id stringlengths 64 64 | problem stringlengths 16 4.03k | gold_answer stringlengths 1 208 | dataset_source stringclasses 3
values | sub_source stringclasses 21
values | domain stringclasses 9
values | math_level int64 | synthetic bool 0
classes | answer_type stringclasses 8
values |
|---|---|---|---|---|---|---|---|---|
9c2c2583b4e1157af57945c31370eb6a7f32d44a27be107f9e996a372ba14e80 | How many positive common divisors do $10^{100}$ and $10^{121}+10^{813}+10$ have? | 4 | numina | olympiads | Number Theory | null | null | integer |
35cb06088304e3a6d6bdfaf3f9a0f9718ad25666fa57d8eeffa8b45bdf272658 | Two identical circles of radius \(r\) are externally tangent to each other and internally tangent to the ellipse \(9x^2 + 4y^2 = 36.\) Determine the radius \(r.\) | 2 | numina | synthetic_math | Geometry | null | null | integer |
ea6682d5027c56c9bd1c19a6672625bb287d340ade0d11d2138c7a257511787c | A set contains 96 distinct blocks. Each block is one of 2 materials (plastic, wood), 3 sizes (small, medium, large), 5 colors (blue, green, red, yellow, purple), and 4 shapes (circle, hexagon, square, triangle). Determine how many blocks in the set differ from the 'plastic medium red circle' in exactly 3 ways. | 26 | numina | synthetic_math | Combinatorics | null | null | integer |
0c746fed208ec533671baf046ce1d68e5d0fd0e6b86e5e7fbfaea93081255e70 | 8. (10 points) Every day, Xiaoming has to pass through a flat section $AB$, an uphill section $BC$, and a downhill section $CD$ (as shown in the figure). It is known that $AB: BC: CD=1: 2: 1$, and Xiaoming's speed ratio on the flat section, uphill section, and downhill section is 3: 2: 4. What is the ratio of the time ... | 19:16 | numina | olympiads | Algebra | null | null | other |
1b9bb4f1d0ad5d5e830fba8619d1d04c32809a4a08fc695ef1eb2fc7f08b4033 | Consider the set of all points $(x,y)$ in the coordinate plane where one coordinate is exactly three times the other. Plot all such points and determine into how many regions the resulting graph would split up the plane. | 4 | numina | synthetic_math | Geometry | null | null | integer |
896d806706a3763b4e12e11a64648d5da67f3c50c07636ebe327074932cfa9e7 | Each pair $(x, y)$ of nonnegative integers is assigned number $f(x, y)$ according the conditions:
$f(0, 0) = 0$;
$f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y)$,
$f(2x + 1, 2y) = f(2x, 2y + 1) = f(x ,y) + 1$ for $x, y \ge 0$.
Let $n$ be a fixed nonnegative integer and let $a$, $b$ be nonnegative integers such that $f(a, b) =... | 2^n | numina | aops_forum | Logic and Puzzles | null | null | expression/text |
c4950d372d1d11716b1f0aecd4543766e12ecf2b1dcdba51448e7d119b5720f7 | Rewrite $\sqrt[3]{2^9 \cdot 3^3 \cdot 7^3}$ as an integer. | 168 | numina | synthetic_math | Number Theory | null | null | integer |
585a355b68cb78c179bca7de267c8710bc06052ae647bb2d09ea97955c295b99 | In a trapezoid, the midline is equal to 7, the height is equal to $\frac{15 \sqrt{3}}{7}$, and the angle between the diagonals opposite the base is $120^{\circ}$.
Find the diagonals of the trapezoid. | 610 | numina | olympiads | Geometry | null | null | integer |
b0a4205b611b027c9ed384377b4e10be077d3e8455d65347b917c87cebefbacb | ## Task 2 - 250522
From the Mathcity train station, a bus departs at every full quarter hour and arrives in Knobelhausen after 2 hours.
From there, buses also depart every quarter hour on the same road to Mathcity, where they arrive after 2 hours of travel.
In the morning, the first bus from Mathcity departs at 5:00... | 16 | numina | olympiads | Logic and Puzzles | null | null | integer |
1e6821c7c8737c52c7d0f8548f6052e061bccaad090ab99c33265f0596251367 | A3. What is the largest integer for which each pair of consecutive digits is a square? | 81649 | numina | olympiads | Number Theory | null | null | integer |
148d95702d330a8d0246ac6148f9fb265a0b0aecf122ed17392ddf68d9d6ee26 | How many integers $m$ satisfy the inequality $-5\pi \le m \le 12\pi$? | 53 | numina | synthetic_math | Inequalities | null | null | integer |
5e5e58f2b774bda22e36ccb0d85c1b69a5bdbf8d737f0c176e5e329457657e35 | What is the value of $x$ if $x = \frac{2021^2 - 2020}{2021} + 7$? | 2027 | numina | synthetic_math | Algebra | null | null | integer |
78ffd2fa84042d6c13047d74af7344f758d52981565517cd2b3e4a46a1e608c0 | 16. $[\mathbf{1 0}]$ Let $R$ be a semicircle with diameter $X Y$. A trapezoid $A B C D$ in which $A B$ is parallel to $C D$ is circumscribed about $R$ such that $A B$ contains $X Y$. If $A D=4, C D=5$, and $B C=6$, determine $A B$. | 10 | numina | olympiads | Geometry | null | null | integer |
01e253874ff85252c5000d87362ab6bf4dd0e56e0f84f8fea08395ddebf47e6c | In hexagon $FIGURE$, $\angle F \cong \angle I \cong \angle U$, $\angle G$ is supplementary to $\angle R$ and $\angle E$ is supplementary to $\angle U$. How many degrees are in the measure of $\angle U$? | 120^\circ | numina | synthetic_math | Geometry | null | null | expression/text |
f7ed13cd26841c460cee774af510a100f657836aaea2d505b5cee8a00e14ed67 | Let $z$ be a complex number that satisfies
\[|z - 3i| + |z - 4| = 5.\]Find the minimum value of $|z|.$ | \frac{12}{5} | math | intermediate_algebra | Algebra | null | null | fraction |
e88070e8219d0c798ced9165cf07accf5d9ce6b19457cd1902826b325fecc8c9 | 3.2. Find the sum of all two-digit numbers for each of which the sum of the squares of the digits is 57 more than the product of the same digits. | 264 | numina | olympiads | Algebra | null | null | integer |
7a31b348b2421cd545f8bd8d3c7d5b1da4b4d807f18cc5ed2c02fdb1d1bbf1e7 | For which positive integers $n$ is the following statement true:
if $a_1, a_2, ... , a_n$ are positive integers, $a_k \le n$ for all $k$ and $\sum\limits_{k=1}^{{n}}{a_k}=2n$
then it is always possible to choose $a_{i1} , a_{i2} , ..., a_{ij}$ in such a way that
the indices $i_1, i_2,... , i_j$ are different numbers... | n | numina | aops_forum | Number Theory | null | null | expression/text |
32475f838e6be15c2917aacfc11da81ab93c1394c25813cbd0d5465770b921f4 | Alice throws six identical darts. Each hits one of five identical dartboards on the wall. After throwing the six darts, she lists the number of darts that hit each board, from greatest to least. How many different lists are possible? | 11 | numina | synthetic_math | Combinatorics | null | null | integer |
8ecf09fd01389fc9acfc214edaa1b2278dbeaafe35487e403526c8eca7b17659 | 6. It is known that a factory produced 10,000 units of a product in January, 12,000 units in February, and 13,000 units in March. In the $n$-th month, the factory produces $a b^{n}+c$ ten thousand units, where $a$, $b$, and $c$ are constants, and $n=1,2, \cdots, 12$. Then, how many ten thousand units will the factory p... | 1.35 | numina | cn_contest | Algebra | null | null | decimal |
29fc8946792caa98aeb788423dc96f0c0060cf48068dd4b4e511019fb260c02b | Heidi can paint a wall in 60 minutes and Linda can paint the same wall in 40 minutes. If they work together, what fractional part of the wall can they paint in 12 minutes? | \frac{1}{2} | numina | synthetic_math | Algebra | null | null | fraction |
c569c49d90ef008a08693feb1a1bcb84b56a782ea0cbddc5d68a56697e52416a | ## 126. Math Puzzle 11/75
A gardener sold half of all his apples and half an apple to the first buyer, half of the remaining apples and another half an apple to the second buyer, half of the remaining apples and half an apple to the third buyer, and so on.
To the seventh buyer, he sold half of the remaining apples an... | 127 | numina | olympiads | Logic and Puzzles | null | null | integer |
8eb191b911547414e0395ee292e334252139afcd6d98c6b06f8a3727e7c7df73 | 9. (14 points) Given the sequence $\left\{a_{n}\right\}$ with the general term
$$
a_{n}=1+2+\cdots+n\left(n \in \mathbf{N}_{+}\right) \text {, }
$$
take all multiples of 3 from this sequence to form a new sequence $b_{1}, b_{2}, \cdots, b_{m}, \cdots$. Find the sum of the first $2 m$ terms of the sequence $\left\{b_{m... | \frac{3}{2} m(m+1)(2 m+1) | numina | cn_contest | Algebra | null | null | expression/text |
6d453e674ac25dc3930af10284cd8ff59160e50131e90aaa9ff9eebcf16a7897 | 7.6 On a circle with a circumference of 1956, what is the minimum number of points that should be selected so that for each of these points, there is exactly 1 point at a distance of 1, and exactly 1 point at a distance of 2 (the distance between two points is measured along the arc)? | 1304 | numina | olympiads | Combinatorics | null | null | integer |
fbfaf3200ab86bc25362aac77f05a2a77cc88ecd92a775b383ceac21a3661479 | 12. Let the lengths of the two legs of a right triangle be $a$ and $b$, and the length of the hypotenuse be $c$. If $a$, $b$, and $c$ are all integers, and $c=\frac{1}{3} a b-(a+b)$, find the number of right triangles that satisfy the condition. | 3 | numina | cn_contest | Number Theory | null | null | integer |
6d84b15ccb345eea2304ee7ad90a78b8408722a4bdae3be8b79361c1aede80eb | Determine the integer \( n \), where
\[
\frac{\tan \frac{\pi}{6} + i}{\tan \frac{\pi}{6} - i}
\]
is a twelfth root of unity, meaning it equals \( \cos \frac{2n \pi}{12} + i \sin \frac{2n \pi}{12} \) for some integer \( n \) between 0 and 11 inclusive. | 1 | numina | synthetic_math | Algebra | null | null | integer |
a07e5ab807fbbc52b91644b84ec0d70af4d3bb862df5821862afaf3b1f9c3b66 | The volume of the cone formed by rotating the right-angled triangle around one of its legs is $800 \pi \mathrm{cm}^{3}$, and the volume of the cone formed by rotating around the other leg is $1920 \pi$ $\mathrm{cm}^{3}$. What is the length of the hypotenuse of this triangle? (in cm) | 26 | numina | cn_contest | Geometry | null | null | integer |
317bd6ae17b67374cbf5605d99a3bd7aa338ea029c274151d8bef24255b9c488 | Abe, Bobby, Charles, Devin, Edwin, and Fiona are participants in a race. How many different 1st-2nd-3rd place outcomes are possible if there are no ties, and Fiona cannot finish in the top three? | 60 | numina | synthetic_math | Combinatorics | null | null | integer |
f0834b02b52f7ebc68faf4a2a733e0ec50d6847f13704423aaae22bfbba541c0 | Simplify $5y + 8y + 2y + 7$. | 15y + 7 | numina | synthetic_math | Algebra | null | null | expression/text |
d43f8c9478520e5a61fb4670532d8975c821fe56eb86e5b73fe07a0ce848f5ac | Find the largest integer less than 74 that leaves a remainder of 3 when divided by 7. | 73 | math | number_theory | Number Theory | null | null | integer |
19b78015c08a6a483ad614fd4bf22569b745f09972e63f304cbcb5052596e50f | On the table lies a rectangular sheet of paper. Sasha cuts it along a straight line into two pieces and puts the pieces on the table. Then he takes one of the pieces, cuts it again along a straight line into two pieces, and puts the pieces back on the table. Then he takes one piece from the table and cuts it again, and... | 2015 | numina | olympiads | Combinatorics | null | null | integer |
dbf9f596649599f1422af1549d5d49aca6371dfefa003cec8a4ba426775a8761 | 44. There are $n k$ stones, which are somehow distributed into $n$ piles. It is allowed to double any pile by moving stones from other piles into it in any way. For which $k$ can such operations always (for any $n$) ensure that all remaining piles have an equal number of stones? | k | numina | olympiads | Combinatorics | null | null | expression/text |
f537edc6ad32145df502ba24c425610b8ec15fb071ee1c2d6baebea4250c1ecc | 70. One year, the ages of Xiao Ming's family of four were: Father 30 years old. Mother is two years younger than Father, Brother's age is three times that of Xiao Ming, this year the total age of the four family members is 114, then Xiao Ming's age this year is $\qquad$ | 14 | numina | olympiads | Algebra | null | null | integer |
2ebea267166ad0eaab3a8e8ab38d131b5c809efd2787ccd6a50ed918390f7654 | 3. In $\triangle A B C$, $A B$ is the longest side, $\sin A \sin B=$ $\frac{2-\sqrt{3}}{4}$. Then the maximum value of $\cos A \cos B$ is $\qquad$ . | \frac{2+\sqrt{3}}{4} | numina | cn_contest | Geometry | null | null | radical |
b59b59433c0b0d4e0dfae517dea6d306649d9b164513e3aee07a037e2416d98e | Find all polynomials $P\in \mathbb{Q}[x]$, which satisfy the following equation:
$P^2 (n)+\frac{1}{4}=P(n^2+\frac{1}{4})$ for $\forall$ $n\in \mathbb{N}$.
| P(x) = f^{[n]}(x) | numina | aops_forum | Algebra | null | null | expression/text |
6d8cf9f403242afd281bd805dcbf3d3c7a13b20fc24b1ee05cc28958e6d2eef5 | Four. (Total 20 points) Find the range of real number $a$ such that the inequality $\sin 2 \theta-(2 \sqrt{2}+\sqrt{2} a) \sin \left(\theta+\frac{\pi}{4}\right)-\frac{2 \sqrt{2}}{\cos \left(\theta-\frac{\pi}{4}\right)}>$ $-3-2 a$, holds for all $\theta \in\left[0, \frac{\pi}{2}\right]$. | a>3 | numina | cn_contest | Inequalities | null | null | expression/text |
4642a35f5af03ae98d44ded6247f4b8994bbf91c2555c769a35010f443e4ddc1 | 3. There is a sequence of numbers, starting from the 2nd number, each number is equal to the sum of its adjacent two numbers. It is known that the 2nd number is $\frac{1}{2}$, the 4th number is $-\frac{5}{6}$, and the 2022nd number is $a$. Then $3a=$ $\qquad$ . | 1 | numina | olympiads | Algebra | null | null | integer |
c162237b1f7b2a039ed98f54d9564fe382868c2441c21e37dbd39de0c6f02204 | What is $2.375$ expressed as a fraction? | \frac{19}{8} | numina | synthetic_math | Algebra | null | null | fraction |
2f27d4bdbe93ce5e269f042fc27d03d70d85eeedad075abebf8898d7ecaff171 | 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 | numina | aops_forum | Combinatorics | null | null | integer |
c2cd9edf930f084d5299afc401304327d7e61c2695a5479f2c40f62de295fec6 | Elmo makes $N$ sandwiches for a school event. For each sandwich he uses $C$ slices of cheese at $3$ cents per slice and $T$ slices of tomato at $4$ cents per slice. The total cost of the cheese and tomato to make all the sandwiches is $\$3.05$. Assume that $C$, $T$, and $N$ are positive integers with $N>1$. What is the... | \$2.00 | numina | synthetic_math | Algebra | null | null | decimal |
beae0ad61db1f7e15da60b7b76a069c71c45f0268140171ba3c1df20601f4c50 | Let $t$ be TNYWR.
Azmi has two fair dice, each with six sides.
The sides of one of the dice are labelled $1,2,3,4,5,6$.
The sides of the other die are labelled $t-10, t, t+10, t+20, t+30, t+40$.
When these two dice are rolled, there are 36 different possible values for the sum of the numbers on the top faces. What ... | 30.5 | numina | olympiads | Algebra | null | null | decimal |
c135174af4dd1a48f1b7c76220b10941c68f97391c92761b1ba959c488c056f1 | How many three-digit numbers are multiples of neither 4 nor 6? | 600 | numina | synthetic_math | Number Theory | null | null | integer |
79132f99a101f406d40ae26ff1bf1fa27fb9c81886444cf6f95b2706a832f301 | Compute $\begin{pmatrix} -4 \\ -1 \end{pmatrix} \cdot \begin{pmatrix} 6 \\ 8 \end{pmatrix}$. | -32 | math | precalculus | Algebra | null | null | integer |
15d203e139a8642f4bb3711a62ddcfc7f77fe6d69a1207514ea45ffb351bbb9a | 10. In the tetrahedron $P-ABC$, $PA=BC=\sqrt{6}$, $PB=AC=\sqrt{8}$, $PC=AB=\sqrt{10}$. Then the radius of the circumscribed sphere of the tetrahedron is $\qquad$. | \sqrt{3} | numina | olympiads | Geometry | null | null | radical |
5720b9624b94d5d5457d3af113613983200e87ad22cf889b028a83719a05c3f4 | 4. If a die is rolled five times in succession, the probability of the event “the numbers appearing in the five rolls are neither all the same nor all different, and from the second roll onwards, each number is not less than the previous one” is | \frac{5}{162} | numina | olympiads | Combinatorics | null | null | fraction |
4da01eebfbde5eca0dcc800e1502be607ceb8b160bddb5698ee9947d5b660e27 | Example 8. Solve the inequality
$$
\sqrt{-x^{2}+6 x-5}>8-2 x
$$ | 3<x\leqslant5 | numina | olympiads | Inequalities | null | null | expression/text |
2179d3b9e4f3ae2f2c5942c5698bf6c59e83f55b15a3cb6abaa1755cb6c28650 | Consider the graph of the function \(f(x) = a(x+2)^2 + 3\). A portion of the graph with 1 unit grid lines is shown, passing through the points \((-2, 3)\) and \((0, 7)\). Determine the value of \(a+3a+2\). | 6 | numina | synthetic_math | Algebra | null | null | integer |
fbde189538eb7cc5d70341d8a33aa72e2694ab47fb548a57057b0ef3f08bc3e6 | 5. There is a ruler 10 cm long without divisions. What is the smallest number of intermediate divisions that need to be made on the ruler so that segments of length 1 cm, 2 cm, 3 cm, ..., 10 cm can be laid off, applying the ruler in each case only once. | 4 | numina | olympiads | Logic and Puzzles | null | null | integer |
a9ac8f63e1f8aaa1bb81a48a309575245a0573d41109ad86d1ae05909c4e4952 | 6. Each of the 10 dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: butter, chocolate, or fruit. First, Snow White asked those who love butter ice cream to raise their hands, and everyone raised their hands, then those who love chocolate ice cream - ... | 4 | numina | olympiads | Logic and Puzzles | null | null | integer |
de98624542ed77e459132fe420cefb5162beca7841303448f431d5e5aaf55a02 | Compute
\[
\sum_{n=1}^\infty \frac{2n^2 - 3n + 2}{n(n+1)(n+2)}.
\] | -4 | numina | synthetic_math | Algebra | null | null | integer |
baabc98f875044bb543d8c289ca4c6f13be78646ddabd06c8d98e067b7a3a433 | What is the smallest possible perimeter, in units, of a triangle whose side-length measures are consecutive integer values? | 9 | math | geometry | Geometry | null | null | integer |
9f7ffed7b34707e1f2be2842422f004e1f23bd79fb9a9f2dfaa852e8be923d64 | Jane is deciding to paint a wall in her living room. The wall is 10 feet high and 15 feet long. However, there is a 3-foot by 5-foot door on this wall that she will not paint. How many square feet of the wall will Jane have to paint? | 135 | numina | synthetic_math | Geometry | null | null | integer |
59dbf803a741655815fbd8a1d052d3ee85249d43c4edebb47c587d845480c290 | Consider a deck of cards arranged in a continuously repeating pattern as follows: $$A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2, A, 2, 3,\dots$$ What is the 73rd card in this sequence? | 8 | numina | synthetic_math | Number Theory | null | null | integer |
4543a2f581481c3d2cdac88ffb7181c3bbbdd25f630aba8955db719cbd3142dc | A circle has a radius of three inches. The distance from the center of the circle to chord $CD$ is two inches. How many inches long is chord $CD$? Express your answer in simplest radical form. | 2\sqrt{5} | math | geometry | Geometry | null | null | radical |
0ee00d1a098476d98adc14255447aafc839423b98c473bc4b8a53c0fd366b167 | 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 o... | 1221_{b}=1596_{b-1} | numina | olympiads | Number Theory | null | null | expression/text |
2289402790762377e92402ba64697f5055cb19b6da7a5786e2f7f8992fe6fac1 | Find the value of $b$ so that the lines with these equations are perpendicular:
\begin{align*}
y &= 3x - 7 \\
8y + bx = 16.
\end{align*} | \frac{8}{3} | numina | synthetic_math | Algebra | null | null | fraction |
6671bb95be201d59720fa686520452befe6f68ccbddc6d5c5c826cd7f146bd74 | Example 5 Given a positive integer $n$ that satisfies the following condition: In any $n$ integers greater than 1 and not exceeding 2009 that are pairwise coprime, at least one is a prime number. Find the minimum value of $n$. ${ }^{[2]}$ | 15 | numina | cn_contest | Number Theory | null | null | integer |
7092f1c348e53df5a9b8b53c59767e898f1068b234db65b3a77f5db15fffdec0 | Two cylindrical tanks have the same volume. The height of one tank is double the height of the other. If the radius of the taller tank is 10 units, what is the radius of the shorter tank? Provide your answer in simplest radical form. | 10\sqrt{2} | numina | synthetic_math | Geometry | null | null | radical |
6a0ff76bb3d0e127a77e34c20217abcfa15288a5cfde98adbc0a40bd7dee90c4 | Mira has a cubical tank with each side measuring 8 feet. She wants to fill it from a nearby spring using a spherical container with a diameter of 6 feet. How many trips must Mira make to the spring in order to fill the tank? | 5 | numina | synthetic_math | Geometry | null | null | integer |
1395d8d8ead0f05a0ef0a50f076696920286064a2c5c6fd3fba8ad8f3382abea | 2. Let $k$ be a real number, and the quadratic equation $x^{2}+k x+k+1=0$ has two real roots $x_{1}$ and $x_{2}$. If $x_{1}+2 x_{2}^{2}=k$, then $k$ equals $\qquad$ . | 5 | numina | cn_contest | Algebra | null | null | integer |
ed72930f2750191c170226d2fa507135fb31226382c611188ea3377da1443d0f | Calculate the area of the portion of the circle defined by \(x^2 + y^2 - 16x - 8y = 0\) that lies above the line \(y = x - 4\) and below the line \(y = -x + 4\). | 40\pi | numina | synthetic_math | Geometry | null | null | with_pi |
336f6c0b7c4acb50af6a24046dc2f8d17d2df999c6863c25985134714017bc8e | 4. Given the equation about $x$: $k x^{2}+\frac{|x|}{x+6}=0$ has four distinct real solutions. Then the range of the real number $k$ is $\qquad$ | k<-\frac{1}{9} | numina | olympiads | Algebra | null | null | expression/text |
6f9be953e457bf6d1e03ad6996feb3ce45f9ba78e26ef9d1f9042a7398d9ec70 | Find the largest positive integer $n$ such that the following is true:
There exists $n$ distinct positive integers $x_1,~x_2,\dots,x_n$ such that whatever the numbers $a_1,~a_2,\dots,a_n\in\left\{-1,0,1\right\}$ are, not all null, the number $n^3$ do not divide $\sum_{k=1}^n a_kx_k$. | 9 | numina | aops_forum | Number Theory | null | null | integer |
83dfcd9a4d5305706da19eb6aac002a32e9f00ee109294c1ef2e740a68005eae | How many integers $n$ satisfy $(n+2)(n-8) \le 0$? | 11 | numina | synthetic_math | Inequalities | null | null | integer |
c50e0f3b3583d254beee8ee87d5a292ade5630343e909fd1ae802f98febad4b4 | 10. 4. In a regular 2009-gon, a non-negative integer not exceeding 100 is placed at each vertex. Adding 1 to the numbers at two adjacent vertices is called an operation on these two adjacent vertices. For any given two adjacent vertices, the operation can be performed at most $k$ times. Find the minimum value of $k$ su... | 100400 | numina | cn_contest | Combinatorics | null | null | integer |
74beb24eb5c1303889f52e0f94913801f4581cd08e76a2d171b3940192d86ca7 | In a circle of center $O$ and radius $r$, a triangle $ABC$ of orthocenter $H$ is inscribed. It is considered a triangle $A'B'C'$ whose sides have by length the measurements of the segments $AB, CH$ and $2r$. Determine the triangle $ABC$ so that the area of the triangle $A'B'C'$ is maximum. | (2\sqrt{2} + 2)r | numina | aops_forum | Geometry | null | null | radical |
17d3710ad23d8a72f456bd2ed4a376e76de668770d0ef8ee641856dd24d7a265 | 7. (3 points) Eight students from a school participate in a math competition, and their average score is 82.5 points, with student $A$ scoring 86 points. If student $A$ only scores 74 points, then their average score would decrease by $\qquad$ points. | 1.5 | numina | olympiads | Algebra | null | null | decimal |
307c4134a1378083d01a03a0ffd2e775be7cc9a7a34807d3fbb9bf863ee37d64 | Example 3 Let $x_{i} \geqslant 0(1 \leqslant i \leqslant n), \sum_{i=1}^{n} x_{i}=1, n \geqslant 2$. Find the maximum value of $F=\sum_{1 \leq i<j \leqslant n} x_{i} x_{j}\left(x_{i}+\right.$ $x_{j}$ ). (32nd IMO Shortlist Problem) | \frac{1}{4} | numina | inequalities | Inequalities | null | null | fraction |
2d114af8c2fe96e8c109e7b07775902e67ec2c8a6c4fe704f1fa2c2232fafb72 | If $a, b, c$ are integers from the set of positive integers less than $8$ such that
\begin{align*}
abc &\equiv 1 \pmod{8}, \\
7c &\equiv 3 \pmod{8}, \\
5b &\equiv 4+b \pmod{8},
\end{align*}
then what is the remainder when $a+b+c$ is divided by $8$? | 3 | numina | synthetic_math | Number Theory | null | null | integer |
9a9ade2f82d68d09d4e7773db0f8846bf5a3850fc4cd7723f59568d7b373b81a | Andrew takes a square sheet of paper $ABCD$ of side length $1$ and folds a kite shape. To do this, he takes the corners at $B$ and $D$ and folds the paper such that both corners now rest at a point $E$ on $AC$. This fold results in two creases $CF$ and $CG$, respectively, where $F$ lies on $AB$ and $G$ lies on $AD$. Co... | 2\sqrt{2} - 2 | numina | aops_forum | Geometry | null | null | radical |
da5ad3d608409467afa22db729d285f8d9218f641cd4bf82a6a96e9d9f6864b8 | A positive integer is called "balanced" if it has four digits, and one of these digits is equal to the average of the other three. For example: the number 2631 is balanced because 3 is the average of 2, 6, and 1; 4444 is also balanced because 4 is the average of 4, 4, and 4.
a) Find the three smallest balanced numbers... | 90 | numina | olympiads | Number Theory | null | null | integer |
7f6b15b3e5a942e1c5051c02b3a892823e61c623092c41764a5dabe29f26b989 | The expression $\log_{y^6}{x}\cdot\log_{x^5}{y^2}\cdot\log_{y^4}{x^3}\cdot\log_{x^3}{y^4}\cdot\log_{y^2}{x^5}$ can be written as $a\log_y{x}$ for what constant $a$? | \frac16 | math | intermediate_algebra | Algebra | null | null | fraction |
616a63e2a09e8b464ec4e4acedfce4149d248cfbbb66221887dc6a5258b1f659 | 2. Solve the equation $\cos 2 x+\cos 6 x+2 \sin ^{2} x=1$.
In the answer, specify the number equal to the sum of the roots of the equation that belong to the interval $A$, rounding this number to two decimal places if necessary.
$$
A=\left[\frac{m \pi}{6} ; \frac{(m+1) \pi}{6}\right], m=5
$$ | 2.88 | numina | olympiads | Algebra | null | null | decimal |
90aa959fd3a9adc454dacbf855b3f80730bbfc015732ac2b8bf83c6020109d33 | Let \( f(x) = ax + b \), where \( a \) and \( b \) are real constants, and \( g(x) = 3x - 6 \). It is known that for all \( x \), \( g(f(x)) = 4x + 5 \). What is the value of \( a + b \)? | 5 | numina | synthetic_math | Algebra | null | null | integer |
6263df8663bc9b4a18648910207cab078d915557d43c02c110875483bcd08358 | Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$,
$I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$
Find $\lim_{n\to\infty} I_n.$
[i]2009 Tokyo Institute of Technology, Master Course in Mathematics[/i] | 0 | numina | aops_forum | Calculus | null | null | integer |
4c659a65164ecfcc181d2da2689d4d551dd20ee7072dc3d1c5909b35efe60a1c | 10 days * * We notice that $6!=8 \cdot 9 \cdot 10$, try to find the largest positive integer $n$ such that $n!$ can be expressed as the product of $n-3$ consecutive natural numbers.
| 23 | numina | olympiads | Number Theory | null | null | integer |
edb0236f1f9869812b115db31bccbd656b7631d7ab4b37a1036011264522b70d | The sequence $ \{ a_n \} _ { n \ge 0 } $ is defined by $ a_0 = 2 , a_1 = 4 $ and
\[ a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1} \]
for all positive integers $ n $. Determine all prime numbers $ p $ for which there exists a positive integer $ m $ such that $ p $ divides the number $ a_m - 1 $. | p > 2 | numina | aops_forum | Number Theory | null | null | expression/text |
f3141e8ee71d61d7fe54a5ffda3fe354d25391aa83867e4c3c4ecaace2611192 | For real numbers $u$, consider the intersection of the lines $2x + 3y = 8u + 4$ and $3x - 2y = 5u - 3$. All intersection points lie on a line. Determine the slope of this line. | \frac{10}{31} | numina | synthetic_math | Algebra | null | null | fraction |
e030a6a5da504d29386c9f33c24f6248eef092971a93e05cf3c42839c996dad4 | 8. Given the hyperbola $x^{2}-y^{2}=t(t>0)$ with its right focus at $F$, any line passing through $F$ intersects the right branch of the hyperbola at points $M$ and $N$. The perpendicular bisector of $M N$ intersects the $x$-axis at point $P$. When $t$ is a positive real number not equal to zero, $\frac{|F P|}{|M N|}=$... | \frac{\sqrt{2}}{2} | numina | olympiads | Geometry | null | null | radical |
eeb782a9d16558bbc7a0ee46fbcb4bb58257e23ac80e4e4dd7ec7321ea751a58 | 131 Given that the function $f(n)$ is a strictly increasing function defined on $\mathbf{N}_{+}$, with its range also in $\mathbf{N}_{+}$, and satisfies $f(f(n))=3n$. Find $f(2003)$. | 3822 | numina | cn_contest | Number Theory | null | null | integer |
4fc4d2cddb6996b8614f4a8e9fee68a1a441fd8c5fcaf578122e8399b25194ce | 1. Given the real number pair $(x, y)$ satisfies the equation $(x-2)^{2}+y^{2}=3$, let the minimum and maximum values of $\frac{y}{x}$ be $m$ and $n$ respectively. Then $m+n=$ | 0 | numina | cn_contest | Geometry | null | null | integer |
afc335fe93edd1f7cd27b8ca1c8408717af1fe19622c783dcba5735f184e8b8f | In Miroslav's kingdom, the cobbler Matěj used to go not only to sing but also to eat and drink well. For one gold piece, he got a whole goose and one jug of wine. Then, however, they increased the prices by $20 \%$, and for a gold piece, he got only half a jug of wine and a whole goose. It is said that after the full m... | 0.96 | numina | olympiads | Algebra | null | null | decimal |
15acb74a68cadac644219fdc75c2adaa8a8a0932ea94bb784eb51f79dd511157 | Consider the integral $$\int_{-1}^1 x^nf(x) \; dx$$ for every $n$-th degree polynomial $f$ with integer coefficients. Let $\alpha_n$ denote the smallest positive real number that such an integral can give. Determine the limit value $$\lim_{n\to \infty} \frac{\log \alpha_n}n.$$ | -\log 2 | numina | aops_forum | Calculus | null | null | expression/text |
f96954c9710b6150c71deed95adac4370f86ea33091843b75fa9b1703edbd624 | Given four points not lying in the same plane. How many planes are there from which all four points are equidistant? | 7 | numina | olympiads | Geometry | null | null | integer |
7aff8c4c66ed0908d5df9ead646b0595eea9c9f8981ea64902a2566702d2ab09 | A projectile is launched upwards and its height in meters above the ground after \( t \) seconds is given by the equation \( h(t) = -20t^2 + 50t + 10 \). What is the maximum height reached by the projectile? | 41.25 | numina | synthetic_math | Algebra | null | null | decimal |
43afb76a05982800fd332e12338324655fc2cee0b7fe88f8040ebce34c5afc0d | Let $ \triangle ABC $ be an equilateral triangle with height $13$, and let $O$ be its center. Point $X$ is chosen at random from all points inside $ \triangle ABC $. Given that the circle of radius $1$ centered at $X$ lies entirely inside $ \triangle ABC $, what is the probability that this circle contains $O$? | \frac{\pi}{121} | numina | aops_forum | Geometry | null | null | with_pi |
77816ec5c4f36e5c90241d6765409517115ad665cde9cf1eb439301c6ec9eea8 | Diameter $A B$ and chord $C D$ intersect at point $M, \angle C M B=73^{\circ}$, the angular magnitude of arc $B C$ is $110^{\circ}$. Find the magnitude of arc $B D$. | 144 | numina | olympiads | Geometry | null | null | integer |
893b5c390cbb4678c190ab0b14df1c15b096eba0604d829f66603dd5551e6a88 | In a right triangle, one of the acute angles $\beta$ satisfies
\[\sin \beta = \frac{1}{\sqrt{3}}.\]
Let $\phi$ be the angle between the altitude and the angle bisector drawn from this acute angle. Find $\tan \phi$. | \tan \phi = \sqrt{\frac{5 - 2\sqrt{6}}{5 + 2\sqrt{6}}} | numina | synthetic_math | Geometry | null | null | radical |
a47cbacc5afab175c8687a2dbd43b3703eaf0fb3588a6c53d9a3b1e62e988bd2 | Let $f(x) = x^3 + 6$ and $g(x) = 3x^2 + 3x + 2$. What is $g(f(-3))$? | 1262 | numina | synthetic_math | Algebra | null | null | integer |
71f9f4c2f71f13c9c1ee57539c3e5e0d98d5c5960d26cfb5fb3671891c8cc53e | 13. Draw two perpendicular chords $O A, O B$ through the vertex of the parabola $y^{2}=4 p x(p>0)$, find the locus of the projection $M$ of the vertex $O$ of the parabola on the line $A B$. | (x-2p)^{2}+y^{2}=4p^{2} | numina | olympiads | Geometry | null | null | expression/text |
8402e855cac755298f6608061cdfc7e64165a982bd3af8800bb4350edb24b5f5 | ## Example. Solve the integral equation
$$
\int_{0}^{x} \cos (x-t) \varphi(t) d t=x
$$ | \varphi(x)=1+\frac{x^{2}}{2} | numina | olympiads | Calculus | null | null | expression/text |
47b2714aec9171dbdfca9fd2a79cbe9bc70ad9a42ff3675a0b44cca4c223ba45 | Let $h(x) = \sqrt[4]{\frac{x+4}{5}}$. For what value of $x$ is $h(3x) = 3h(x)$? Express your answer in simplest form. | -\frac{160}{39} | numina | synthetic_math | Algebra | null | null | fraction |
5ee0d62d2c5b21fe29edac57c2b75f4d2bf7488204a439589767c85c84b1aa1f | Cassidy rolls a fair icosahedral die labeled with numbers $1, 2, 3, \ldots, 20$. What is the expected number of digits in the number she obtains? Express your answer as a decimal. | 1.6 | numina | synthetic_math | Number Theory | null | null | decimal |
a8fa92d283011273e579c6b477a6486111855e9374d3a7480b736b4301a01f83 | Given that $\theta$ and $\phi$ are acute angles such that $\tan \theta = \frac{1}{3}$ and $\sin \phi = \frac{3}{5},$ find $\theta + 3 \phi,$ measured in radians. | \theta + 3 \phi \approx \pi - \arctan\left(\frac{199}{93}\right) | numina | synthetic_math | Algebra | null | null | with_pi |
081990e05c1eab6dde7e5c7dd0d4a72f5511af8e3d645538abafd17ced354afe | Two congruent cones with radius 12 cm and height 12 cm are enclosed within a cylinder. The base of each cone is a base of the cylinder, and the height of the cylinder is 24 cm. What is the number of cubic centimeters in the volume of the cylinder not occupied by the cones? Express your answer in terms of $\pi$. | 2304\pi | math | geometry | Geometry | null | null | with_pi |
5db72068ac69236141ba1332860a356ab95cbcdb3357ca2a1e41eae92efbdc52 | Determine the value of $a$ that satisfies the equation $375_{a} + 592_{a} = 9C7_{a}$, where $C_{a}=12_{10}$. | 13 | numina | synthetic_math | Number Theory | null | null | integer |
86f5eb9195558b6d1cb27f73c25be89e0678226caccca3613d1132355cee6697 | Three. (50 points) Let $S=\{1,2, \cdots, 2016\}$. For any non-empty finite sets of real numbers $A$ and $B$, find
$$
f=|A \Delta S|+|B \Delta S|+|C \Delta S|
$$
the minimum value, where,
$$
X \Delta Y=\{a \in X \mid a \notin Y\} \cup\{a \in Y \mid a \notin X\}
$$
is the symmetric difference of sets $X$ and $Y$, and
$... | 2017 | numina | olympiads | Combinatorics | null | null | integer |
d9c9858b0a0716d866628bacfd9216328da0e84e44e74411623b0a4ed35c284f | Let $n \geqslant 1$ be an integer. We have $n$ cubes with sides $1, 2, \cdots, n$. We want to stack them in a certain order, such that a cube of side $k$ can only be placed on a cube of side $\ell$ with $\ell \geqslant k-2$. How many such different stackings exist? | 3^{n-1}\times2 | numina | olympiads | Combinatorics | null | null | expression/text |
bd4b9af525a0b4929e66e150f27f16da23a5a39e19d123254fd92abb25b16225 | Find $\left|\left(1+i\right)^6\right|$. | 8 | math | intermediate_algebra | Algebra | null | null | integer |
1bf8898f7a29ddedcaf2de7d2fd77398346e3905153be81eecebb4c31289733b | For how many digits $C$ is the positive three-digit number $1C3$ a multiple of 3? | 3 | math | prealgebra | Algebra | null | null | integer |
5696eaaa05a75e59fd1a33d88b62a54fb9e43cfebb323d919953753e2f19cccd | 1. If the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{1}{2}, a_{n+1}=\frac{2 a_{n}}{3 a_{n}+2}, n \in \mathbf{N}_{+}$, then $a_{2017}=$ $\qquad$ . | \frac{1}{3026} | numina | olympiads | Algebra | null | null | fraction |
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