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Numbers from 1 to 100 are written in a vertical row in ascending order. Fraction bars of different sizes are inserted between them. The calculation starts with the smallest fraction bar and ends with the largest one, for example, $\frac{1}{\frac{5}{3}}=\frac{15}{4}$. What is the greatest possible value that the resulting fraction can have? | 100 | 1/8 |
A farmer had an enclosure with a fence 50 rods long, which could only hold 100 sheep. Suppose the farmer wanted to expand the enclosure so that it could hold twice as many sheep.
How many additional rods will the farmer need? | 21 | 3/8 |
Find all sets of four positive real numbers \((a, b, c, d)\) satisfying
\[
\begin{array}{l}
a + b + c + d = 1, \\
\text{and} \\
\max \left\{\frac{a^{2}}{b}, \frac{b^{2}}{a}\right\} \max \left\{\frac{c^{2}}{d}, \frac{d^{2}}{c}\right\} = (\min \{a+b, c+d\})^{4}.
\end{array}
\] | (\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}) | 2/8 |
Given that Erin the ant starts at a given corner of a hypercube (4-dimensional cube) and crawls along exactly 15 edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point, determine the number of paths that Erin can follow to meet these conditions. | 24 | 1/8 |
The triangle ABC satisfies \(0 \leq AB \leq 1 \leq BC \leq 2 \leq CA \leq 3\). What is the maximum area it can have? | 1 | 6/8 |
Find all pairs of positive integers \((a, n)\) such that \(a \geq n\), and \((a+1)^{n} + a - 1\) is a power of 2. | (4,3) | 1/8 |
At lunch, Abby, Bart, Carl, Dana, and Evan share a pizza divided radially into 16 slices. Each one takes takes one slice of pizza uniformly at random, leaving 11 slices. The remaining slices of pizza form "sectors" broken up by the taken slices, e.g. if they take five consecutive slices then there is one sector, but if none of them take adjacent slices then there will be five sectors. What is the expected number of sectors formed? | \frac{11}{3} | 3/8 |
For which values of \(a\) does the equation \(x \cdot |x-a| = 1\) have three distinct solutions? | 2 | 1/8 |
A circle with a radius of 3 is centered at the midpoint of one side of an equilateral triangle each side of which has a length of 9. Determine the difference between the area inside the circle but outside the triangle and the area inside the triangle but outside the circle. | 9\pi - \frac{81\sqrt{3}}{4} | 3/8 |
Let $b_n$ be the number obtained by writing the integers $1$ to $n$ from left to right and then subtracting $n$ from the resulting number. For example, $b_4 = 1234 - 4 = 1230$ and $b_{12} = 123456789101112 - 12 = 123456789101100$. For $1 \le k \le 100$, how many $b_k$ are divisible by 9? | 22 | 1/8 |
Let \( A B C \) be a triangle with circumcircle \( \omega \). The internal angle bisectors of \( \angle A B C \) and \( \angle A C B \) intersect \( \omega \) at \( X \neq B \) and \( Y \neq C \), respectively. Let \( K \) be a point on \( C X \) such that \( \angle K A C = 90^{\circ} \). Similarly, let \( L \) be a point on \( B Y \) such that \( \angle L A B = 90^{\circ} \). Let \( S \) be the midpoint of the arc \( C A B \) of \( \omega \). Prove that \( S K = S L \). | SK=SL | 1/8 |
Let \( x, y, z, u, v \in \mathbf{R}_{+} \). The maximum value of
\[
f = \frac{x y + y z + z u + u v}{2 x^{2} + y^{2} + 2 z^{2} + u^{2} + 2 v^{2}}
\]
is $\qquad$ . | \frac{\sqrt{6}}{4} | 5/8 |
A construction team has 6 projects (A, B, C, D, E, F) that need to be completed separately. Project B must start after project A is completed, project C must start after project B is completed, and project D must immediately follow project C. Determine the number of different ways to schedule these 6 projects. | 20 | 2/8 |
Let \( p \) be a given prime number, and let \( a_1, \cdots, a_k \) be \( k \) integers (\( k \geqslant 3 \)), none of which are divisible by \( p \) and are pairwise incongruent modulo \( p \). Define the set
\[
S = \{ n \mid 1 \leq n \leq p-1, \, (n a_1)_p < \cdots < (n a_k)_p \},
\]
where \((b)_p\) denotes the remainder when the integer \( b \) is divided by \( p \). Prove that
\[ |S| < \frac{2p}{k+1}. \] | |S|<\frac{2p}{k+1} | 1/8 |
A regular triangular prism \(A B C A_{1} B_{1} C_{1}\) with base \(A B C\) and lateral edges \(A A_{1}, B B_{1}, C C_{1}\) is inscribed in a sphere. The segment \(C D\) is the diameter of this sphere, and point \(K\) is the midpoint of edge \(A A_{1}\). Find the volume of the prism if \(C K = 2 \sqrt{3}\) and \(D K = 2 \sqrt{2}\). | 9\sqrt{2} | 6/8 |
In the Cartesian coordinate system, with the origin as the pole and the positive x-axis as the polar axis, the polar equation of line $l$ is $$ρ\cos(θ+ \frac {π}{4})= \frac { \sqrt {2}}{2}$$, and the parametric equation of curve $C$ is $$\begin{cases} x=5+\cos\theta \\ y=\sin\theta \end{cases}$$, (where $θ$ is the parameter).
(Ⅰ) Find the Cartesian equation of line $l$ and the general equation of curve $C$;
(Ⅱ) Curve $C$ intersects the x-axis at points $A$ and $B$, with $x_A < x_B$, $P$ is a moving point on line $l$, find the minimum perimeter of $\triangle PAB$. | 2+ \sqrt {34} | 4/8 |
A function \( f(x) \) defined on the set of real numbers \(\mathbf{R}\) satisfies the following three conditions:
1. \( f(x) > 0 \) when \( x > 0 \),
2. \( f(1) = 2 \),
3. For any \( m, n \in \mathbf{R} \), \( f(m+n) = f(m) + f(n) \).
Let the sets be defined as:
\[
A = \left\{ (x, y) \mid f(3x^2) + f(4y^2) \leq 24 \right\},
\]
\[
B = \{ (x, y) \mid f(x) - f(ay) + f(3) = 0 \},
\]
\[
C = \left\{ (x, y) \mid f(x) = \frac{1}{2} f(y^2) + f(a) \right\}.
\]
If \( A \cap B \neq \varnothing \) and \( A \cap C \neq \varnothing \), determine the range of the real number \( a \). | [-\frac{13}{6},-\frac{\sqrt{15}}{3}]\cup[\frac{\sqrt{15}}{3},2] | 2/8 |
Today, Ivan the Confessor prefers continuous functions $f:[0,1]\to\mathbb{R}$ satisfying $f(x)+f(y)\geq |x-y|$ for all pairs $x,y\in [0,1]$ . Find the minimum of $\int_0^1 f$ over all preferred functions.
(Proposed by Fedor Petrov, St. Petersburg State University) | \frac{1}{4} | 3/8 |
On the side \( BC \) of triangle \( ABC \), point \( K \) is marked.
It is known that \( \angle B + \angle C = \angle AKB \), \( AK = 4 \), \( BK = 9 \), and \( KC = 3 \).
Find the area of the circle inscribed in triangle \( ABC \). | \frac{35}{13}\pi | 1/8 |
In a $15 \times 15$ grid, there is a non-self-intersecting closed polygonal chain that consists of several segments connecting the centers of adjacent smaller squares (two smaller squares sharing a common edge are called adjacent). This chain is symmetric with respect to some diagonal of the grid. Prove that the length of this closed polygonal chain does not exceed 200. | 200 | 2/8 |
Rectangle PQRS and right triangle SRT share side SR and have the same area. Rectangle PQRS has dimensions PQ = 4 and PS = 8. Find the length of side RT. | 16 | 5/8 |
If Greg rolls four fair six-sided dice, what is the probability that he rolls more 1's than 6's? | \dfrac{421}{1296} | 7/8 |
In the forest, all trees are between 10 and 50 meters high, and the distance between any two trees is no more than the difference in their heights. Prove that this forest can be enclosed with a fence of length $80 \mathrm{~m}$. | 80 | 3/8 |
Given the parabola \(\Gamma: y^{2}=8 x\) with focus \(F\), a line \(l\) passing through \(F\) intersects parabola \(\Gamma\) at points \(A\) and \(B\). Tangents to parabola \(\Gamma\) at \(A\) and \(B\) intersect the \(y\)-axis at points \(P\) and \(Q\) respectively. Find the minimum area of the quadrilateral \(APQB\). | 12 | 7/8 |
Given an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity $e = \frac{\sqrt{3}}{3}$. The left and right foci are $F_1$ and $F_2$, respectively, with $F_2$ coinciding with the focus of the parabola $y^2 = 4x$.
(I) Find the standard equation of the ellipse;
(II) If a line passing through $F_1$ intersects the ellipse at points $B$ and $D$, and another line passing through $F_2$ intersects the ellipse at points $A$ and $C$, with $AC \perp BD$, find the minimum value of $|AC| + |BD|$. | \frac{16\sqrt{3}}{5} | 3/8 |
In a new diagram below, we have $\cos \angle XPY = \frac{3}{5}$. A point Z is placed such that $\angle XPZ$ is a right angle. What is $\sin \angle YPZ$?
[asy]
pair X, P, Y, Z;
P = (0,0);
X = Rotate(-aCos(3/5))*(-2,0);
Y = (2,0);
Z = Rotate(-90)*(2,0);
dot("$Z$", Z, S);
dot("$Y$", Y, S);
dot("$X$", X, W);
dot("$P$", P, S);
draw(X--P--Y--Z--cycle);
[/asy] | \frac{3}{5} | 6/8 |
Calculate the value of the expression: $3 - 7 + 11 - 15 + 19 - \cdots - 59 + 63 - 67 + 71$. | -36 | 3/8 |
In a triangle $ABC$ , let $I$ denote the incenter. Let the lines $AI,BI$ and $CI$ intersect the incircle at $P,Q$ and $R$ , respectively. If $\angle BAC = 40^o$ , what is the value of $\angle QPR$ in degrees ? | 20 | 1/8 |
Given the polynomial
$$
\begin{aligned}
P(x)= & x^{15}-2008 x^{14}+2008 x^{13}-2008 x^{12}+2008 x^{11} \\
& -\cdots+2008 x^{3}-2008 x^{2}+2008 x,
\end{aligned}
$$
determine \( P(2007) \). | 2007 | 7/8 |
Let $ABC$ be an acute triangle and $AD$ a height. The angle bissector of $\angle DAC$ intersects $DC$ at $E$ . Let $F$ be a point on $AE$ such that $BF$ is perpendicular to $AE$ . If $\angle BAE=45º$ , find $\angle BFC$ . | 135 | 1/8 |
Find the smallest six-digit number that is divisible by 11, where the sum of the first and fourth digits is equal to the sum of the second and fifth digits, and equal to the sum of the third and sixth digits. | 100122 | 5/8 |
Given that the answer to this problem can be expressed as $a\cdot b\cdot c$ , where $a$ , $b$ , and $c$ are pairwise relatively prime positive integers with $b=10$ , compute $1000a+100b+10c$ .
*Proposed by Ankit Bisain* | 203010 | 6/8 |
For what value(s) of $k$ does the pair of equations $y=x^2$ and $y=3x+k$ have two identical solutions?
$\textbf{(A)}\ \frac{4}{9}\qquad \textbf{(B)}\ -\frac{4}{9}\qquad \textbf{(C)}\ \frac{9}{4}\qquad \textbf{(D)}\ -\frac{9}{4}\qquad \textbf{(E)}\ \pm\frac{9}{4}$ | \textbf{(D)}\-\frac{9}{4} | 1/8 |
What is the minimum number of points that can be chosen on a circle with a circumference of 1956 so that for each of these points there is exactly one chosen point at a distance of 1 and exactly one at a distance of 2 (distances are measured along the circle)? | 1304 | 2/8 |
A positive integer $n$ is infallible if it is possible to select $n$ vertices of a regular 100-gon so that they form a convex, non-self-intersecting $n$-gon having all equal angles. Find the sum of all infallible integers $n$ between 3 and 100, inclusive. | 262 | 1/8 |
In triangle \( \triangle ABC \), \( BC = 4 \) and \( \angle BAC = 60^\circ \). Let \( I \) be the incentre of \( \triangle ABC \). The circle passing through \( B, I \), and \( C \) meets the perpendicular bisector of \( BC \) at a point \( X \) inside \( \triangle ABC \). Find \( AX \). | \frac{4\sqrt{3}}{3} | 1/8 |
Given a triangle $ABC$ with $\angle BAC = 60^\circ$, let $H$, $I$, and $O$ denote its orthocenter, its incenter, and its circumcenter respectively. Show that $IO = IH$. | IO=IH | 1/8 |
Suppose $S_n$ is the set of positive divisors of $n$ , and denote $|X|$ as the number of elements in a set $X$ . Let $\xi$ be the set of positive integers $n$ where $|S_n| = 2m$ is even, and $S_n$ can be partitioned evenly into pairs $\{a_i, b_i\}$ for integers $1 \le i \le m$ such that the following conditions hold: $\bullet$ $a_i$ and $b_i$ are relatively prime for all integers $1 \le i \le m$ $\bullet$ There exists a $j$ where $6$ divides $a^2_j+ b^2_j+ 1$ $\bullet$ $|S_n| \ge 20.$ Determine the number of positive divisors $d | 24!$ such that $d \in \xi$ . | 64 | 2/8 |
In how many ways can eight out of nine digits $1,2,3,4,5,6,7,8$, and $9$ be placed in a $4 \times 2$ table (4 rows, 2 columns) such that the sum of the digits in each row, starting from the second, is 1 more than in the previous row? | 64 | 5/8 |
Using toothpicks of equal length, a rectangular grid is constructed. The grid measures 25 toothpicks in height and 15 toothpicks in width. Additionally, there is an internal horizontal partition at every fifth horizontal line starting from the bottom. Calculate the total number of toothpicks used. | 850 | 2/8 |
Determine the number of arrangements of the letters a, b, c, d, e in a sequence such that neither a nor b is adjacent to c. | 36 | 5/8 |
Let $P$ be a $10$ -degree monic polynomial with roots $r_1, r_2, . . . , r_{10} \ne $ and let $Q$ be a $45$ -degree monic polynomial with roots $\frac{1}{r_i}+\frac{1}{r_j}-\frac{1}{r_ir_j}$ where $i < j$ and $i, j \in \{1, ... , 10\}$ . If $P(0) = Q(1) = 2$ , then $\log_2 (|P(1)|)$ can be written as $a/b$ for relatively prime integers $a, b$ . Find $a + b$ . | 19 | 5/8 |
Let the set
\[ S = \{m \mid m \in \mathbf{Z}_{+}, \text{each prime factor of } m \text{ is less than } 10\}. \]
Find the smallest positive integer \( n \) such that in any \( n \)-element subset of \( S \), there exist four distinct numbers whose product is a perfect square. | 9 | 1/8 |
If $$\sin\alpha= \frac {4}{7} \sqrt {3}$$ and $$\cos(\alpha+\beta)=- \frac {11}{14}$$, and $\alpha$, $\beta$ are acute angles, then $\beta= \_\_\_\_\_\_$. | \frac {\pi}{3} | 4/8 |
Let $S(n)$ denote the sum of digits of a natural number $n$ . Find all $n$ for which $n+S(n)=2004$ . | 2001 | 7/8 |
When triangle $EFG$ is rotated by an angle $\arccos(_{1/3})$ around point $O$, which lies on side $EG$, vertex $F$ moves to vertex $E$, and vertex $G$ moves to point $H$, which lies on side $FG$. Find the ratio in which point $O$ divides side $EG$. | 3:1 | 4/8 |
Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD=\angle CMD$. What is the degree measure of $\angle AMD$? | 75 | 7/8 |
10 white and 20 black chips are arranged in a circle. It is allowed to swap any two chips that have exactly three chips between them. Two arrangements of chips (in these 30 positions) are called equivalent if one can be transformed into the other by several such swaps. How many non-equivalent arrangements are there? | 11 | 4/8 |
Three lines were drawn through the point $X$ in space. These lines crossed some sphere at six points. It turned out that the distances from point $X$ to some five of them are equal to $2$ cm, $3$ cm, $4$ cm, $5$ cm, $6$ cm. What can be the distance from point $X$ to the sixth point?
(Alexey Panasenko) | 2.4 | 1/8 |
Find $A^2_{}$, where $A^{}_{}$ is the sum of the absolute values of all roots of the following equation:
$x = \sqrt{19} + \frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{x}}}}}}}}}$ | 383 | 4/8 |
An old clock's minute and hour hands overlap every 66 minutes of standard time. Calculate how much the old clock's 24 hours differ from the standard 24 hours. | 12 | 2/8 |
Acute triangle $ABC$ has altitudes $AD$ , $BE$ , and $CF$ . Point $D$ is projected onto $AB$ and $AC$ to points $D_c$ and $D_b$ respectively. Likewise, $E$ is projected to $E_a$ on $BC$ and $E_c$ on $AB$ , and $F$ is projected to $F_a$ on $BC$ and $F_b$ on $AC$ . Lines $D_bD_c$ , $E_cE_a$ , $F_aF_b$ bound a triangle of area $T_1$ , and lines $E_cF_b$ , $D_bE_a$ , $F_aD_c$ bound a triangle of area $T_2$ . What is the smallest possible value of the ratio $T_2/T_1$ ? | 25 | 1/8 |
Let \( S = \{1, 2, \cdots, 2005\} \). Find the minimum value of \( n \) such that any set of \( n \) pairwise coprime elements from \( S \) contains at least one prime number. | 16 | 6/8 |
Ten balls of equal radius are stacked in the shape of a triangular pyramid such that each ball touches at least three others. Find the radius of the sphere into which the pyramid of balls is inscribed, given that the radius of the ball inscribed at the center of the pyramid of balls, that touches six identical balls, is $\sqrt{6}-1$. | 5(\sqrt{2}+1) | 1/8 |
ABC is an equilateral triangle. D is on the side AB and E is on the side AC such that DE touches the incircle. Show that AD/DB + AE/EC = 1. | 1 | 5/8 |
What is the maximum number of natural numbers that can be written in a row such that the sum of any three consecutive numbers is even, and the sum of any four consecutive numbers is odd? | 5 | 2/8 |
A manufacturer built a machine which will address $500$ envelopes in $8$ minutes. He wishes to build another machine so that when both are operating together they will address $500$ envelopes in $2$ minutes. The equation used to find how many minutes $x$ it would require the second machine to address $500$ envelopes alone is:
$\textbf{(A)}\ 8-x=2 \qquad \textbf{(B)}\ \dfrac{1}{8}+\dfrac{1}{x}=\dfrac{1}{2} \qquad \textbf{(C)}\ \dfrac{500}{8}+\dfrac{500}{x}=500 \qquad \textbf{(D)}\ \dfrac{x}{2}+\dfrac{x}{8}=1 \qquad\\ \textbf{(E)}\ \text{None of these answers}$ | \textbf{(B)}\\frac{1}{8}+\frac{1}{x}=\frac{1}{2} | 1/8 |
For points \( A_{1}, B_{1}, C_{1} \) to lie on a straight line, it is necessary and sufficient that \( R=1 \) (see Problem 191) and that an even number (i.e., zero or two) of the points \( A_{1}, B_{1}, C_{1} \) lie on the sides of the triangle, rather than on their extensions. | 1 | 7/8 |
Four black $1 \times 1 \times 1$ cubes and four white $1 \times 1 \times 1$ cubes can be assembled into $\qquad$ different $2 \times 2 \times 2$ cubes (cubes that are identical after rotation are considered the same). | 7 | 3/8 |
Let $k$ be an integer greater than 1. Suppose $a_0 > 0$, and define \[a_{n+1} = a_n + \frac{1}{\sqrt[k]{a_n}}\] for $n > 0$. Evaluate \[\lim_{n \to \infty} \frac{a_n^{k+1}}{n^k}.\] | \left( \frac{k+1}{k} \right)^k | 6/8 |
Several stones are placed in 5 piles. It is known that:
- There are six times as many stones in the fifth pile as in the third pile.
- There are twice as many stones in the second pile as in the third and fifth piles combined.
- There are three times fewer stones in the first pile than in the fifth pile, and 10 fewer than in the fourth pile.
- There are twice as few stones in the fourth pile as in the second pile.
How many stones are there in total in these five piles? | 60 | 7/8 |
How many perfect squares are between 100 and 400? | 11 | 2/8 |
Find the dimensions of the cone that can be formed from a $300^{\circ}$ sector of a circle with a radius of 12 by aligning the two straight sides. | 12 | 2/8 |
In a circle with centre at $O$ and diameter $AB$ , two chords $BD$ and $AC$ intersect at $E$ . $F$ is a point on $AB$ such that $EF \perp AB$ . $FC$ intersects $BD$ in $G$ . If $DE = 5$ and $EG =3$ , determine $BG$ . | 12 | 1/8 |
The sum of the areas of all triangles whose vertices are also vertices of a $1$ by $1$ by $1$ cube is $m + \sqrt{n} + \sqrt{p},$ where $m, n,$ and $p$ are integers. Find $m + n + p.$
| 348 | 1/8 |
Given a triangle with acute angle $ \angle BEC,$ let $ E$ be the midpoint of $ AB.$ Point $ M$ is chosen on the opposite ray of $ EC$ such that $ \angle BME \equal{} \angle ECA.$ Denote by $ \theta$ the measure of angle $ \angle BEC.$ Evaluate $ \frac{MC}{AB}$ in terms of $ \theta.$ | \cos\theta | 2/8 |
In a checkered square with a side length of 2018, some cells are painted white and the rest are black. It is known that from this square, one can cut out a 10x10 square where all the cells are white, and a 10x10 square where all the cells are black. What is the smallest value for which it is guaranteed that one can cut out a 10x10 square in which the number of black and white cells differ by no more than? | 10 | 1/8 |
From a \(6 \times 6\) square grid, gray triangles were cut out. What is the area of the remaining shape? The length of each side of the cells is 1 cm. Provide your answer in square centimeters. | 27 | 1/8 |
A garden fence, similar to the one shown in the picture, had in each section (between two vertical posts) the same number of columns, and each vertical post (except for the two end posts) divided one of the columns in half. When we absentmindedly counted all the columns from end to end, counting two halves as one whole column, we found that there were a total of 1223 columns. We also noticed that the number of sections was 5 more than twice the number of whole columns in each section.
How many columns were there in each section? | 23 | 4/8 |
A plane flies from city A to city B against a wind in 120 minutes. On the return trip with the wind, it takes 10 minutes less than it would in still air. Determine the time in minutes for the return trip. | 110 | 1/8 |
For the functions \( s(x) \) and \( t(x) \), it is known that \( s(0) = t(0) > 0 \) and \( s^{\prime}(x) \sqrt{t^{\prime}(x)} = 5 \) for any \( x \in [0, 1] \). Prove that if \( x \in [0, 1] \), then \( 2s(x) + 5t(x) > 15x \). | 2s(x)+5t(x)>15x | 7/8 |
Given a real number \(a\), and for any \(k \in [-1, 1]\), when \(x \in (0, 6]\), the inequality \(6 \ln x + x^2 - 8x + a \leq kx\) always holds. Determine the maximum value of \(a\). | 6 - 6 \ln 6 | 2/8 |
In triangle $ABC$, where $AB = \sqrt{34}$ and $AC = 5$, the angle $B$ is $90^\circ$. Calculate $\tan A$. | \frac{3}{5} | 5/8 |
A rectangular prism has 4 green faces, 2 yellow faces, and 6 blue faces. What's the probability that when it is rolled, a blue face will be facing up? | \frac{1}{2} | 4/8 |
Given that a normal vector of the straight line $l$ is $\overrightarrow{n} = (1, -\sqrt{3})$, find the size of the inclination angle of this straight line. | \frac{\pi}{6} | 2/8 |
There are 19 candy boxes arranged in a row, with the middle box containing $a$ candies. Moving to the right, each box contains $m$ more candies than the previous one; moving to the left, each box contains $n$ more candies than the previous one ($a$, $m$, and $n$ are all positive integers). If the total number of candies is 2010, then the sum of all possible values of $a$ is. | 105 | 5/8 |
There are 4 numbers, not all of which are identical. If you take any two of them, the ratio of the sum of these two numbers to the sum of the other two numbers will be the same value \( \mathrm{k} \). Find the value of \( \mathrm{k} \). Provide at least one set of four numbers that satisfies the condition. Describe all possible sets of such numbers and determine their quantity. | -1 | 7/8 |
A gumball machine contains $9$ red, $7$ white, and $8$ blue gumballs. The least number of gumballs a person must buy to be sure of getting four gumballs of the same color is
$\text{(A)}\ 8 \qquad \text{(B)}\ 9 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 18$ | (C)\10 | 1/8 |
Given the real numbers \( x \) and \( y \) satisfy the equations:
\[ 2^x + 4x + 12 = \log_2{(y-1)^3} + 3y + 12 = 0 \]
find the value of \( x + y \). | -2 | 4/8 |
The positive numbers \(a, b,\) and \(c\) satisfy the condition \(abc(a+b+c) = 3\). Prove the inequality \((a+b)(b+c)(c+a) \geq 8\). | 8 | 7/8 |
Points \( A, B, C \), and \( D \) are located on a line such that \( AB = BC = CD \). Segments \( AB \), \( BC \), and \( CD \) serve as diameters of circles. From point \( A \), a tangent line \( l \) is drawn to the circle with diameter \( CD \). Find the ratio of the chords cut on line \( l \) by the circles with diameters \( AB \) and \( BC \). | \sqrt{6}: 2 | 1/8 |
Maria ordered a certain number of televisions for the stock of a large store, paying R\$ 1994.00 per television. She noticed that in the total amount to be paid, the digits 0, 7, 8, and 9 do not appear. What is the smallest number of televisions she could have ordered? | 56 | 4/8 |
Does there exist a convex hexagon and a point \( M \) inside it such that all sides of the hexagon are greater than 1, and the distance from \( M \) to any vertex is less than 1? | 0 | 1/8 |
Given that \(\sin x + \sin y = 1\), determine the range of values for \(\cos x + \cos y\). | [-\sqrt{3},\sqrt{3}] | 7/8 |
Amelia has a coin that lands heads with probability $\frac{1}{3}\,$, and Blaine has a coin that lands on heads with probability $\frac{2}{5}$. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $q-p$?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$ | \textbf{(D)}\4 | 1/8 |
Find the maximum value of
\[\frac{2x + 3y + 4}{\sqrt{x^2 + 4y^2 + 2}}\]
over all real numbers \( x \) and \( y \). | \sqrt{29} | 1/8 |
In an acute-angled triangle \(SAP\), the altitude \(AK\) is drawn. A point \(L\) is selected on the side \(PA\), and a point \(M\) is chosen on the extension of the side \(SA\) beyond point \(A\) such that \(\angle LSP = \angle LPS\) and \(\angle MSP = \angle MPS\). The lines \(SL\) and \(PM\) intersect the line \(AK\) at points \(N\) and \(O\) respectively. Prove that \(2ML = NO\). | 2ML=NO | 2/8 |
Comprehensive exploration: When two algebraic expressions containing square roots are multiplied together and the product does not contain square roots, we call these two expressions rationalizing factors of each other. For example, $\sqrt{2}+1$ and $\sqrt{2}-1$, $2\sqrt{3}+3\sqrt{5}$ and $2\sqrt{3}-3\sqrt{5}$ are all rationalizing factors of each other. When performing calculations involving square roots, using rationalizing factors can eliminate square roots in the denominator. For example: $\frac{1}{\sqrt{2}+1}=\frac{1\times(\sqrt{2}-1)}{(\sqrt{2}+1)(\sqrt{2}-1)}=\sqrt{2}-1$; $\frac{1}{\sqrt{3}+\sqrt{2}}=\frac{1\times(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}=\sqrt{3}-\sqrt{2}$. Based on the above information, answer the following questions:
$(1)$ $\sqrt{2023}-\sqrt{2022}$ and ______ are rationalizing factors of each other;
$(2)$ Please guess $\frac{1}{\sqrt{n+1}+\sqrt{n}}=\_\_\_\_\_\_$; ($n$ is a positive integer)
$(3)$ $\sqrt{2023}-\sqrt{2022}$ ______ $\sqrt{2022}-\sqrt{2021}$ (fill in "$>$", "$<$", or "$=$");
$(4)$ Calculate: $(\frac{1}{\sqrt{3}+1}+\frac{1}{\sqrt{5}+\sqrt{3}}+\frac{1}{\sqrt{7}+\sqrt{5}}+\ldots +\frac{1}{\sqrt{2023}+\sqrt{2021}})\times (\sqrt{2023}+1)$. | 1011 | 7/8 |
Given a triangular pyramid \(ABCD\). In this pyramid, \(R\) is the radius of the circumscribed sphere, \(r\) is the radius of the inscribed sphere, \(a\) is the length of the largest edge, \(h\) is the length of the shortest height (to some face). Prove that \(\frac{R}{r} > \frac{a}{h}\). | \frac{R}{r}>\frac{}{} | 2/8 |
My three-digit code is 023. Reckha can't choose a code that is the same as mine in two or more of the three digit-positions, nor that is the same as mine except for switching the positions of two digits (so 320 and 203, for example, are forbidden, but 302 is fine). Reckha can otherwise choose any three-digit code where each digit is in the set $\{0, 1, 2, ..., 9\}$. How many codes are available for Reckha? | 969 | 7/8 |
Given the ellipse $C:\dfrac{x^2}{m^2}+y^2=1$ (where $m > 1$ is a constant), $P$ is a moving point on curve $C$, and $M$ is the right vertex of curve $C$. The fixed point $A$ has coordinates $(2,0)$.
$(1)$ If $M$ coincides with $A$, find the coordinates of the foci of curve $C$;
$(2)$ If $m=3$, find the maximum and minimum values of $|PA|$. | \dfrac{\sqrt{2}}{2} | 6/8 |
The numbers \(1, 2, 3, \ldots, 29, 30\) are written in a row in a random order, and their partial sums are calculated: the first sum \(S_1\) is equal to the first number, the second sum \(S_2\) is equal to the sum of the first and second numbers, \(S_3\) is equal to the sum of the first, second, and third numbers, and so on. The last sum \(S_{30}\) is equal to the sum of all the numbers. What is the highest possible count of odd numbers among the sums \(S_{1}, S_{2}, \ldots, S_{30}\)? | 23 | 1/8 |
Use the Horner's method to calculate the value of the polynomial $f(x) = 10 + 25x - 8x^2 + x^4 + 6x^5 + 2x^6$ at $x = -4$, then determine the value of $v_3$. | -36 | 7/8 |
In what ratio does the line $TH$ divide the side $BC$? | 1:1 | 2/8 |
The diagonal of a right trapezoid and its lateral side are equal. Find the length of the midline if the height of the trapezoid is 2 cm, and the lateral side is 4 cm. | 3\sqrt{3}\, | 1/8 |
Monsieur and Madame Dubois are traveling from Paris to Deauville, where their children live. Each is driving their own car. They depart together and arrive in Deauville at the same time. However, Monsieur Dubois spent on stops one-third of the time during which his wife continued driving, while Madame Dubois spent on stops one-quarter of the time during which her husband was driving.
What is the ratio of the average speeds of each of their cars? | 8/9 | 2/8 |
For any positive integer $a$ , define $M(a)$ to be the number of positive integers $b$ for which $a+b$ divides $ab$ . Find all integer(s) $a$ with $1\le a\le 2013$ such that $M(a)$ attains the largest possible value in the range of $a$ . | 1680 | 1/8 |
Find the volume of the tetrahedron with vertices \((5, 8, 10)\), \((10, 10, 17)\), \((4, 45, 46)\), \((2, 5, 4)\). | 0 | 6/8 |
Given the function \( f(x) = \frac{\ln x}{x} \):
1. Let \( k \) be a real number such that \( f(x) < kx \) always holds. Determine the range of \( k \).
2. Let \( g(x) = f(x) - kx \) (where \( k \) is a real number). If the function \( g(x) \) has two zeros in the interval \(\left[\frac{1}{e}, e^{2}\right] \), determine the range of \( k \). | [\frac{2}{e^4},\frac{1}{2e}) | 2/8 |
Given $f(x)= \frac{\ln x+2^{x}}{x^{2}}$, find $f'(1)=$ ___. | 2\ln 2 - 3 | 5/8 |
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