problem
stringlengths 10
7.44k
| answer
stringlengths 1
270
| difficulty
stringclasses 8
values |
|---|---|---|
The number of zeros of \( y = \tan 1935 x - \tan 2021 x + \tan 2107 x \) in the interval \([0, \pi]\) is \(\qquad\). | 2022 | 1/8 |
A beam of light shines from point \( S \), reflects off a reflector at point \( P \), and reaches point \( T \) such that \( P T \) is perpendicular to \( R S \). Then \( x \) is:
(A) \( 26^{\circ} \)
(B) \( 13^{\circ} \)
(C) \( 64^{\circ} \)
(D) \( 32^{\circ} \)
(E) \( 58^{\circ} \) | 32 | 1/8 |
Two fair octahedral dice, each with the numbers 1 through 8 on their faces, are rolled. Let \( N \) be the remainder when the product of the numbers showing on the two dice is divided by 8. Find the expected value of \( N \). | \frac{11}{4} | 3/8 |
In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\frac{\sin B\sin C}{\sin A}=\frac{3\sqrt{7}}{2}$, $b=4a$, and $a+c=5$, find the area of $\triangle ABC$. | \frac{3\sqrt{7}}{4} | 7/8 |
Let \( E \) and \( F \) be points on the sides \( AD \) and \( CD \) of the square \( ABCD \), respectively, such that \(\angle AEB = \angle AFB = 90^{\circ}\). Additionally, \( EG \parallel AB \), and \( EG \) intersects \( BF \) at point \( G \). If \( AF \) intersects \( BE \) at point \( H \) and \( DH \) intersects \( BC \) at point \( I \), prove that \( FI \perp GH \). | FI\perpGH | 1/8 |
The center of ellipse \( \Gamma \) is at the origin \( O \). The foci of the ellipse lie on the x-axis, and the eccentricity \( e = \sqrt{\frac{2}{3}} \). A line \( l \) intersects the ellipse \( \Gamma \) at two points \( A \) and \( B \) such that \( \overrightarrow{C A} = 2 \overrightarrow{B C} \), where \( C \) is a fixed point \((-1, 0)\). When the area of the triangle \( \triangle O A B \) is maximized, find the equation of the ellipse \( \Gamma \). | x^2+3y^2=5 | 4/8 |
Solve the equation \(\cos 2x + \cos 6x + 2 \sin^2 x = 1\).
In your answer, specify the number equal to the sum of the equation's roots that belong to the segment \(A\), rounding this number to two decimal places if necessary.
$$
A=\left[\frac{5 \pi}{6} ; \frac{6 \pi}{6}\right]
$$ | 2.88 | 7/8 |
Find the area of quadrilateral \(ABCD\) if \(AB = BC = 3\sqrt{3}\), \(AD = DC = \sqrt{13}\), and vertex \(D\) lies on a circle of radius 2 inscribed in the angle \(ABC\), where \(\angle ABC = 60^\circ\). | 3\sqrt{3} | 1/8 |
A circle with a radius of 15 is tangent to two adjacent sides \( AB \) and \( AD \) of square \( ABCD \). On the other two sides, the circle intercepts segments of 6 and 3 cm from the vertices, respectively. Find the length of the segment that the circle intercepts from vertex \( B \) to the point of tangency. | 12 | 5/8 |
Given $f(x)=x^{2}-ax$, $g(x)=\ln x$, $h(x)=f(x)+g(x)$,
(1) Find the range of values for the real number $a$ such that $f(x) \geq g(x)$ holds true for any $x$ within their common domain;
(2) Suppose $h(x)$ has two critical points $x_{1}$, $x_{2}$, with $x_{1} \in (0, \frac{1}{2})$, and if $h(x_{1}) - h(x_{2}) > m$ holds true, find the maximum value of the real number $m$. | \frac{3}{4} - \ln 2 | 2/8 |
A school has 1200 students, and each student participates in exactly \( k \) clubs. It is known that any group of 23 students all participate in at least one club in common, but no club includes all 1200 students. Find the minimum possible value of \( k \). | 23 | 1/8 |
Given that there are \( c \) prime numbers less than 100 such that their unit digits are not square numbers, find the values of \( c \). | 15 | 3/8 |
Given the line $x-y+2=0$ and the circle $C$: $(x-3)^{2}+(y-3)^{2}=4$ intersect at points $A$ and $B$. The diameter through the midpoint of chord $AB$ is $MN$. Calculate the area of quadrilateral $AMBN$. | 4\sqrt{2} | 7/8 |
Oleg drew an empty 50Γ50 table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, 50 of which are rational and the remaining 50 are irrational. Then, in each cell of the table, he recorded the product of the numbers written near its row and its column (a "multiplication table"). What is the maximum number of products in this table that could turn out to be rational numbers? | 1250 | 1/8 |
Given the function $f(x) = \begin{cases} \log_{10} x, & x > 0 \\ x^{-2}, & x < 0 \end{cases}$, if $f(x\_0) = 1$, find the value of $x\_0$. | 10 | 4/8 |
Let \( ABCD \) be a quadrilateral such that \(\angle ABC = \angle CDA = 90^\circ\), and \( BC = 7 \). Let \( E \) and \( F \) be on \( BD \) such that \( AE \) and \( CF \) are perpendicular to \( BD \). Suppose that \( BE = 3 \). Determine the product of the smallest and largest possible lengths of \( DF \). | 9 | 2/8 |
A volleyball net is in the shape of a rectangle with dimensions of $50 \times 600$ cells.
What is the maximum number of strings that can be cut so that the net does not fall apart into pieces? | 30000 | 7/8 |
Let \( AB \) be a focal chord of the parabola \( y^2 = 2px \) (where \( p > 0 \)), and let \( O \) be the origin. Then:
1. The minimum area of triangle \( OAB \) is \( \frac{p^2}{2} \).
2. The tangents at points \( A \) and \( B \) intersect at point \( M \), and the minimum area of triangle \( MAB \) is \( p^2 \). | p^2 | 5/8 |
Alan, Beth, Carla, and Dave weigh themselves in pairs. Together, Alan and Beth weigh 280 pounds, Beth and Carla weigh 230 pounds, Carla and Dave weigh 250 pounds, and Alan and Dave weigh 300 pounds. How many pounds do Alan and Carla weigh together? | 250 | 1/8 |
Let $p \ge 3$ be a prime. For $j = 1,2 ,... ,p - 1$ , let $r_j$ be the remainder when the integer $\frac{j^{p-1}-1}{p}$ is divided by $p$ .
Prove that $$ r_1 + 2r_2 + ... + (p - 1)r_{p-1} \equiv \frac{p+1}{2} (\mod p) $$ | \frac{p+1}{2} | 1/8 |
In a certain sequence, the first term is \(a_1 = 1010\) and the second term is \(a_2 = 1011\). The values of the remaining terms are chosen so that \(a_n + a_{n+1} + a_{n+2} = 2n\) for all \(n \geq 1\). Determine \(a_{1000}\). | 1676 | 4/8 |
If $\frac{a}{b}<-\frac{c}{d}$ where $a$, $b$, $c$, and $d$ are real numbers and $bd \not= 0$, then:
$\text{(A)}\ a \; \text{must be negative} \qquad \text{(B)}\ a \; \text{must be positive} \qquad$
$\text{(C)}\ a \; \text{must not be zero} \qquad \text{(D)}\ a \; \text{can be negative or zero, but not positive } \\ \text{(E)}\ a \; \text{can be positive, negative, or zero}$ | \textbf{(E)}\; | 1/8 |
Let $x$ be a real number selected uniformly at random between 100 and 300. If $\lfloor \sqrt{x} \rfloor = 14$, find the probability that $\lfloor \sqrt{100x} \rfloor = 140$.
A) $\frac{281}{2900}$
B) $\frac{4}{29}$
C) $\frac{1}{10}$
D) $\frac{96}{625}$
E) $\frac{1}{100}$ | \frac{281}{2900} | 1/8 |
Masha surveyed her friends from her ensemble and received the following responses: 25 of them study mathematics, 30 have been to Moscow, 28 traveled by train. Among those who traveled by train, 18 study mathematics and 17 have been to Moscow. 16 friends study mathematics and have been to Moscow, and among them, 15 also traveled by train. Furthermore, there are a total of 45 girls in the ensemble. Is this possible? | No | 5/8 |
In $\triangle ABC$, $\angle A= \frac {\pi}{3}$, $BC=3$, $AB= \sqrt {6}$, find $\angle C=$ \_\_\_\_\_\_ and $AC=$ \_\_\_\_\_\_. | \frac{\sqrt{6} + 3\sqrt{2}}{2} | 2/8 |
Given the quadratic function \( y = ax^2 + bx + c \) with its graph intersecting the \( x \)-axis at points \( A \) and \( B \), and its vertex at point \( C \):
(1) If \( \triangle ABC \) is a right-angled triangle, find the value of \( b^2 - 4ac \).
(2) Consider the quadratic function
\[ y = x^2 - (2m + 2)x + m^2 + 5m + 3 \]
with its graph intersecting the \( x \)-axis at points \( E \) and \( F \), and it intersects the linear function \( y = 3x - 1 \) at two points, with the point having the smaller \( y \)-coordinate denoted as point \( G \).
(i) Express the coordinates of point \( G \) in terms of \( m \).
(ii) If \( \triangle EFG \) is a right-angled triangle, find the value of \( m \). | -1 | 4/8 |
Arrange 1999 positive integers in a row such that the sum of any consecutive n terms (where n = 1, 2, ..., 1999) is not equal to 119. Find the minimum possible sum of these 1999 numbers. | 3903 | 1/8 |
Given that $\alpha$ is an acute angle and satisfies $\cos(\alpha+\frac{\pi}{4})=\frac{\sqrt{3}}{3}$.
$(1)$ Find the value of $\sin(\alpha+\frac{7\pi}{12})$.
$(2)$ Find the value of $\cos(2\alpha+\frac{\pi}{6})$. | \frac{2\sqrt{6}-1}{6} | 7/8 |
Given the four-digit number $\overline{ABCD}$, it satisfies the following properties: $\overline{AB}$, $\overline{BC}$, and $\overline{CD}$ are all perfect squares (a perfect square is a number that can be expressed as the square of an integer. For example, $4=2^{2}, 81=9^{2}$, so we call $4$ and $81$ perfect squares). Find the sum of all four-digit numbers that satisfy these properties. | 13462 | 7/8 |
Let \( A B C D \) be a tetrahedron having each sum of opposite sides equal to 1. Prove that
$$
r_A + r_B + r_C + r_D \leq \frac{\sqrt{3}}{3},
$$
where \( r_A, r_B, r_C, r_D \) are the inradii of the faces, with equality holding only if \( A B C D \) is regular. | \frac{\sqrt{3}}{3} | 1/8 |
It is given that $x = -2272$ , $y = 10^3+10^2c+10b+a$ , and $z = 1$ satisfy the equation $ax + by + cz = 1$ , where $a, b, c$ are positive integers with $a < b < c$ . Find $y.$ | 1987 | 3/8 |
Sergey arranged several (more than two) pairwise distinct real numbers in a circle in such a way that each number is equal to the product of its neighbors. How many numbers could Sergey have arranged? | 6 | 6/8 |
Given a pair $(a_0, b_0)$ of real numbers, we define two sequences $a_0, a_1, a_2,...$ and $b_0, b_1, b_2, ...$ of real numbers by $a_{n+1}= a_n + b_n$ and $b_{n+1}=a_nb_n$ for all $n = 0, 1, 2,...$ . Find all pairs $(a_0, b_0)$ of real numbers such that $a_{2022}= a_0$ and $b_{2022}= b_0$ . | (,0) | 4/8 |
Diagonals of a polygon. Find the maximum possible number of intersections of diagonals in a planar convex $n$-gon. | \frac{n(n-1)(n-2)(n-3)}{24} | 7/8 |
Find maximal positive integer $p$ such that $5^7$ is sum of $p$ consecutive positive integers | 125 | 1/8 |
In the plane, there is a circle and a point. Starting from point \( A \), we traverse a closed polygonal line where each segment lies on a tangent of the circle, eventually returning to the starting point. We consider segments on which we are approaching the circle as positive and those on which we are receding from the circle as negative.
Prove that the sum of these signed segments is zero. | 0 | 2/8 |
The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle, with the vertices around the circumference in the given order. Given that the polygon $P_1P_3P_5P_7$ is a square of area 5, and the polygon $P_2P_4P_6P_8$ is a rectangle of area 4, find the maximum possible area of the octagon. | 3\sqrt{5} | 1/8 |
Squares of a chessboard are painted in 3 colors - white, gray, and black - in such a way that adjacent squares sharing a side differ in color; however, a sharp change in color (i.e., adjacency of white and black squares) is prohibited. Find the number of such colorings of the chessboard (colorings that are identical upon rotating the board by 90 and 180 degrees are considered different). | 2^{33} | 1/8 |
Let $S = \{1, 2, \dots , 2021\}$ , and let $\mathcal{F}$ denote the set of functions $f : S \rightarrow S$ . For a function $f \in \mathcal{F},$ let
\[T_f =\{f^{2021}(s) : s \in S\},\]
where $f^{2021}(s)$ denotes $f(f(\cdots(f(s))\cdots))$ with $2021$ copies of $f$ . Compute the remainder when
\[\sum_{f \in \mathcal{F}} |T_f|\]
is divided by the prime $2017$ , where the sum is over all functions $f$ in $\mathcal{F}$ . | 255 | 1/8 |
Arrange the numbers 3, 4, 5, 6 into two natural numbers A and B, so that the product AΓB is maximized. Find the value of AΓB. | \left( 3402 \right) | 2/8 |
Let $f(x)=x^4+14x^3+52x^2+56x+16$. Let $z_1,z_2,z_3,z_4$ be the four roots of $f$. Find the smallest possible value of $|z_{a}z_{b}+z_{c}z_{d}|$ where $\{a,b,c,d\}=\{1,2,3,4\}$. | 8 | 2/8 |
In triangle \(ABC\), the interior and exterior angle bisectors of \(\angle BAC\) intersect the line \(BC\) in \(D\) and \(E\), respectively. Let \(F\) be the second point of intersection of the line \(AD\) with the circumcircle of the triangle \(ABC\). Let \(O\) be the circumcenter of the triangle \(ABC\) and let \(D'\) be the reflection of \(D\) in \(O\). Prove that \(\angle D'FE = 90^\circ\). | 90 | 1/8 |
For integer $n$ , let $I_n=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos (2n+1)x}{\sin x}\ dx.$ (1) Find $I_0.$ (2) For each positive integer $n$ , find $I_n-I_{n-1}.$ (3) Find $I_5$ . | \frac{1}{2}\ln2-\frac{8}{15} | 4/8 |
Lil writes one of the letters \( \text{P}, \text{Q}, \text{R}, \text{S} \) in each cell of a \( 2 \times 4 \) table. She does this in such a way that, in each row and in each \( 2 \times 2 \) square, all four letters appear. In how many ways can she do this? | 24 | 1/8 |
Suppose we have an (infinite) cone $\mathcal{C}$ with apex $A$ and a plane $\pi$. The intersection of $\pi$ and $\mathcal{C}$ is an ellipse $\mathcal{E}$ with major axis $BC$, such that $B$ is closer to $A$ than $C$, and $BC=4, AC=5, AB=3$. Suppose we inscribe a sphere in each part of $\mathcal{C}$ cut up by $\mathcal{E}$ with both spheres tangent to $\mathcal{E}$. What is the ratio of the radii of the spheres (smaller to larger)? | \frac{1}{3} | 1/8 |
For \( x, y \in (0,1] \), find the maximum value of the expression
\[
A = \frac{\left(x^{2} - y\right) \sqrt{y + x^{3} - x y} + \left(y^{2} - x\right) \sqrt{x + y^{3} - x y} + 1}{(x - y)^{2} + 1}
\] | 1 | 6/8 |
Let $A_{1} A_{2} \ldots A_{100}$ be the vertices of a regular 100-gon. Let $\pi$ be a randomly chosen permutation of the numbers from 1 through 100. The segments $A_{\pi(1)} A_{\pi(2)}, A_{\pi(2)} A_{\pi(3)}, \ldots, A_{\pi(99)} A_{\pi(100)}, A_{\pi(100)} A_{\pi(1)}$ are drawn. Find the expected number of pairs of line segments that intersect at a point in the interior of the 100-gon. | \frac{4850}{3} | 1/8 |
Carefully observe the arrangement pattern of the following hollow circles ($β$) and solid circles ($β$): $ββββββββββββββββββββββββββββ¦$. If this pattern continues, a series of $β$ and $β$ will be obtained. The number of $β$ in the first $100$ of $β$ and $β$ is $\_\_\_\_\_\_\_$. | 12 | 7/8 |
Given is a positive integer $n$ . There are $2n$ mutually non-attacking rooks placed on a grid $2n \times 2n$ . The grid is splitted into two connected parts, symmetric with respect to the center of the grid. What is the largest number of rooks that could lie in the same part? | 2n-1 | 1/8 |
Two ferries cross a river with constant speeds, turning at the shores without losing time. They start simultaneously from opposite shores and meet for the first time 700 feet from one shore. They continue to the shores, return, and meet for the second time 400 feet from the opposite shore. Determine the width of the river. | 1400 | 1/8 |
Two circles touch each other externally. The four points of tangency of their external common tangents $A, B, C, D$ are successively connected. Show that a circle can be inscribed in the quadrilateral $\boldsymbol{A} \boldsymbol{B} C D$ and find its radius, given that the radii of the given circles are $\boldsymbol{R}$ and $\boldsymbol{r}$. | \frac{2Rr}{R+r} | 1/8 |
In the multiplication shown, $P, Q,$ and $R$ are all different digits such that
$$
\begin{array}{r}
P P Q \\
\times \quad Q \\
\hline R Q 5 Q
\end{array}
$$
What is the value of $P + Q + R$? | 17 | 7/8 |
In the addition problem below, the same Chinese characters represent the same digits, different Chinese characters represent different digits, and none of the digits are zero. What is the minimum difference between "ζ°ε¦θ§£ι’" (math problem solving) and "θ½ε" (ability)?
ζ°ε¦θ§£ι’ (xue jie ti)
θ½ε (neng li)
+ ε±η€Ί (zhan shi)
----------
2010 | 1747 | 1/8 |
Solve the system
$$
\left\{\begin{array}{l}
\operatorname{tg}^{3} x + \operatorname{tg}^{3} y + \operatorname{tg}^{3} z = 36 \\
\operatorname{tg}^{2} x + \operatorname{tg}^{2} y + \operatorname{tg}^{2} z = 14 \\
\left(\operatorname{tg}^{2} x + \operatorname{tg} y\right)(\operatorname{tg} x + \operatorname{tg} z)(\operatorname{tg} y + \operatorname{tg} z) = 60
\end{array}\right.
$$
In the answer, specify the sum of the minimum and maximum values of \(\operatorname{tg} x\) that solve the system. | 4 | 5/8 |
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ $\mathbf{d}$ be four distinct unit vectors in space such that
\[\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} = \mathbf{b} \cdot \mathbf{c} =\mathbf{b} \cdot \mathbf{d} = \mathbf{c} \cdot \mathbf{d} = -\frac{1}{11}.\]Find $\mathbf{a} \cdot \mathbf{d}.$ | -\frac{53}{55} | 4/8 |
In the triangle \( \triangle ABC \), \( BD = DC \). A piece of cheese is placed on side \( AC \) at the point which is one-fourth the distance from point \( C \). On \( AD \), there are three mirrors \( W_1, W_2, W_3 \) which divide \( AD \) into four equal parts. A very paranoid mouse is crawling on \( AB \) (moving from \( A \) to \( B \)), where \( AB = 400 \) meters. When the mouse, one of the mirrors, and the cheese are collinear, the mouse can see the cheese. Due to its paranoia, the mouse wants to see the cheese multiple times to ensure that the cheese has not been eaten by another mouse before it reaches it. The mouse crawls forward 80 meters in the first minute, then retreats 20 meters in the second minute, crawls forward 80 meters in the third minute, retreats 20 meters in the fourth minute, and so on. When the mouse reaches point \( B \), it will directly run along \( BC \) to eat the cheese.
How many times in total can the mouse see the cheese while on segment \( AB \)? | 5 | 1/8 |
Three congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circle centered at $Q$ has diameter $4$ and passes through points $P$ and $R$. The area of the rectangle is | 32 | 6/8 |
Each cell of a 5x5 table is painted in one of several colors. Lada shuffled the rows of the table so that no row remained in its original place. Then Lera shuffled the columns so that no column remained in its original place. Surprisingly, the girls noticed that the resulting table was the same as the original one. What is the maximum number of different colors that can be used to paint this table? | 7 | 2/8 |
Find the minimum value of the expression
$$
\sqrt{x^{2}-\sqrt{3} \cdot|x|+1}+\sqrt{x^{2}+\sqrt{3} \cdot|x|+3}
$$
and the values of \( x \) at which it is achieved. | \sqrt{7} | 6/8 |
Given \(0 < a < 1\) and \(x^{2} + y = 0\), prove \(\log_{a}\left(a^{x} + a^{y}\right) \leq \log_{a} 2 + \frac{1}{8}\). | \log_{}(^x+^y)\le\log_{}2+\frac{1}{8} | 4/8 |
a) In how many different ways can the teacher call the students to the front of the board?
b) In how many of these sequences are the students NOT in alphabetical order?
c) In how many of these sequences are Arnaldo and Bernaldo called in consecutive positions? | 48 | 3/8 |
In a triangle $ABC$ , let $A'$ be a point on the segment $BC$ , $B'$ be a point on the segment $CA$ and $C'$ a point on the segment $AB$ such that $$ \frac{AB'}{B'C}= \frac{BC'}{C'A} =\frac{CA'}{A'B}=k, $$ where $k$ is a positive constant. Let $\triangle$ be the triangle formed by the interesctions of $AA'$ , $BB'$ and $CC'$ . Prove that the areas of $\triangle $ and $ABC$ are in the ratio $$ \frac{(k-1)^{2}}{k^2 +k+1}. $$ | \frac{(k-1)^2}{k^2+k+1} | 4/8 |
A triangle $\bigtriangleup ABC$ has vertices lying on the parabola defined by $y = x^2 + 4$. Vertices $B$ and $C$ are symmetric about the $y$-axis and the line $\overline{BC}$ is parallel to the $x$-axis. The area of $\bigtriangleup ABC$ is $100$. $A$ is the point $(2,8)$. Determine the length of $\overline{BC}$. | 10 | 1/8 |
$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$ , are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$ | x^2+2py=0 | 1/8 |
There are eight points on a circle that divide the circumference equally. Count the number of acute-angled triangles or obtuse-angled triangles that can be formed with these division points as vertices. | 32 | 2/8 |
Inside the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is the center $O$ of a sphere with a radius of 10. The sphere intersects the face $A A_{1} D_{1} D$ by a circle with a radius of 1, the face $A_{1} B_{1} C_{1} D_{1}$ by a circle with a radius of 1, and the face $C D D_{1} C_{1}$ by a circle with a radius of 3. Find the length of the segment $O D_{1}$. | 17 | 7/8 |
How many of the numbers
\[
a_1\cdot 5^1+a_2\cdot 5^2+a_3\cdot 5^3+a_4\cdot 5^4+a_5\cdot 5^5+a_6\cdot 5^6
\]
are negative if $a_1,a_2,a_3,a_4,a_5,a_6 \in \{-1,0,1 \}$ ? | 364 | 7/8 |
Let \( x, y, z \geq 0 \) and \( yz + zx + xy = 1 \). Show that:
\[ x(1 - y)^2(1 - z^2) + y(1 - z^2)(1 - x^2) + z(1 - x^2)(1 - y^2) \leq \frac{4}{9} \sqrt{3} \]. | \frac{4}{9}\sqrt{3} | 1/8 |
Masha and the Bear ate a basket of raspberries and 60 pies, starting and finishing at the same time. Initially, Masha ate raspberries while the Bear ate pies, and then they switched at some point. The Bear ate raspberries 6 times faster than Masha and pies 3 times faster. How many pies did the Bear eat if the Bear ate twice as many raspberries as Masha? | 54 | 7/8 |
In a rectangular parallelepiped \(ABCD A_{1} B_{1} C_{1} D_{1}\) with bases \(ABCD\) and \(A_{1} B_{1} C_{1} D_{1}\), it is known that \(AB = 29\), \(AD = 36\), \(BD = 25\), and \(AA_{1} = 48\). Find the area of the cross-section \(AB_{1} C_{1} D\). | 1872 | 1/8 |
Suppose that $n\ge3$ is a natural number. Find the maximum value $k$ such that there are real numbers $a_1,a_2,...,a_n \in [0,1)$ (not necessarily distinct) that for every natural number like $j \le k$ , sum of some $a_i$ -s is $j$ .
*Proposed by Navid Safaei* | n-2 | 1/8 |
What is the hundreds digit of $(20!-15!)?$
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }5$ | \textbf{(A)}\0 | 1/8 |
In a convex quadrilateral \(ABCD\), there are two circles of the same radius \(r\), each tangent to the other externally. The center of the first circle is located on the segment connecting vertex \(A\) to the midpoint \(F\) of side \(CD\), and the center of the second circle is located on the segment connecting vertex \(C\) to the midpoint \(E\) of side \(AB\). The first circle is tangent to sides \(AB\), \(AD\), and \(CD\), while the second circle is tangent to sides \(AB\), \(BC\), and \(CD\).
Find \(AC\). | 2r\sqrt{5} | 1/8 |
A point $P$ is randomly selected from the rectangular region with vertices $(0,0), (2,0)$, $(2,1),(0,1)$. What is the probability that $P$ is closer to the origin than it is to the point $(3,1)$? | \frac{3}{4} | 5/8 |
In $\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\angle BAC$, and $BN \perp AN$. If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then find $MN$. | \frac{5}{2} | 5/8 |
Evaluate the expression $3 - (-3)^{-\frac{2}{3}}$. | 3 - \frac{1}{\sqrt[3]{9}} | 2/8 |
Given the ellipse \( C: \frac{y^{2}}{a^{2}} + \frac{x^{2}}{b^{2}} = 1 \) with \( a > b > 0 \) and an eccentricity of \( \frac{1}{2} \). The upper and lower foci are \( F_{1} \) and \( F_{2} \), and the right vertex is \( D \). A line through \( F_{1} \) perpendicular to \( D F_{2} \) intersects the ellipse \( C \) at points \( A \) and \( B \), such that \( |B D| - |A F_{1}| = \frac{8 \sqrt{3}}{39} \).
(1) Find the value of \( |A D| + |B D| \).
(2) Lines tangent to the ellipse at points \( A \) and \( B \) intersect at point \( E \). If \( F_{1} E \) intersects the \( x \)-axis at point \( P \) and \( F_{2} E \) intersects the \( x \)-axis at point \( Q \), find the value of \( |P Q| \). | \frac{16}{5} | 1/8 |
The right focus of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (with $a>0$, $b>0$) is $F$, and $B$ is a point on the left branch of the hyperbola. The line segment $BF$ intersects with one asymptote of the hyperbola at point $A$, and it is given that $(\vec{OF} - \vec{OB}) \cdot \vec{OA} = 0$ and $2\vec{OA} = \vec{OB} + \vec{OF}$ (where $O$ is the origin). Find the eccentricity $e$ of the hyperbola. | \sqrt{5} | 6/8 |
Li Gang delivered honeycomb briquettes to Grandma Wang, a dependent of a military personnel. The first time, he delivered $\frac{3}{8}$ of the total. The second time, he delivered 50 pieces. At this point, the amount delivered was exactly $\frac{5}{7}$ of the amount not yet delivered. How many pieces of honeycomb briquettes have not been delivered yet? | 700 | 5/8 |
Oleg drew an empty $50 \times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 being rational and the remaining 50 being irrational. Then, in each cell of the table, he recorded the sum of the numbers written next to its row and column ("addition table"). What is the maximum number of sums in this table that could be rational numbers? | 1250 | 2/8 |
Given that $a,b$ are positive real numbers, and $({(a-b)}^{2}=4{{(ab)}^{3}})$, find the minimum value of $\dfrac{1}{a}+\dfrac{1}{b}$ . | 2\sqrt{2} | 5/8 |
The decimal representation of $m/n,$ where $m$ and $n$ are relatively prime positive integers and $m < n,$ contains the digits $2, 5$, and $1$ consecutively, and in that order. Find the smallest value of $n$ for which this is possible.
| 127 | 1/8 |
In the plane Cartesian coordinate system $xOy$, find the area of the shape formed by the points whose coordinates satisfy the condition $\left(x^{2}+y^{2}+2x+2y\right)\left(4-x^{2}-y^{2}\right) \geq 0$. | 2\pi+4 | 5/8 |
Given a parabola with its vertex at the origin \( O \) and focus \( F \), \( PQ \) is a chord that passes through \( F \), where \( |OF|=a \) and \( |PQ|=b \). Find the area \( S_{\triangle OPQ} \). | \sqrt{} | 5/8 |
Let \(\triangle XYZ\) be a right triangle with \(\angle XYZ = 90^\circ\). Suppose there exists an infinite sequence of equilateral triangles \(X_0Y_0T_0\), \(X_1Y_1T_1\), \ldots such that \(X_0 = X\), \(Y_0 = Y\), \(X_i\) lies on the segment \(XZ\) for all \(i \geq 0\), \(Y_i\) lies on the segment \(YZ\) for all \(i \geq 0\), \(X_iY_i\) is perpendicular to \(YZ\) for all \(i \geq 0\), \(T_i\) and \(Y\) are separated by line \(XZ\) for all \(i \geq 0\), and \(X_i\) lies on segment \(Y_{i-1}T_{i-1}\) for \(i \geq 1\).
Let \(\mathcal{P}\) denote the union of the equilateral triangles. If the area of \(\mathcal{P}\) is equal to the area of \(\triangle XYZ\), find \(\frac{XY}{YZ}\). | 1 | 4/8 |
Let \( S = \{1, 2, \ldots, n\} \). Define \( A \) as an arithmetic sequence with at least two terms and a positive common difference, where all terms are in \( S \). Also, adding any other element of \( S \) to \( A \) does not result in an arithmetic sequence with the same common difference as \( A \). Find the number of such sequences \( A \) (a sequence of only two terms is also considered an arithmetic sequence). | \lfloor\frac{n^2}{4}\rfloor | 1/8 |
[IMO 2007 HKTST 1](http://www.mathlinks.ro/Forum/viewtopic.php?t=107262)
Problem 1
Let $p,q,r$ and $s$ be real numbers such that $p^{2}+q^{2}+r^{2}-s^{2}+4=0$ . Find the maximum value of $3p+2q+r-4|s|$ . | -2\sqrt{2} | 3/8 |
Given a polynomial of degree $n$ with real coefficients,
$$
p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x + a_{0},
$$
where $n \geq 3$, and all of its (real and complex) roots are in the left half-plane, meaning their real parts are negative. Prove that for any $0 \leq k \leq n-3$,
$$
a_{k} a_{k+3}<a_{k+1} a_{k+2}.
$$ | a_ka_{k+3}<a_{k+1}a_{k+2} | 3/8 |
Nine digits: \(1, 2, 3, \ldots, 9\) are written in a certain order (forming a nine-digit number). Consider all consecutive triples of digits, and find the sum of the resulting seven three-digit numbers. What is the largest possible value of this sum? | 4648 | 2/8 |
Let $S(k)$ denote the sum of the digits of a natural number $k$. A natural number $a$ is called $n$-good if there exists a sequence of natural numbers $a_{0}, a_{1}, \ldots, a_{n}$ such that $a_{n} = a$ and $a_{i+1} = a_{i} - S(a_{i})$ for all $i = 0, 1, \ldots, n-1$. Is it true that for any natural number $n$, there exists a natural number that is $n$-good but not $(n+1)$-good?
(A. Antropov) | Yes | 1/8 |
The cafΓ© has enough chairs to seat $310_5$ people. If $3$ people are supposed to sit at one table, how many tables does the cafΓ© have? | 26 | 2/8 |
Prove that there are infinitely many positive integers \( n \) such that \( n, n+1 \), and \( n+2 \) can each be written as the sum of two squares. | 2 | 1/8 |
Let the line \( y = kx + m \) passing through any point \( P \) on the ellipse \( \frac{x^{2}}{4} + y^{2} = 1 \) intersect the ellipse \( \frac{x^{2}}{16} + \frac{y^{2}}{4} = 1 \) at points \( A \) and \( B \). Let the ray \( PO \) intersect the ellipse \( \frac{x^{2}}{16} + \frac{y^{2}}{4} = 1 \) at point \( Q \). Find the value of the ratio \( \frac{S_{\triangle ABQ}}{S_{\triangle ABO}} \). | 3 | 1/8 |
How many six-letter words formed from the letters of AMC do not contain the substring AMC? (For example, AMAMMC has this property, but AAMCCC does not.) | 622 | 1/8 |
Find the number of ordered pairs of positive integers $(x, y)$ with $x, y \leq 2020$ such that $3 x^{2}+10 x y+3 y^{2}$ is the power of some prime. | 29 | 2/8 |
Given a triangle \(ABC\). Point \(P\) is the center of the inscribed circle. Find the angle \(B\), given that \(R_{ABC} = R_{APC}\), where \(R_{ABC}\) and \(R_{APC}\) are the circumradii of triangles \(ABC\) and \(APC\) respectively. | 60 | 7/8 |
Let the line \( l: y = kx + m \) (where \( k \) and \( m \) are integers) intersect the ellipse \( \frac{x^2}{16} + \frac{y^2}{12} = 1 \) at two distinct points \( A \) and \( B \), and intersect the hyperbola \( \frac{x^2}{4} - \frac{y^2}{12} = 1 \) at two distinct points \( C \) and \( D \). Determine if there exists such a line \( l \) such that the vector \( \overrightarrow{AC} + \overrightarrow{BD} = 0 \). If such a line exists, specify how many such lines there are. If no such line exists, provide reasoning. | 9 | 4/8 |
Find the maximum number \( E \) such that the following holds: there is an edge-colored graph with 60 vertices and \( E \) edges, with each edge colored either red or blue, such that in that coloring, there is no monochromatic cycle of length 3 and no monochromatic cycle of length 5. | 1350 | 1/8 |
We have a calculator with two buttons that displays an integer \( x \). Pressing the first button replaces \( x \) by \( \left\lfloor \frac{x}{2} \right\rfloor \), and pressing the second button replaces \( x \) by \( 4x + 1 \). Initially, the calculator displays 0. How many integers less than or equal to 2014 can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, \( \lfloor y \rfloor \) denotes the greatest integer less than or equal to the real number \( y \).) | 233 | 1/8 |
On the board, a fraction $\frac{a x + b}{c x + d}$ was written, as well as all other fractions obtained from it by permuting the numbers $a, b, c, d$, except those with an identically zero denominator. Could it happen that among the listed fractions there are exactly 5 distinct ones? | Yes | 1/8 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.