(2018^{2018^{2018}}\right)$. | 7 | 58 | 1 |
math | 1. Larry can swim from Harvard to MIT (with the current of the Charles River) in 40 minutes, or back (against the current) in 45 minutes. How long does it take him to row from Harvard to MIT, if he rows the return trip in 15 minutes? (Assume that the speed of the current and Larry's swimming and rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss. | 14:24 | 96 | 5 |
math | 3. Let $1997=2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{n}}$, where $a_{1}, a_{2}, \cdots, a_{n}$ are distinct non-negative integers, calculate the value of $a_{1}+a_{2}+\cdots+a_{n}$. | 45 | 79 | 2 |
math | 4. Simplify $\sqrt{13+4 \sqrt{3}}+\sqrt{13-4 \sqrt{3}}$. | 4\sqrt{3} | 29 | 6 |
math | $f$ is a polynomial of degree $n$ with integer coefficients and $f(x)=x^2+1$ for $x=1,2,\cdot ,n$. What are the possible values for $f(0)$? | A (-1)^n n! + 1 | 48 | 11 |
math | B3. The doubling sum function is defined by
$$
D(a, n)=\overbrace{a+2 a+4 a+8 a+\ldots}^{n \text { terms }} .
$$
For example, we have
$$
D(5,3)=5+10+20=35
$$
and
$$
D(11,5)=11+22+44+88+176=341 .
$$
Determine the smallest positive integer $n$ such that for every integer $i$ between 1 and 6 , inclusive, there exists a positive integer $a_{i}$ such that $D\left(a_{i}, i\right)=n$. | 9765 | 158 | 4 |
math | Example 1 As shown, in Rt $\triangle ABC$, the hypotenuse $AB=5, CD \perp AB$. It is known that $BC, AC$ are the two roots of the quadratic equation $x^{2}-(2 m-1) x+4(m-1)=0$. Then the value of $m$ is $\qquad$. | 4 | 76 | 1 |
math | (15) (50 points) Let integers $a, b, c$ satisfy $1 \leqslant a \leqslant b \leqslant c$, and $a \mid b+c+1$, $b \mid c+a+1$, $c \mid a+b+1$. Find all triples $(a, b, c)$. | (1,1,1),(1,2,2),(3,4,4),(1,1,3),(2,2,5),(1,2,4),(2,3,6),(4,5,10),(1,4,6),(2,6,9),(3,8,12),(6,14,21) | 79 | 77 |
math | 【Example 3.42】If $P_{1}(x)=x^{2}-2, P_{i}(x)=P_{1}\left[P_{i-1}(x)\right], i=$ $2,3, \cdots, n$. Solve $P_{n}(x)=x$. | 2\cos\frac{2\pi}{2^{n}-1},\quad=0,1,3,\cdots,2^{n-1}-1,\quad2\cos\frac{2k\pi}{2^{n}+1},\quadk=0,1,3,\cdots,2^{n-1}-1 | 66 | 75 |
math | ## Task 1 - V00501
The fencing of a square garden is being renewed. It costs 992.00 DM. One meter of fencing is charged at 4.00 DM.
Calculate the area of this garden and convert the result into hectares. | 0.38 | 60 | 4 |
math | 7. (CZS 1) Find all real solutions of the system of equations
$$ \begin{aligned} x_{1}+x_{2}+\cdots+x_{n} & =a, \\ x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2} & =a^{2}, \\ \ldots \cdots \cdots \cdots+x_{n}^{n} & =a^{n} . \end{aligned} $$ | a, 0, 0, \ldots, 0 | 110 | 14 |
math | 3. The boy thought of a number. He added 5 to that number, then divided the sum by 2, multiplied the quotient by 9, subtracted 6 from the product, divided the difference by 7, and got the number 3. What number did the boy think of? | 1 | 62 | 1 |
math | 4. Six numbers are written in a row on the board. It is known that each number, starting from the third, is equal to the product of the two preceding numbers, and the fifth number is equal to 108. Find the product of all six numbers in this row. | 136048896 | 59 | 9 |
math | One. (20 points) Divide a cube with an edge length of a positive integer into 99 smaller cubes, among which, 98 smaller cubes are unit cubes. Find the surface area of the original cube. | 150 | 46 | 3 |
math | Given a triangle $ABC$ with integer side lengths, where $BD$ is an angle bisector of $\angle ABC$, $AD=4$, $DC=6$, and $D$ is on $AC$, compute the minimum possible perimeter of $\triangle ABC$. | 25 | 54 | 2 |
math | 8. (10 points) In the expression $(x+y+z)^{2020}+(x-y-z)^{2020}$, the brackets were expanded and like terms were combined. How many monomials $x^{a} y^{b} z^{c}$ with a non-zero coefficient were obtained? | 1022121 | 69 | 7 |
math | At the end-of-year concert of a music school, four violinists performed. Whenever one of them was not playing, they took a seat among the audience. In at least how many pieces did the violinists perform, if each of them had the opportunity to watch any of their (violinist) colleagues from the auditorium? | 4 | 67 | 1 |
math | 2. Points $M$ and $N$ divide side $A C$ of triangle $A B C$ into three parts, each of which is equal to 5, such that $A B \perp B M, B C \perp B N$. Find the area of triangle $A B C$. | \frac{} | 65 | 3 |
math | $\left[\begin{array}{l}\text { Evenness and Oddness }\end{array}\right]$
Several consecutive natural numbers are written on the board. Exactly 52\% of them are even. How many even numbers are written on the board? | 13 | 55 | 2 |
math | 8,9 Find the perimeter of quadrilateral $A B C D$, in which $A B=C D=a, \angle B A D=\angle B C D=\alpha<90^{\circ}, B C \neq A D$. | 2(1+\cos\alpha) | 51 | 8 |
math | $2 n$ men and $2 n$ women can be paired into $n$ mixed (quartet) groups in how many ways? (Tennis game!)
Translating the text as requested, while preserving the original formatting and line breaks. | \frac{(2n!)^{2}}{n!2^{n}} | 53 | 16 |
math | Example 5 Let $x, y, z, w$ be four real numbers, not all zero. Find:
$S=\frac{x y+2 y z+z w}{x^{2}+y^{2}+z^{2}+w^{2}}$'s maximum value. | \frac{1}{2}(1+\sqrt{2}) | 62 | 13 |
math | Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12}$, find all possible values of $x$. | 4, 6 | 54 | 4 |
math | ## Task 13/83
We are looking for all triples $(a ; b ; c)$ of positive integers with $c>1$, that satisfy the Diophantine equation $a^{2 c}-b^{2 c}=665$. | (3;2;3) | 53 | 7 |
math | Find the number of positive integers $n$ such that a regular polygon with $n$ sides has internal angles with measures equal to an integer number of degrees. | 22 | 32 | 2 |
math | 5.1. Mother gives pocket money to her children: 1 ruble to Anya, 2 rubles to Borya, 3 rubles to Vitya, then 4 rubles to Anya, 5 rubles to Borya and so on until she gives 202 rubles to Anya, and 203 rubles to Borya. By how many rubles will Anya receive more than Vitya? | 68 | 98 | 2 |
math | 3. (35 points) Journalists have found out that
a) in the lineage of Tsar Pafnuty, all descendants are male: the tsar himself had 2 sons, 60 of his descendants also had 2 sons each, and 20 had 1 son each;
b) in the lineage of Tsar Zinovy, all descendants are female: the tsar himself had 4 daughters, 35 of his descendants had 3 daughters each, and another 35 had 1 daughter each.
The rest of the descendants of both tsars had no children. Who had more descendants? | 144 | 131 | 3 |
math | ## Task B-1.2.
Solve the inequality $\left(1-\frac{4 x^{3}-x}{x-2 x^{2}}\right)^{-3}>0$. | x\in\langle-1,+\infty\rangle\backslash{0,\frac{1}{2}} | 41 | 25 |
math | Let's explain the following facts. The sequence of digits "1221" is first considered as a number in the base-13, then in the base-12, followed by the base-11, and finally in the base-10 number system. In each case, we convert our number to the number system with the base one less. In the first three cases, the result of the conversion is the same sequence of digits (the same digits in the same order). Why can the final conversion not result in the same sequence as the first three cases? - Also provide another such sequence of digits. | 1221_{b}=1596_{b-1} | 127 | 16 |
math | Example 4 (1992 "Friendship Cup" International Mathematics Competition Question) Find the largest natural number $x$, such that for every natural number $y$, $x$ divides $7^{y}+12 y-1$. | 18 | 51 | 2 |
math | 8. $11 n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns:
$$
\begin{array}{l}
a_{11} a_{12} a_{13} a_{14} \cdots a_{1 n} \\
a_{21} a_{22} a_{23} a_{24} \cdots a_{2 n} \\
a_{31} a_{32} a_{33} a_{34} \cdots a_{3 n} \\
a_{41} a_{42} a_{43} a_{44} \cdots a_{4 n} \\
\cdots \cdots \\
a_{n 1} a_{n 2} a_{n 3} a_{n 4} \cdots a_{m n}
\end{array}
$$
where the numbers in each row form an arithmetic sequence, and the numbers in each column form a geometric sequence, with all common ratios being equal. Given
$$
a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16},
$$
find $\quad a_{11}+a_{22}+\cdots+a_{m}$. | 2-\frac{n+2}{2^{n}} | 293 | 11 |
math | Folklore
Solve the equation: $2 \sqrt{x^{2}-16}+\sqrt{x^{2}-9}=\frac{10}{x-4}$. | 5 | 39 | 1 |
math | Example 4 Given the sequence $\left\{a_{n}\right\}$ with the sum of the first $n$ terms $S_{n}=p^{n}+q(p \neq 0, p \neq 1)$, find the necessary and sufficient condition for the sequence $\left\{a_{n}\right\}$ to be a geometric sequence. | -1 | 79 | 2 |
math | Let $E(n)$ denote the largest integer $k$ such that $5^k$ divides $1^{1}\cdot 2^{2} \cdot 3^{3} \cdot \ldots \cdot n^{n}.$ Calculate
$$\lim_{n\to \infty} \frac{E(n)}{n^2 }.$$ | \frac{1}{8} | 75 | 7 |
math | Let $n \geqslant 3$ be an integer. For each pair of prime numbers $p$ and $q$ such that $p<q \leqslant n$, Morgane writes the sum $p+q$ on the board. She then notes $\mathcal{P}(n)$ as the product of all these sums. For example, $\mathcal{P}(5)=(2+3) \times(2+5) \times(3+5)=280$.
Find all values of $n \geqslant 3$ for which $n$ ! divides $\mathcal{P}(n)$.
Note: If two sums $p+q$ formed from two different pairs are equal to each other, Morgane writes them both. For example, if $n=13$, she writes both sums $3+13$ and $5+11$. | 7 | 192 | 1 |
math | 2. The sum of the first term and the second term of a geometric sequence is 30, the sum of the third term and the fourth term is 120, find the sum of the fifth term and the sixth term. | 480 | 49 | 3 |
math | Snow White has a row of 101 plaster dwarfs arranged by weight from the heaviest to the lightest in her garden, with the weight difference between any two adjacent dwarfs being the same. One day, Snow White weighed the dwarfs and found that the first, heaviest, dwarf weighed exactly $5 \mathrm{~kg}$. What surprised Snow White the most was that when she placed the 76th to 80th dwarfs on the scale, they weighed as much as the 96th to 101st dwarfs. What is the weight of the lightest dwarf?
(M. Mach) | 2.5 | 132 | 3 |
math | 6. The sequence satisfies $a_{0}=\frac{1}{4}$, and for natural number $n$, $a_{n+1}=a_{n}^{2}+a_{n}$. Then the integer part of $\sum_{n=0}^{2011} \frac{1}{a_{n}+1}$ is | 3 | 76 | 1 |
math | Example 9 Let $n=1990$. Then
$$
\frac{1}{2^{n}}\left(1-3 \mathrm{C}_{n}^{2}+3^{2} \mathrm{C}_{n}^{4}-\cdots+3^{99} \mathrm{C}_{n}^{108}-3^{90} \mathrm{C}_{n}^{900}\right) \text {. }
$$
(1990, National High School Mathematics Competition) | -\frac{1}{2} | 114 | 7 |
math | 8. (9) In the country, there are 101 airports. Each possible route is served by exactly one airline (in both directions). It is known that no airline can organize a round trip that includes more than two cities without repeating any city. What is the minimum possible number of airlines? | 51 | 62 | 2 |
math | Eight 1 Ft coins, one of which is fake. It's a very good counterfeit, differing from the others only in that it is lighter. Using a balance scale, how can you find the fake 1 Ft coin in 2 weighings? | 2 | 51 | 1 |
math | Example 7 A positive integer $n$ cannot be divisible by $2$ or $3$, and there do not exist non-negative integers $a$, $b$ such that $\left|2^{a}-3^{b}\right|=n$. Find the minimum value of $n$.
(2003, National Training Team Problem) | 35 | 71 | 2 |
math | Find the positive integer $n$ such that $10^n$ cubic centimeters is the same as 1 cubic kilometer.
| 15 | 27 | 2 |
math | Example 6. Find the mathematical expectation of a random variable $X$, having an exponential distribution. | M(X)=\frac{1}{\alpha},\sigma(x)=\frac{1}{\alpha} | 20 | 23 |
math | 6. Define an operation $*$ on the set of positive real numbers, with the rule: when $a \geqslant b$, $a * b=b^{a}$; when $a<b$, $a * b=b^{2}$. According to this rule, the solution to the equation $3 * x$ $=27$ is $\qquad$ . | 3,3\sqrt{3} | 78 | 8 |
math | Consider finitely many points in the plane with no three points on a line. All these points can be coloured red or green such that any triangle with vertices of the same colour contains at least one point of the other colour in its interior.
What is the maximal possible number of points with this property? | 8 | 61 | 1 |
math | 19. A bank issues ATM cards to its customers. Each card is associated with a password, which consists of 6 digits with no three consecutive digits being the same. It is known that no two cards have the same password. What is the maximum number of ATM cards the bank has issued?
A bank issues ATM cards to its customers. Each card is associated with a password, which consists of 6 digits with no three consecutive digits being the same. It is known that no two cards have the same password. What is the maximum number of ATM cards the bank has issued? | 963090 | 117 | 6 |
math | 5. Simplify $C_{n}^{0} C_{n}^{1}+C_{n}^{1} C_{n}^{2}+\cdots+C_{n}^{n-1} C_{n}^{n}=$ | C_{2n}^{n-1} | 54 | 10 |
math | 8. For some subsets of the set $\{1,2,3, \cdots, 100\}$, no number is twice another number. The maximum number of elements in such a subset is $\qquad$ . | 67 | 49 | 2 |
math | Given a positive integer $n$ greater than 2004, fill the numbers $1, 2, \cdots, n^2$ into an $n \times n$ chessboard (consisting of $n$ rows and $n$ columns of squares) such that each square contains exactly one number. If a number in a square is greater than the numbers in at least 2004 other squares in its row and at least 2004 other squares in its column, then this square is called a "super square." Determine the maximum number of "super squares" on the chessboard.
(Feng Yuefeng, problem contributor) | n(n-2004) | 138 | 8 |
math | Example 11 Given a positive integer $n$, find the number of ordered quadruples of integers $(a, b, c, d)$ (where $0 \leqslant a \leqslant b \leqslant c \leqslant d \leqslant n$). | \mathrm{C}_{n+4}^{4} | 65 | 12 |
math | Let's calculate the value of the expression under a) and simplify the expressions under b) and c):
a) $0.027^{-\frac{1}{3}}-\left(-\frac{1}{6}\right)^{-2}+256^{0.75}+0.25^{0}+(-0.5)^{-5}-3^{-1}$.
b) $\left[\frac{\left(a^{\frac{3}{4}}-b^{\frac{3}{4}}\right)\left(a^{\frac{3}{4}}+b^{\frac{3}{4}}\right)}{a^{\frac{1}{2}}-b^{\frac{1}{2}}}-\sqrt{a b}\right] \cdot \frac{2 \cdot \sqrt{2.5}(a+b)^{-1}}{(\sqrt{1000})^{\frac{1}{3}}}$.
c) $\left\{1-\left[x\left(1+x^{2}\right)^{-\frac{1}{2}}\right]^{2}\right\}^{-1} \cdot\left(1+x^{2}\right)^{-1} \cdot\left[x^{0}\left(1+x^{2}\right)^{-\frac{1}{2}}-x^{2}\left(1+x^{2}\right)^{-\frac{1}{2}}\right]$. | 0 | 312 | 1 |
math | Find all positive integers $x, y, z$ such that
$$
45^{x}-6^{y}=2019^{z}
$$ | (2,1,1) | 33 | 7 |
math | Suppose an integer $x$, a natural number $n$ and a prime number $p$ satisfy the equation $7x^2-44x+12=p^n$. Find the largest value of $p$. | 47 | 46 | 2 |
math | 3. At the mathematics olympiad in two rounds, it is necessary to solve 14 problems. For each correctly solved problem, 7 points are given, and for each incorrectly solved problem, 12 points are deducted. How many problems did the student solve correctly if he scored 60 points? | 12 | 64 | 2 |
math | 1 We write $\{a, b, c\}$ for the set of three different positive integers $a, b$, and $c$. By choosing some or all of the numbers $a, b$ and $c$, we can form seven nonempty subsets of $\{a, b, c\}$. We can then calculate the sum of the elements of each subset. For example, for the set $\{4,7,42\}$ we will find sums of $4,7,42,11,46,49$, and 53 for its seven subsets. Since 7,11 , and 53 are prime, the set $\{4,7,42\}$ has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors, 1 and themselves. In particular, the number 1 is not prime.)
What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set $\{a, b, c\}$ that has that number of subsets with prime sums, and explain why no other three-element set could have more. | 5 | 248 | 1 |
math | ## Zadatak B-2.1.
Koliko je $1+z^{2}+z^{4}+\cdots+z^{2 \cdot 2019}$, ako je $z=\frac{1+\sqrt{3} i}{2}$ ?
| 1 | 58 | 1 |
math | 2. Calculate the value of the expression $\frac{a^{2}}{a b+b^{2}}$ if the ratio of the numbers $a$ and $b$ is $2: 5$. | \frac{4}{35} | 43 | 8 |
math | Five. (Full marks 20 points) Let the constant $a>1>b>0$. Then, under what relationship between $a$ and $b$ is the solution set of $\lg \left(a^{x}-b^{x}\right)>0$ $(1, +\infty)$? | a=b+1 | 64 | 4 |
math | 13. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=1, a_{n}=2 a_{n-1}+n-2(n \geqslant 2) \text {. }
$$
Find the general term $a_{n}$. | a_{n}=2^{n}-n(n \geqslant 1) | 67 | 18 |
math | Five, it is known that the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ passes through the fixed point $A(1, 0)$, and its foci are on the $x$-axis. The ellipse intersects the curve $|y|=x$ at points $B$ and $C$. There is a parabola with $A$ as its focus, passing through points $B$ and $C$, and opening to the left, with the vertex coordinates of the parabola being $M(m, 0)$. When the eccentricity $e$ of the ellipse satisfies $\frac{2}{3}<e^{2}<1$, find the range of the real number $m$. | m>1 | 163 | 3 |
math | 2. [5 points] Several pairwise distinct natural numbers are written on the board. If the smallest number is multiplied by 30, the sum of the numbers on the board will become 450. If the largest number is multiplied by 14, the sum of the numbers on the board will also become 450. What numbers could have been written on the board | 13,14,17,29or13,15,16,29 | 79 | 23 |
math | 5. How many five-digit natural numbers are there in which all digits are different, and among which the digits 1 and 2 appear in adjacent decimal places?
The use of a pocket calculator or any reference materials is not allowed.
# Ministry of Science and Education of the Republic of Croatia Agency for Education and Education Croatian Mathematical Society
COUNTY COMPETITION IN MATHEMATICS February 26, 2024. 6th grade - elementary school
Each task is worth 10 points. In addition to the final result, the process is also graded. To earn all points, it is necessary to find all solutions, determine that there are no others, document the process, and justify your conclusions. | 2436 | 147 | 4 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{\sqrt{n^{2}+3 n-1}+\sqrt[3]{2 n^{2}+1}}{n+2 \sin n}$ | 1 | 58 | 1 |
math | 8. (i) (Grade 11) Given the function
$$
f(x)=2 \cos \left(\frac{k}{4} x+\frac{\pi}{3}\right)
$$
the smallest positive period is no greater than 2. Then the smallest positive integer value of $k$ is $\qquad$ .
(ii) (Grade 12) The line $y=k x-2$ intersects the parabola $y^{2}$ $=8 x$ at points $A$ and $B$. If the x-coordinate of the midpoint of $A B$ is 2, then $|A B|=$ $\qquad$ | 2 \sqrt{15} | 138 | 7 |
math | 3. Given the complex number $z$ satisfies $|z+\sqrt{3} i|+|z-\sqrt{3} i|=4$, then the minimum value of $|z-\mathrm{i}|$ is $\qquad$ . | \frac{\sqrt{6}}{3} | 51 | 10 |
math | What is the largest perfect square that can be written as the product of three different one-digit positive integers? | 144 | 21 | 3 |
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