problem
stringlengths 18
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The coordinates of the vertices of isosceles trapezoid $PQRS$ are all integers, with $P=(10,50)$ and $S=(11,51)$. The trapezoid has no horizontal or vertical sides, and $\overline{PQ}$ and $\overline{RS}$ are the only parallel sides. Determine the sum of the absolute values of all possible slopes for $\overline{PQ}$.
|
1
| 0.083333 |
Let $z_1$ and $z_2$ be the complex roots of $z^2 + az + b = 0,$ where $a$ and $b$ are complex numbers. In the complex plane, the triangle formed by the points 0, $z_1,$ and $z_2$ is an equilateral triangle but rotated, such that $z_2 = \omega z_1$, where $\omega = e^{2\pi i/3}$. Find $\frac{a^2}{b}$.
|
1
| 0.583333 |
Circle $\Gamma$ is the incircle of $\triangle ABC$ and also the circumcircle of $\triangle XYZ$. The point $X$ is on $\overline{BC}$, the point $Y$ is on $\overline{AB}$, and the point $Z$ is on $\overline{AC}$. If $\angle A = 50^\circ$, $\angle B = 70^\circ$, let $\angle C$ be computed based on this data. What is the measure of $\angle YZX$?
|
60^\circ
| 0.166667 |
In the diagram, $ABCD$ is a square with side length $8,$ and $WXYZ$ is a rectangle with $ZY=12$ and $XY=8.$ Also, $AD$ and $WX$ are perpendicular. If the shaded area is equal to half of the area of $WXYZ,$ what is the length of $AP?$
[asy]
draw((0,0)--(12,0)--(12,8)--(0,8)--cycle,black+linewidth(1));
draw((2,2)--(10,2)--(10,10)--(2,10)--cycle,black+linewidth(1));
filldraw((2,2)--(10,2)--(10,8)--(2,8)--cycle,gray,black+linewidth(1));
label("$W$",(0,8),NW);
label("$X$",(12,8),NE);
label("$Y$",(12,0),SE);
label("$Z$",(0,0),SW);
label("$A$",(2,10),NW);
label("$B$",(10,10),NE);
label("$C$",(10,2),E);
label("$D$",(2,2),W);
label("$P$",(2,8),SW);
label("8",(2,10)--(10,10),N);
label("8",(12,0)--(12,8),E);
label("12",(0,0)--(12,0),S);
[/asy]
|
AP = 2
| 0.166667 |
Shown below are rows 1, 2, and 3 of Pascal's triangle.
\[
\begin{array}{ccccccc}
& & 1 & & 1 & & \\
& 1 & & 2 & & 1 & \\
1 & & 3 & & 3 & & 1
\end{array}
\]
Let \((d_i), (e_i), (f_i)\) be the sequence, from left to right, of elements in the 2010th, 2011th, and 2012th rows, respectively, with the leftmost element occurring at \(i = 0.\) Compute
\[
\sum_{i = 0}^{2010} \frac{e_i}{f_i} - \sum_{i = 0}^{2009} \frac{d_i}{e_i}.
\]
|
\frac{1}{2}
| 0.083333 |
In the United Kingdom, coins have the following thicknesses: 1p, 1.65 mm; 2p, 2.05 mm; 5p, 1.85 mm; 10p, 1.95 mm. If a stack of these coins is precisely 19 mm high, consisting of 2p and 10p coins only, how many coins are in the stack?
|
10 \text{ coins}
| 0.166667 |
The graph of the function $y=g(x)$ is given. For all $x > 3$, it is true that $g(x) > 0.3$. If $g(x) = \frac{x^2}{Dx^2 + Ex + F}$, where $D, E,$ and $F$ are integers, find $D+E+F$. Assume the graph shows vertical asymptotes at $x = -3$ and $x = 2$, and the horizontal asymptote is below $y = 1$ but above $y = 0.3$.
|
-8
| 0.333333 |
Express 1000 as a sum of at least two distinct powers of 2. What would be the least possible sum of the exponents of these powers?
|
38
| 0.75 |
Let's consider two mathematicians, Alice and Bob, each thinking of a polynomial. Each polynomial is monic, has a degree of 4, and both polynomials have the same positive constant term and the same coefficient of $x^3$. The product of their polynomials is given by \[x^8 + 2x^7 + 3x^6 + 2x^5 + 5x^4 + 4x^3 + 3x^2 + 2x + 9.\] What is the constant term of Bob's polynomial?
|
3
| 0.916667 |
Let $a < b < c$ be three integers such that $a, b, c$ is an arithmetic progression and $c, a, b$ is a geometric progression. Further assume that $a, b, c$ are multiples of 5. What is the smallest possible value of $c$ where all numbers are positive?
|
20
| 0.583333 |
Determine the number of solutions to
\[ |\sin x| = \left(\frac{1}{2}\right)^x \]
on the interval \((0, 200\pi)\).
|
400
| 0.5 |
Find the smallest possible value of the expression $$\frac{(a+b)^2+(b+c)^2+(c-a)^2}{c^2},$$ where $c > b > a$ are real numbers, and $c \neq 0.$
|
2
| 0.25 |
Let \( S = \{x + iy : -2 \leq x \leq 2, -2 \leq y \leq 2\} \). A complex number \( z = x + iy \) is chosen uniformly at random from \( S \). Compute the probability that the transformation \( \left(\frac{1}{2} + \frac{i}{2}\right)z \) results in a number that remains within \( S \).
|
1
| 0.583333 |
In triangle $XYZ$, sides $XY$, $YZ$, and $ZX$ are tangent to an external circle with center $A$. Given that $\angle XYZ = 80^\circ$ and $\angle YZA = 20^\circ$, find the measure of angle $\angle YXZ$, in degrees.
|
60^\circ
| 0.416667 |
Let $x_1, x_2, \dots, x_{50}$ be real numbers such that $x_1 + x_2 + \dots + x_{50} = 2$ and
\[
\frac{x_1}{1 - x_1} + \frac{x_2}{1 - x_2} + \dots + \frac{x_{50}}{1 - x_{50}} = 2.
\]
Find
\[
\frac{x_1^2}{1 - x_1} + \frac{x_2^2}{1 - x_2} + \dots + \frac{x_{50}^2}{1 - x_{50}}.
\]
|
0
| 0.5 |
The number of games won by five basketball teams is shown in a bar chart. The teams' names are not displayed. The following clues provide information about the teams:
1. The Hawks won more games than the Falcons.
2. The Warriors won more games than the Knights, but fewer games than the Royals.
3. The Knights won more than 22 games.
How many games did the Warriors win? The win numbers given in the bar chart are 23, 28, 33, 38, and 43 games respectively.
|
33
| 0.666667 |
Let \(a,\) \(b,\) and \(c\) be positive real numbers such that \(a + b + c = 3\) and no one of these numbers is more than three times any other. Find the minimum value of the product \(abc.\)
|
\frac{81}{125}
| 0.083333 |
Let $\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\triangle ABC$, so that $DABC$ is a pyramid. Suppose each edge of $DABC$ has length 10, 24, or 24, but no face of $DABC$ is equilateral. Determine the surface area of $DABC$.
|
20\sqrt{551}
| 0.416667 |
Let $0 \le x, y, z, w \le 1$. Determine the possible values of the expression
\[
\sqrt{x^2 + (1 - y)^2} + \sqrt{y^2 + (1 - z)^2} + \sqrt{z^2 + (1 - w)^2} + \sqrt{w^2 + (1 - x)^2}.
\]
|
[2\sqrt{2}, 4]
| 0.083333 |
What is the smallest positive integer that is a multiple of both 36 and 45 but not a multiple of 10?
|
180
| 0.083333 |
The numbers $a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3$ are equal to the numbers $1, 2, 3, \dots, 9$ in some order. Each triplet $(a_1, a_2, a_3)$, $(b_1, b_2, b_3)$, and $(c_1, c_2, c_3)$ forms an arithmetic sequence. Find the smallest possible value of
\[
a_1 a_2 a_3 + b_1 b_2 b_3 + c_1 c_2 c_3.
\]
|
270
| 0.333333 |
In circle $O$, four points $P, A, B, Q$ lie on the circumference such that tangents to $O$ at points $A$, $B$, and $Q$ meet at point $P$ outside the circle, and $\angle APB = 60^\circ$. Determine $\angle AQB$.
|
60^\circ
| 0.333333 |
Let \(x, y, z\) be nonzero real numbers such that \(x + y + z = 0\) and \(xy + xz + yz \neq 0\). Find all possible values of
\[
\frac{x^6 + y^6 + z^6}{xyz (xy + xz + yz)}.
\]
|
6
| 0.25 |
Find the common ratio of the infinite geometric series: $$\frac{4}{7} - \frac{16}{21} - \frac{64}{63} - \dots$$
|
-\frac{4}{3}
| 0.916667 |
In $\triangle ABC,$ $AB=20$, $AC=24$, and $BC=18$. Points $D,E,$ and $F$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively. Angle $\angle BAC = 60^\circ$ and $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $ADEF$?
|
44
| 0.333333 |
In triangle $ABC$, $\angle C = 90^\circ$, $AC = 7$ and $BC = 24$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED = 90^\circ$. If $DE = 10$, then what is the length of $BD$?
|
\frac{250}{7}
| 0.083333 |
Determine the product of the real parts of the complex solutions to the equation $x^2 + 2x = -i$.
|
\frac{1-\sqrt{2}}{2}
| 0.5 |
Let $p<q<r$ be three integers such that $p,q,r$ is a geometric progression and $p,r,q$ is an arithmetic progression. Find the smallest possible value of $r$.
|
4
| 0.333333 |
Let $S$ be the region in the complex plane defined by:
\[ S = \{x + iy: -2 \leq x \leq 2, -2 \leq y \leq 2\} \]
A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\left(\frac12 + \frac12i\right)z$ is also in $S$?
|
1
| 0.75 |
Find the sum of all positive integers such that their expression in base $5$ digits is the reverse of their expression in base $11$ digits. Express your answer in base $10$.
|
10
| 0.25 |
What is the least whole number that is divisible by 9, but leaves a remainder of 1 when divided by any integer 2 through 6 and 8?
|
721
| 0.583333 |
In triangle $ABC$, $AB = 4$, $AC = 7$, $BC = 9$, and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC$. Additionally, the area of triangle $ABC$ is $12$ square units. Find $\cos \angle BAD.$
|
\frac{\sqrt{70}}{14}
| 0.083333 |
Linda bought 50 items each priced at 50 cents, 2 dollars, or 4 dollars. If her total purchase price was $\$$50.00, how many 50-cent items did she purchase?
|
40
| 0.083333 |
Let $AB$ be a diameter of a circle centered at $O$. Let $F$ be a point on the circle such that $F$ is not on the semicircle containing $E$. Let the tangent at $B$ intersect the tangent at $F$ and $AF$ at points $C'$ and $D'$ respectively. If $\angle BAF = 30^\circ$, find $\angle C'ED'$, in degrees.
|
60^\circ
| 0.666667 |
Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=10$. A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P$, which is 6 units from $\overline{BG}$ and 3 units from $\overline{BC}$. The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube is given by $m\sqrt{n}$, where $m$ and $n$ are integers and $n$ is not divisible by the square of any prime. Find $m+n$.
|
155
| 0.083333 |
Let $T$ be the set of complex numbers of the form $x + yi$, where $x$ and $y$ are real numbers, such that
\[\frac{\sqrt{3}}{2} \le x \le \frac{2}{\sqrt{3}}.\]
Find the smallest positive integer $m$ such that for all positive integers $n \ge m,$ there exists a complex number $z \in T$ such that $z^n = 1.$
|
12
| 0.416667 |
Determine the number of angles between 0 and $2 \pi,$ excluding integer multiples of $\frac{\pi}{2},$ such that $\sin^2 \theta,$ $\cos \theta$, and $\tan \theta$ form a geometric sequence in some order.
|
2
| 0.166667 |
For each positive integer $n$, let $f(n) = n^4 - 400n^2 + 600$. What is the sum of all values of $f(n)$ that are prime numbers?
|
0
| 0.5 |
Given the property for any positive integer $n$, the matrix $\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}^n$ is equal to $\begin{pmatrix} G_{n + 1} & 2G_n \\ 2G_n & G_{n - 1} \end{pmatrix}$, where $G_n$ corresponds to a sequence defined by $G_{n} = 2G_{n - 1} + G_{n - 2}$ with $G_1 = 1$ and $G_2 = 2$. Calculate $G_{100} G_{102} - 4G_{101}^2$.
|
-3^{101}
| 0.333333 |
The inscribed circle of triangle $XYZ$ is tangent to $\overline{XY}$ at $P,$ and its radius is $15$. Given that $XP=18$ and $PY=24$, and the height from $Z$ to $\overline{XY}$ is $36$, find the area of triangle $XYZ$.
|
756
| 0.916667 |
In a slightly larger weekend softball tournament, five teams (A, B, C, D, E) are participating. On Saturday, Team A plays Team B, Team C plays Team D, and Team E will automatically advance to the semi-final round. On Sunday, the winners of A vs B and C vs D play each other (including E), resulting in one winner, while the remaining two teams (one from initial losers and Loser of semifinal of E's match) play for third and fourth places. The sixth place is reserved for the loser of the losers' game. One possible ranking of the teams from first place to sixth place at the end of this tournament is the sequence AECDBF. What is the total number of possible six-team ranking sequences at the end of the tournament?
|
32
| 0.083333 |
If
\[1 \cdot 1500 + 2 \cdot 1499 + 3 \cdot 1498 + \dots + 1499 \cdot 2 + 1500 \cdot 1 = 1500 \cdot 751 \cdot x,\]
compute the integer $x.$
|
501
| 0.166667 |
I have 5 shirts, 5 pairs of pants, and 7 hats. The pants come in tan, black, blue, gray, and green. The shirts and hats come in those colors, and also white and yellow. I refuse to wear an outfit in which the shirt and pants are the same color. How many choices for outfits, consisting of one shirt, one hat, and one pair of pants, do I have?
|
140
| 0.5 |
Find the minimum value of the function:
\[ f(x) = x + \frac{1}{x} + \frac{1}{x^2 + \frac{1}{x}} \quad \text{for } x > 0. \]
|
\frac{5}{2}
| 0.25 |
Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the small right triangles is $n$ times the area of the square. What is the ratio of the area of the other small right triangle to the area of the square? Express your answer as a common fraction in terms of $n$.
|
\frac{1}{4n}
| 0.083333 |
In rectangle $JKLM$, points $N$ and $P$ bisect $\overline{JK}$ and points $Q$ and $R$ bisect $\overline{LM}$. Rectangle $JKLM$ has dimensions $JK = 6$ and $JL = 4$. Find the area of quadrilateral $STUV$ formed by connecting midpoints of sides in the following manner: $S$ is the midpoint of $\overline{JN}$, $T$ is the midpoint of $\overline{KP}$, $U$ is the midpoint of $\overline{LQ}$, and $V$ is the midpoint of $\overline{MR}$.
|
6
| 0.416667 |
Find the smallest positive integer $n$ such that all the roots of $z^5 - z^3 + 1 = 0$ are $n^{\text{th}}$ roots of unity.
|
30
| 0.583333 |
Let \(a, b, c\) be nonzero real numbers, and define
\[
x = \frac{b}{c} + 2\frac{c}{b}, \quad y = \frac{a}{c} + 2\frac{c}{a}, \quad z = \frac{a}{b} + 2\frac{b}{a}.
\]
Simplify \(x^2 + y^2 + z^2 - xyz\).
|
4
| 0.416667 |
In rectangle $ABCD$, $AB=7$ and $BC =4$. Points $F$ and $G$ are on $\overline{CD}$ so that $DF = 2$ and $GC=1$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$.
|
\frac{98}{3}
| 0.5 |
Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right in reverse order. For example, $b_4 = 4321$ and $b_{12} = 121110987654321$. For $1 \le k \le 150$, how many $b_k$ are divisible by 9?
|
32
| 0.083333 |
Let $f(n)$ be the sum of all the divisors of a positive integer $n$. Determine the number of positive integers $n$ for which $f(f(n)) = n + 3$.
|
0
| 0.5 |
Given a matrix for projecting onto a certain line $m$, which passes through the origin, is represented by the matrix
\[\renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{5}{21} & -\frac{2}{21} & -\frac{2}{7} \\ -\frac{2}{21} & \frac{1}{42} & \frac{1}{14} \\ -\frac{2}{7} & \frac{1}{14} & \frac{8}{14} \end{pmatrix} \renewcommand{\arraystretch}{1}.\]
Find the direction vector of line $m$ in the form $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$, where $a, b, c$ are integers, $a > 0$, and $\gcd(|a|,|b|,|c|) = 1$.
|
\begin{pmatrix} 5 \\ -2 \\ -6 \end{pmatrix}
| 0.25 |
How many rectangles can be formed when the vertices are chosen from points on a 4x4 grid (having 16 points)?
|
36
| 0.166667 |
When the digits in the number $8019$ are reversed, we obtain the number $9108.$ Factorize $9108$ such that it is the product of three distinct primes $x,$ $y,$ and $z.$ How many other positive integers are the products of exactly three distinct primes $q_1,$ $q_2,$ and $q_3$ such that $q_1 + q_2 + q_3 = x+y+z$?
|
6 \text{ distinct integers}
| 0.166667 |
In rectangle $ADEH$, points $B$ and $C$ bisect $\overline{AD}$, and points $G$ and $F$ bisect $\overline{HE}$. In addition, $AH=AC=3$. What is the area of quadrilateral $WXYZ$?
|
\frac{9}{4}
| 0.166667 |
Find the largest constant \( m \), so that for any positive real numbers \( a, b, c, d, \) and \( e \),
\[
\sqrt{\frac{a}{b+c+d+e}} + \sqrt{\frac{b}{a+c+d+e}} + \sqrt{\frac{c}{a+b+d+e}} + \sqrt{\frac{d}{a+b+c+e}} + \sqrt{\frac{e}{a+b+c+d}} > m.
\]
|
2
| 0.333333 |
For all integers $x$ and $y$, define the operation $\diamond$ such that $x \diamond 0 = x$, $x \diamond y = y \diamond x$, and $(x + 2) \diamond y = (x \diamond y) + 2y + 3$. Calculate the value of $8 \diamond 3$.
|
39
| 0.666667 |
Let $a = \log 8$ and $b = \log 27$. Compute
\[2^{a/b} + 3^{b/a}.\]
|
5
| 0.916667 |
Henry's little brother now has $10$ identical stickers and $5$ identical sheets of paper. How many ways are there for him to distribute all the stickers across the sheets of paper, given that only the number of stickers on each sheet matters?
|
30
| 0.083333 |
Triangle $PQR$ has side lengths $PQ=160, QR=300$, and $PR=240$. Lines $m_P, m_Q$, and $m_R$ are drawn parallel to $\overline{QR}, \overline{RP}$, and $\overline{PQ}$, respectively, such that the intersections of $m_P, m_Q$, and $m_R$ with the interior of $\triangle PQR$ are segments of lengths $75, 60$, and $20$, respectively. Find the perimeter of the triangle whose sides lie on lines $m_P, m_Q$, and $m_R$.
|
155
| 0.083333 |
Let $\triangle ABC$ have side lengths $AB = 40$, $BC = 24$, and $AC = 32$. Point $Y$ lies in the interior of $\overline{AC}$, and points $I_1$ and $I_2$ are the incenters of $\triangle ABY$ and $\triangle BCY$, respectively. Find the minimum possible area of $\triangle BI_1I_2$ as $Y$ varies along $\overline{AC}$.
|
96
| 0.5 |
The matrix for projecting onto a certain line $m,$ which passes through the origin, is given by
\[\renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{18} & \frac{1}{9} & -\frac{1}{6} \\ \frac{1}{9} & \frac{1}{6} & -\frac{1}{3} \\ -\frac{1}{6} & -\frac{1}{3} & \frac{2}{3} \end{pmatrix} \renewcommand{\arraystretch}{1}.\]
Find the direction vector of line $m.$ Enter your answer in the form $\begin{pmatrix} a \\ b \\ c \end{pmatrix},$ where $a,$ $b,$ and $c$ are integers, $a > 0,$ and $\gcd(|a|,|b|,|c|) = 1.$
|
\begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}
| 0.166667 |
Fido's leash is tied to a stake at the center of his yard, which is now in the shape of a regular octagon. His leash is exactly long enough to reach the midpoint of each side of this octagonal yard. If the fraction of the area of Fido's yard that he can reach while on his leash is expressed in simplest radical form as $\frac{\sqrt{a}}{b}\pi$, what is the value of $a \times b$?
|
16
| 0.583333 |
Find all real values of $x$ that satisfy \(\frac{1}{x(x+1)} - \frac{1}{(x+1)(x+2)} < \frac{1}{4}\). (Give your answer in interval notation.)
|
(-\infty, -2) \cup (-1, 0) \cup (2, \infty)
| 0.083333 |
Find the minimum value of
\[ g(x) = x + \frac{2x}{x^2 + 2} + \frac{x(x + 5)}{x^2 + 3} + \frac{3(x + 3)}{x(x^2 + 3)} \]
for \( x > 0 \).
|
6
| 0.083333 |
Let \(x\), \(y\), and \(z\) be real numbers such that
\[3x^2 + 9xy + 6y^2 + z^2 = 1.\]
Let \(n\) and \(N\) be the minimum and maximum values of \(x^2 + 4xy + 3y^2 + z^2\), respectively. Find the product \(nN.\)
|
\frac{2}{3}
| 0.083333 |
Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^3}$ is a real number and $|\alpha - \beta| = 6$. Find $|\alpha|$.
|
3\sqrt{2}
| 0.583333 |
Find the least positive integer $n$ such that
$$\frac{1}{\sin 30^\circ \sin 31^\circ} + \frac{1}{\sin 32^\circ \sin 33^\circ} + \cdots + \frac{1}{\sin 148^\circ \sin 149^\circ} = \frac{a}{\sin n^\circ},$$
where $a = 2\sin 1^\circ$.
|
1
| 0.166667 |
How many distinct two-digit numbers can appear as the last two digits of an integral perfect-cube number?
|
44
| 0.083333 |
There are two cubes. The mass of the second cube is $25\%$ less than the mass of the first cube, and the edge length of the second cube is $25\%$ greater than that of the first cube. By what percentage does the density of the second cube differ from the density of the first cube?
|
61.6\%
| 0.375 |
An urn contains 9 balls, numbered from 1 to 9. José and Maria each simultaneously draw one ball from the urn. They form a two-digit number, with the number on José's ball being the tens digit and the number on Maria's ball being the units digit. What is the probability that this number is even?
|
\frac{4}{9}
| 0.875 |
At the World Meteorological Conference, each participant took turns announcing the average monthly temperature in their home city. At that moment, all other participants recorded the product of the temperatures in his and their cities. A total of 50 positive and 60 negative numbers were recorded. What is the minimum number of times a positive temperature could have been announced?
|
5
| 0.875 |
A reception hall in the palace of the thirteenth kingdom comprises points on a plane whose coordinates satisfy the condition \(4|x| + 5|y| \leq 20\). How many identical tiles of double-sided parquet, each in the shape of a right triangle with legs 1 and \(5/4\), are required to tile the floor of the hall? Tiling is considered to be done without any gaps, overlaps, and without extending beyond the area.
|
64
| 0.875 |
Solve the equation \( 2021 \cdot \sqrt[202]{x^{2020}} - 1 = 2020x \) for \( x \geq 0 \). (10 points)
|
x=1
| 0.375 |
Let \( R \) be a point on the curve such that \( OMRN \) is a square. If \( r \) is the \( x \)-coordinate of \( R \), find the value of \( r \).
|
r = 1
| 0.875 |
Find the sum of all four-digit numbers (written in base 10) which contain only the digits 1, 2, 3, 4, and 5, and contain no digit more than once.
|
399960
| 0.5 |
Several people were seated around a round table such that the distances between neighboring people were equal. One of them was given a card with the number 1, and the rest were given cards with numbers 2, 3, and so on, in a clockwise direction.
The person with the card numbered 31 noticed that the distance from him to the person with the card numbered 7 is the same as the distance to the person with the card numbered 14. How many people are seated at the table in total?
|
41
| 0.5 |
Find the probability that a randomly selected two-digit number is divisible by either 2, 5, or both.
|
0.6
| 0.125 |
Let \( x \) and \( y \) be the integral and fractional parts of \( \sqrt{37 - 20 \sqrt{3}} \). Find the value of \( x + y + \frac{4}{y} \).
|
9
| 0.875 |
In an $8 \times 8$ chessboard, how many ways are there to select 56 squares so that all the black squares are selected, and each row and each column has exactly seven squares selected?
|
576
| 0.375 |
As shown in the figure, $A$ and $B$ are the two endpoints of a diameter of a circular track. Three miniature robots, labeled as A, B, and C, start at the same time from point $A$ and perform uniform circular motion on the track. Robots A and B start from $A$, while robot C starts from $B$. Robot B moves in a clockwise direction, while robots A and C move in a counterclockwise direction. After 12 seconds, robot A reaches $B$. After an additional 9 seconds, robot A catches up with robot C for the first time and simultaneously meets robot B for the first time. Calculate how many seconds after robot C first reaches $A$ it will take for robot B to reach $B$ for the first time.
|
56
| 0.75 |
The sought-after three-digit number ends with the digit 1. If this digit is erased and then reattached as the first digit of the number, the resulting new three-digit number will be smaller than the original by \(10 a^{\log _{\sqrt{a}} 3}\). Find this number.
|
211
| 0.875 |
The product of three natural numbers is equal to 60. What is the largest possible value of their sum?
|
62
| 0.875 |
In a regular hexagon with an area of $144 \mathrm{~cm}^{2}$, all its diagonals were drawn. The hexagon thus "split" into several triangles and quadrilaterals.
a) Determine into how many parts the diagonals have divided the hexagon.
b) Determine the area of the regular hexagon formed by combining all the quadrilateral parts of the given hexagon.
|
48 \text{ cm}^2
| 0.25 |
In a convex pentagon \(ABCDE\), \(\angle A = 60^\circ\), and the other angles are equal to each other. It is known that \(AB = 6\), \(CD = 4\), and \(EA = 7\). Find the distance from point \(A\) to the line \(CD\).
|
\frac{9\sqrt{3}}{2}
| 0.125 |
Given complex numbers \( z, z_{1}, z_{2} \left( z_{1} \neq z_{2} \right) \) such that \( z_{1}^{2}=z_{2}^{2}=-2-2 \sqrt{3} \mathrm{i} \), and \(\left|z-z_{1}\right|=\left|z-z_{2}\right|=4\), find \(|z|=\ \ \ \ \ .\)
|
2\sqrt{3}
| 0.875 |
Donald Duck and Mickey Mouse are competing in a 10,000-meter race. Mickey Mouse runs at a speed of 125 meters per minute, while Donald Duck runs at a speed of 100 meters per minute. Donald Duck has an electronic remote control that can force Mickey Mouse to move backward. When this remote control is used for the \( n \)-th time, Mickey Mouse will move backward for one minute at a speed that is \( n \times 10\% \) of his original speed, then continue running forward at his original speed. What is the minimum number of times Donald Duck needs to use the remote control in order to win the race?
|
13
| 0.625 |
Three positive integers greater than 1000 satisfy the condition that the unit digit of the sum of any two of them is equal to the unit digit of the third number. What are the possible values for the last three digits of the product of these three numbers?
|
000, 250, 500, 750
| 0.375 |
In the plane, there are 2020 points, some of which are black and the rest are green.
For each black point, there are exactly two green points that are at a distance of 2020 from this black point.
Determine the minimum possible number of green points.
|
45
| 0.125 |
Find the sum of the first 10 elements that appear both in the arithmetic progression $\{4, 7, 10, 13, \ldots\}$ and the geometric progression $\{10, 20, 40, 80, \ldots\}$. (10 points)
|
3495250
| 0.125 |
Compute the limit of the function:
$$\lim _{x \rightarrow 0} \frac{\ln \left(x^{2}+1\right)}{1-\sqrt{x^{2}+1}}$$
|
-2
| 0.875 |
Let \(\alpha\), \(\beta\), and \(\gamma\) be real numbers satisfying \(\alpha + \beta + \gamma = 2\) and \(\alpha \beta \gamma = 4\). Let \(v\) be the minimum value of \(|\alpha| + |\beta| + |\gamma|\). Find the value of \(v\).
|
6
| 0.875 |
Two positive integers are written on the blackboard, one being 2002 and the other being a number less than 2002. If the arithmetic mean of the two numbers is an integer $m$, then the following operation can be performed: one of the numbers is erased and replaced by $m$. What is the maximum number of times this operation can be performed?
|
10
| 0.75 |
Find the pairs of positive integers \((x, y)\) that satisfy \(x > y\) and \((x - y)^{xy} = x^y y^x\).
|
(4, 2)
| 0.625 |
Let \( a, b \), and \( c \) be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: \( ax^2 + bx + c \), \( bx^2 + cx + a \), and \( cx^2 + ax + b \).
|
4
| 0.625 |
In city $\mathrm{N}$, there are exactly three monuments. One day, a group of 42 tourists arrived in this city. Each tourist took no more than one photograph of each of the three monuments. It turned out that any two tourists together had photographs of all three monuments. What is the minimum number of photographs that all the tourists together could have taken?
|
123
| 0.625 |
Let \(ABCD\) be a quadrilateral such that \(AD = BC\), \( (AB) \) and \( (CD) \) are parallel, and \(AB > CD\). Let \(E\) be the midpoint of \([AC]\), and \(F\) the point of intersection of the diagonals \((AC)\) and \((BD)\). The line parallel to \((BD)\) passing through \(E\) intersects the line \((CD)\) at point \(G\).
1. Show that triangle \(CGA\) is a right triangle at \(G\).
2. Let \(CD = b\) and \(AB = a\). Calculate the ratio \(\frac{EG}{CF}\) as a function of \(a\) and \(b\).
|
\frac{a + b}{2b}
| 0.625 |
Let \( R \) be the circumradius of \( \triangle ABC \), \( r \) be the inradius of \( \triangle ABC \), and \( d \) be the distance between the circumcenter \( O \) and the incenter \( I \). Then \( d^{2} = R^{2} - 2Rr \).
|
d^2 = R^2 - 2 R r
| 0.25 |
Given \(\theta = \arctan \frac{5}{12}\), find the principal value of the argument of the complex number \(z = \frac{\cos 2\theta + i \sin 2\theta}{239 + i}\).
|
\frac{\pi}{4}
| 0.75 |
Using the digits 1 to 5 and a multiplication sign $\times$, Clara forms the product of two numbers with the $\times$ sign between them. How should Clara arrange the digits to obtain the largest possible product?
|
22412
| 0.125 |
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