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Let \(\alpha, \beta\) be the roots of \(x^{2}+bx-2=0\). If \(\alpha>1\) and \(\beta<-1\), and \(b\) is an integer, find the value of \(b\). | 0 | 7/8 |
Suppose that \( x \) and \( y \) are positive real numbers such that \( x^{2} - x y + 2 y^{2} = 8 \). Find the maximum possible value of \( x^{2} + x y + 2 y^{2} \). | \frac{72+32\sqrt{2}}{7} | 6/8 |
A polygon of area \( S \), containing the center of a circle of radius \( R \), is inscribed in the circle, and a point is chosen on each of its sides. Prove that the perimeter of the convex polygon with vertices at the chosen points is not less than \( \frac{2S}{R} \). | \frac{2S}{R} | 1/8 |
On the sides \(A B\) and \(A C\) of triangle \(A B C\), points \(E\) and \(F\) are taken respectively. Segments \(B F\) and \(C E\) intersect at point \(D\). Find the area of triangle \(A E F\) if the areas of triangles \(B C D, B D E\), and \(C D F\) are \(1, \frac{1}{3}\), and \(\frac{1}{5}\), respectively. | \frac{4}{35} | 2/8 |
Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly 50 of her first 100 shots? | \(\frac{1}{99}\) | 3/8 |
Choose some different numbers from 1, 2, 3, 4, 5, 6, 7 (order of selection does not matter), such that the sum of the even numbers equals the sum of the odd numbers. How many such selections are possible? | 7 | 4/8 |
Given the parabola \( C: y^{2} = 4x \), and the point \( A(1,2) \). Draw chords \( AP \) and \( AQ \) on the parabola passing through point \( A \).
(1) If \( AP \perp AQ \), prove that the line \( QP \) passes through a fixed point and find the coordinates of this fixed point.
(2) Assuming the line \( PQ \) passes through point \( T(5,-2) \), determine whether there exists an isosceles triangle \( APQ \) with \( PQ \) as the base. If it exists, find the number of such triangles \( \triangle QAP \); if it does not exist, provide a reason. | 1 | 2/8 |
Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$ , defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$ . And for $n \geq 1$ we have:
\[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\]
Show that this sequence has a finite limit. Determine this limit. | 3 | 2/8 |
Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$. | n = 2^k | 1/8 |
Find all increasing sequences $a_1,a_2,a_3,...$ of natural numbers such that for each $i,j\in \mathbb N$ , number of the divisors of $i+j$ and $a_i+a_j$ is equal. (an increasing sequence is a sequence that if $i\le j$ , then $a_i\le a_j$ .) | a_n=n | 1/8 |
Let positive real numbers \( a, b, c \) satisfy
$$
(a+c)\left(b^{2}+a c\right)=4a.
$$
Find the maximum value of \( b+c \). | 2 | 3/8 |
The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] draw(Circle((0,0), 4)); draw(Circle((0,0), 3)); draw((0,4)--(0,3)); draw((0,-4)--(0,-3)); draw((-2.598, 1.5)--(-3.4641, 2)); draw((-2.598, -1.5)--(-3.4641, -2)); draw((2.598, -1.5)--(3.4641, -2)); draw((2.598, 1.5)--(3.4641, 2)); [/asy] | 732 | 5/8 |
$N $ different numbers are written on blackboard and one of these numbers is equal to $0$ .One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between $-2016$ and $2016$ were written on blackboard(and some other numbers maybe). Find the smallest possible value of $N $ . | 2 | 1/8 |
Let $(b_1, b_2, ... b_{12})$ be a list of the 12 integers from 4 to 15 inclusive such that for each $2 \le i \le 12$, either $b_i + 1$ or $b_i - 1$ or both appear somewhere before $b_i$ in the list. How many such lists are there? | 2048 | 4/8 |
The route from point A to point B consists only of uphill and downhill segments, with a total distance of 21 kilometers. If the uphill speed is 4 km/h and the downhill speed is 6 km/h, and the journey from point A to point B takes 4.25 hours, how long will it take to travel from point B to point A? | 4.5\, | 1/8 |
A number is called good if it has no repeating digits and is divisible by the sum of its digits. Come up with at least two good two-digit numbers that remain good after each of their digits is increased by 1. | 70 | 1/8 |
There are exactly 120 ways to color five cells in a \( 5 \times 5 \) grid such that each row and each column contains exactly one colored cell.
There are exactly 96 ways to color five cells in a \( 5 \times 5 \) grid without the corner cell such that each row and each column contains exactly one colored cell.
How many ways are there to color five cells in a \( 5 \times 5 \) grid without two corner cells such that each row and each column contains exactly one colored cell? | 78 | 4/8 |
The side of rhombus \(ABCD\) is equal to 5. A circle with a radius of 2.4 is inscribed in this rhombus.
Find the distance between the points where this circle touches the sides \(AB\) and \(BC\), if the diagonal \(AC\) is less than the diagonal \(BD\). | 3.84 | 1/8 |
Define the sequence $f_1,f_2,\ldots :[0,1)\to \mathbb{R}$ of continuously differentiable functions by the following recurrence: $$ f_1=1; \qquad \quad f_{n+1}'=f_nf_{n+1} \quad\text{on $(0,1)$ }, \quad \text{and}\quad f_{n+1}(0)=1. $$ Show that $\lim\limits_{n\to \infty}f_n(x)$ exists for every $x\in [0,1)$ and determine the limit function. | \frac{1}{1-x} | 6/8 |
Flight number 608 of "Aeroflot" departs from Moscow at 12:00 and arrives in Bishkek at 18:00 (local time).
The return flight number 607 departs at 8:00 and arrives at 10:00. How long does the flight last? | 4 | 2/8 |
Given the function $f(x)=\cos (2x-φ)- \sqrt {3}\sin (2x-φ)(|φ| < \dfrac {π}{2})$, its graph is shifted to the right by $\dfrac {π}{12}$ units and is symmetric about the $y$-axis. Find the minimum value of $f(x)$ in the interval $\[- \dfrac {π}{2},0\]$. | - \sqrt {3} | 7/8 |
Billy and Bobby are located at points $A$ and $B$ , respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point $A$ ; the second time they meet, they are located 10 units from point $B$ . Find all possible values for the distance between $A$ and $B$ .
*Proposed by Isabella Grabski* | 15 | 1/8 |
Consider the two triangles $ ABC$ and $ PQR$ shown below. In triangle $ ABC, \angle ADB \equal{} \angle BDC \equal{} \angle CDA \equal{} 120^\circ$ . Prove that $ x\equal{}u\plus{}v\plus{}w$ .
[asy]unitsize(7mm);
defaultpen(linewidth(.7pt)+fontsize(10pt));
pair C=(0,0), B=4*dir(5);
pair A=intersectionpoints(Circle(C,5), Circle(B,6))[0];
pair Oc=scale(sqrt(3)/3)*rotate(30)*(B-A)+A;
pair Ob=scale(sqrt(3)/3)*rotate(30)*(A-C)+C;
pair D=intersectionpoints(Circle(Ob,length(Ob-C)), Circle(Oc,length(Oc-B)))[1];
real s=length(A-D)+length(B-D)+length(C-D);
pair P=(6,0), Q=P+(s,0), R=rotate(60)*(s,0)+P;
pair M=intersectionpoints(Circle(P,length(B-C)), Circle(Q,length(A-C)))[0];
draw(A--B--C--A--D--B);
draw(D--C);
label(" $B$ ",B,SE);
label(" $C$ ",C,SW);
label(" $A$ ",A,N);
label(" $D$ ",D,NE);
label(" $a$ ",midpoint(B--C),S);
label(" $b$ ",midpoint(A--C),WNW);
label(" $c$ ",midpoint(A--B),NE);
label(" $u$ ",midpoint(A--D),E);
label(" $v$ ",midpoint(B--D),N);
label(" $w$ ",midpoint(C--D),NNW);
draw(P--Q--R--P--M--Q);
draw(M--R);
label(" $P$ ",P,SW);
label(" $Q$ ",Q,SE);
label(" $R$ ",R,N);
label(" $M$ ",M,NW);
label(" $x$ ",midpoint(P--R),NW);
label(" $x$ ",midpoint(P--Q),S);
label(" $x$ ",midpoint(Q--R),NE);
label(" $c$ ",midpoint(R--M),ESE);
label(" $a$ ",midpoint(P--M),NW);
label(" $b$ ",midpoint(Q--M),NE);[/asy] | u+v+w | 4/8 |
Compute the $\textit{number}$ of ordered quadruples $(w,x,y,z)$ of complex numbers (not necessarily nonreal) such that the following system is satisfied:
\begin{align*}
wxyz &= 1
wxy^2 + wx^2z + w^2yz + xyz^2 &=2
wx^2y + w^2y^2 + w^2xz + xy^2z + x^2z^2 + ywz^2 &= -3
w^2xy + x^2yz + wy^2z + wxz^2 &= -1\end{align*} | 24 | 1/8 |
In the cyclic quadrilateral $ABCD$, let $E$ be the intersection point of the diagonals. Let $K$ and $M$ be the midpoints of sides $AB$ and $CD$, respectively. Points $L$ and $N$ are the perpendicular projections of $E$ onto sides $BC$ and $AD$. Prove that lines $KM$ and $LN$ are perpendicular to each other. | KM\perpLN | 2/8 |
A cylinder with a volume of 21 is inscribed in a cone. The plane of the upper base of this cylinder cuts off a truncated cone with a volume of 91 from the original cone. Find the volume of the original cone. | 94.5 | 1/8 |
Urn A contains 4 white balls and 2 red balls. Urn B contains 3 red balls and 3 black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ball. What is the probability that the first ball drawn was red, given that the second ball drawn was black? | 7/15 | 4/8 |
When I saw Eleonora, I found her very pretty. After a brief trivial conversation, I told her my age and asked how old she was. She answered:
- When you were as old as I am now, you were three times older than me. And when I will be three times older than I am now, together our ages will sum up to exactly a century.
How old is this capricious lady? | 15 | 4/8 |
Place two $a$s and two $b$s into a $4 \times 4$ grid, with at most one letter per cell, such that no two identical letters are in the same row or column. Determine the number of possible arrangements. | 3960 | 2/8 |
Given the function $f$ mapping from set $M$ to set $N$, where $M=\{a, b, c\}$ and $N=\{-3, -2, -1, 0, 1, 2, 3\}$, calculate the number of mappings $f$ that satisfy the condition $f(a) + f(b) + f(c) = 0$. | 37 | 6/8 |
Let $\mathbf{N}^{+}$ denote the set of positive integers, $\mathbf{R}$ denote the set of real numbers, and $S$ be the set of functions $f: \mathbf{N}^{+} \rightarrow \mathbf{R}$ satisfying the following conditions:
1. \( f(1) = 2 \)
2. \( f(n+1) \geq f(n) \geq \frac{n}{n+1} f(2n) \) for \( n = 1, 2, \cdots \)
Find the smallest positive integer \( M \) such that for any \( f \in S \) and any \( n \in \mathbf{N}^{+} \), \( f(n) < M \). | 10 | 1/8 |
A tetrahedron \(ABCD\) is divided into two parts by a plane parallel to \(AB\) and \(CD\). It is known that the ratio of the distance from \(AB\) to this plane to the distance from \(CD\) to the plane is \(k\). Find the ratio of the volumes of the two parts of the tetrahedron. | \frac{k^2(k+3)}{3k+1} | 1/8 |
Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a diamond, and the third card is a Jack? | \frac{1}{650} | 1/8 |
If the function $f(x) = x^2$ has a domain $D$ and its range is $\{0, 1, 2, 3, 4, 5\}$, how many such functions $f(x)$ exist? (Please answer with a number). | 243 | 7/8 |
Triangle $ABC$ is a right triangle with $AC = 7,$ $BC = 24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD = BD = 15.$ Given that the area of triangle $CDM$ may be expressed as $\frac {m\sqrt {n}}{p},$ where $m,$ $n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m + n + p.$
| 578 | 5/8 |
The integer points $(x, y)$ in the first quadrant satisfy $x + y > 8$ and $x \leq y \leq 8$. How many such integer points $(x, y)$ are there?
A. 16
B. 18
C. 20
D. 24 | 20 | 1/8 |
Azar and Carl play a game of tic-tac-toe. Azar places an \(X\) in one of the boxes in a \(3\)-by-\(3\) array of boxes, then Carl places an \(O\) in one of the remaining boxes. After that, Azar places an \(X\) in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third \(O\). How many ways can the board look after the game is over? | 148 | 2/8 |
A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was $84\%$ of the original price, by what percent was the price increased and decreased$?$
$\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40$ | \textbf{(E)}\40 | 1/8 |
For natural numbers \(a > b > 1\), define the sequence \(x_1, x_2, \ldots\) by the formula \(x_n = \frac{a^n - 1}{b^n - 1}\). Find the smallest \(d\) such that for any \(a\) and \(b\), this sequence does not contain \(d\) consecutive terms that are prime numbers. | 3 | 1/8 |
Seven students stand in a row for a photo, among them, students A and B must stand next to each other, and students C and D must not stand next to each other. The total number of different arrangements is. | 960 | 7/8 |
Given a positive integer \( m \geqslant 17 \), there are \( 2m \) players participating in a round-robin tournament. In each round, the \( 2m \) players are divided into \( m \) pairs, and each pair competes. In the next round, the pairings are rearranged. The tournament lasts for \( 2m-1 \) rounds, ensuring that each player competes against every other player exactly once. Determine the smallest possible positive integer \( n \) such that after some \( n \) rounds of a feasible tournament program, for any set of four players, either all of them have never competed against each other, or they have competed in at least two matches in total. | -1 | 1/8 |
New smartwatches cost 2019 rubles. Namzhil has $\left(500^{2}+4 \cdot 500+3\right) \cdot 498^{2}-500^{2} \cdot 503 \cdot 497$ rubles. Will he have enough money to buy the smartwatches? | No | 3/8 |
Prove that there exists a number \(\alpha\) such that for any triangle \(ABC\) the inequality
\[
\max \left(h_{A}, h_{B}, h_{C}\right) \leq \alpha \cdot \min \left(m_{A}, m_{B}, m_{C}\right)
\]
holds, where \(h_{A}, h_{B}, h_{C}\) denote the lengths of the altitudes and \(m_{A}, m_{B}, m_{C}\) denote the lengths of the medians. Find the smallest possible value of \(\alpha\). | 2 | 3/8 |
Let \( f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z} \) be a function with the following properties:
(i) \( f(1)=0 \),
(ii) \( f(p)=1 \) for all prime numbers \( p \),
(iii) \( f(xy)=y f(x)+x f(y) \) for all \( x, y \in \mathbb{Z}_{>0} \).
Determine the smallest integer \( n \geq 2015 \) that satisfies \( f(n)=n \).
(Gerhard J. Woeginger) | 3125 | 1/8 |
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \cos ^{8} x \, dx
$$ | \frac{35 \pi}{8} | 6/8 |
Suppose $f$ and $g$ are differentiable functions such that \[xg(f(x))f^\prime(g(x))g^\prime(x)=f(g(x))g^\prime(f(x))f^\prime(x)\] for all real $x$ . Moreover, $f$ is nonnegative and $g$ is positive. Furthermore, \[\int_0^a f(g(x))dx=1-\dfrac{e^{-2a}}{2}\] for all reals $a$ . Given that $g(f(0))=1$ , compute the value of $g(f(4))$ . | e^{-16} | 5/8 |
The numbers \(a_1, a_2, a_3, a_4,\) and \(a_5\) form a geometric progression. Among them, there are both rational and irrational numbers. What is the maximum number of terms in this progression that can be rational numbers? | 3 | 7/8 |
A sequence of numbers is written on the blackboard: \(1, 2, 3, \cdots, 50\). Each time, the first 4 numbers are erased, and the sum of these 4 erased numbers is written at the end of the sequence, creating a new sequence. This operation is repeated until there are fewer than 4 numbers remaining on the blackboard. Determine:
1. The sum of the numbers remaining on the blackboard at the end: $\qquad$
2. The last number written: $\qquad$ | 755 | 1/8 |
Susie thinks of a positive integer \( n \). She notices that, when she divides 2023 by \( n \), she is left with a remainder of 43. Find how many possible values of \( n \) there are. | 19 | 3/8 |
Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles? | \frac{\sqrt{3}}{3} | 3/8 |
There are no more than 100 workers in the workshop, a third of them are women, and 8% of the workers have a reduced workday. How many workers are there in the workshop? How many of them are women, and how many people have a reduced workday? | 6 | 4/8 |
The points of intersection of the angle bisectors of the interior angles of a parallelogram are the vertices of some quadrilateral. Prove that this quadrilateral is a rectangle. | Rectangle | 6/8 |
**(a)** Prove that every positive integer $n$ can be written uniquely in the form \[n=\sum_{j=1}^{2k+1}(-1)^{j-1}2^{m_j},\] where $k\geq 0$ and $0\le m_1<m_2\cdots <m_{2k+1}$ are integers.
This number $k$ is called *weight* of $n$ .**(b)** Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight. | 2^{1009} | 1/8 |
The price of Margit néni's favorite chocolate was increased by 30%, and at the same time her pension increased by 15%. By what percentage does Margit néni's chocolate consumption decrease if she can spend only 15% more on chocolate? | 11.54 | 7/8 |
Let \(ABC\) be a triangle with \(AB=13\), \(BC=14\), and \(CA=15\). The altitude from \(A\) intersects \(BC\) at \(D\). Let \(\omega_1\) and \(\omega_2\) be the incircles of \(ABD\) and \(ACD\), and let the common external tangent of \(\omega_1\) and \(\omega_2\) (other than \(BC\)) intersect \(AD\) at \(E\). Compute the length of \(AE\). | 7 | 6/8 |
Let $ ABC$ be an acute triangle, $ CC_1$ its bisector, $ O$ its circumcenter. The perpendicular from $ C$ to $ AB$ meets line $ OC_1$ in a point lying on the circumcircle of $ AOB$ . Determine angle $ C$ . | 60 | 1/8 |
In the Cartesian coordinate system $xOy$, a line segment of length $\sqrt{2}+1$ has its endpoints $C$ and $D$ sliding on the $x$-axis and $y$-axis, respectively. It is given that $\overrightarrow{CP} = \sqrt{2} \overrightarrow{PD}$. Let the trajectory of point $P$ be curve $E$.
(I) Find the equation of curve $E$;
(II) A line $l$ passing through point $(0,1)$ intersects curve $E$ at points $A$ and $B$, and $\overrightarrow{OM} = \overrightarrow{OA} + \overrightarrow{OB}$. When point $M$ is on curve $E$, find the area of quadrilateral $OAMB$. | \frac{\sqrt{6}}{2} | 6/8 |
In an isosceles triangle \(ABC\) with \(AB = AC\), points \(P\) and \(Q\) are taken on the sides \(AB\) and \(BC\), respectively, such that \(P\) is the midpoint of side \(AB\) and the angles \(\angle PQB\) and \(\angle AQC\) are equal. Let \(M\) be the foot of the altitude of triangle \(BPQ\) drawn from vertex \(P\). Find the ratio of the lengths of segments \(CQ\) to \(QM\). If the answer is a fractional number, it should be written as a decimal with a dot, for example, "0.15". | 8 | 6/8 |
Let \( n \geq 2 \) be a fixed integer. Find the least constant \( C \) such that the inequality
\[ \sum_{i<j} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq C\left(\sum_{i} x_{i}\right)^{4} \]
holds for every \( x_{1}, \ldots, x_{n} \geq 0 \) (the sum on the left consists of \(\binom{n}{2}\) summands). For this constant \( \bar{C} \), characterize the instances of equality. | \frac{1}{8} | 3/8 |
Point $P$ is inside a square $A B C D$ such that $\angle A P B=135^{\circ}, P C=12$, and $P D=15$. Compute the area of this square. | 123+6\sqrt{119} | 1/8 |
For which integers $n > 2$ is the following statement true? "Any convex $n$-gon has a side such that neither of the two angles at its endpoints is an acute angle." | n\ge7 | 1/8 |
Given that \(\alpha\) and \(\beta\) are two acute angles satisfying \(\sin^2 \alpha + \sin^2 \beta = \sin (\alpha + \beta)\), prove that \(\alpha + \beta = \frac{\pi}{2}\). | \alpha+\beta=\frac{\pi}{2} | 1/8 |
In a chess tournament, 30 players participated, each playing exactly one game with every other player. A win was awarded 1 point, a draw 0.5 points, and a loss 0 points. What is the maximum number of players that could have exactly 5 points at the end of the tournament? | 11 | 5/8 |
The acute angle between the 2 hands of a clock at 3:30 a.m. is \( p^{\circ} \). Find \( p \).
In \(\triangle ABC, \angle B = \angle C = p^{\circ} \). If \( q = \sin A \), find \( q \).
The 3 points \((1,3), (a, 5), (4,9)\) are collinear. Find \( a \).
The average of \( 7, 9, x, y, 17 \) is 10. If \( m \) is the average of \( x+3, x+5, y+2, 8, y+18 \), find \( m \). | 14 | 7/8 |
How many pairs of integers \((x, y)\) satisfy the equation
$$
\sqrt{x} + \sqrt{y} = \sqrt{200600}?
$$ | 11 | 3/8 |
What is the minimum number of different numbers that must be chosen from $1, 2, 3, \ldots, 1000$ to ensure that among the chosen numbers, there exist 3 different numbers that can form the side lengths of a triangle? | 16 | 7/8 |
Let $P$ be any point in the first quadrant on the ellipse $C: \frac{x^{2}}{5} + y^{2} = 1$. $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse, respectively. The lines $P F_{1}$ and $P F_{2}$ intersect the ellipse $C$ at points $M$ and $N$, respectively. Given that $\overrightarrow{P F_{1}} = \lambda_{1} \overrightarrow{F_{1} M}$ and $\overrightarrow{P F_{2}} = \lambda_{2} \overrightarrow{F_{2} N}$, find the coordinates of point $P$ such that the slope of the line segment $MN$ is $-\frac{1}{9}$. | (\frac{5\sqrt{6}}{6},\frac{\sqrt{6}}{6}) | 2/8 |
A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A$.
| 384 | 3/8 |
The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $50^{\mbox{th}}$ term of this sequence. | 327 | 7/8 |
Xinjiang region has a dry climate and is one of the three major cotton-producing areas in China, producing high-quality long-staple cotton. In an experiment on the germination rate of a certain variety of long-staple cotton seeds, research institute staff selected experimental fields with basically the same conditions, sowed seeds simultaneously, and determined the germination rate, obtaining the following data:
| Number of<br/>cotton seeds| $100$ | $200$ | $500$ | $1000$ | $2000$ | $5000$ | $10000$ |
|---|---|---|---|---|---|---|---|
| Number of<br/>germinated seeds| $98$ | $192$ | $478$ | $953$ | $1902$ | $4758$ | $9507$ |
Then the germination rate of this variety of long-staple cotton seeds is approximately ______ (rounded to $0.01$). | 0.95 | 4/8 |
Given an ellipse in the Cartesian coordinate system $xOy$, its center is at the origin, the left focus is $F(-\sqrt{3},0)$, and the right vertex is $D(2,0)$. Let point $A\left( 1,\frac{1}{2} \right)$.
(Ⅰ) Find the standard equation of the ellipse;
(Ⅱ) If a line passing through the origin $O$ intersects the ellipse at points $B$ and $C$, find the maximum value of the area of $\triangle ABC$. | \sqrt {2} | 7/8 |
Consider the graphs of $y=2\log{x}$ and $y=\log{2x}$. We may say that:
$\textbf{(A)}\ \text{They do not intersect}\qquad \\ \textbf{(B)}\ \text{They intersect at 1 point only}\qquad \\ \textbf{(C)}\ \text{They intersect at 2 points only} \qquad \\ \textbf{(D)}\ \text{They intersect at a finite number of points but greater than 2} \qquad \\ \textbf{(E)}\ \text{They coincide}$ | \textbf{(B)}\ | 1/8 |
The ratio of the area of the rectangle to the area of the decagon can be calculated given that a regular decagon $ABCDEFGHIJ$ contains a rectangle $AEFJ$. | \frac{2}{5} | 7/8 |
How many five-character license plates consist of a consonant, followed by a vowel, followed by a consonant, a digit, and then a special character from the set {$, #, @}? (For this problem, consider Y as both a consonant and a vowel.) | 79,380 | 1/8 |
Let $g(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $g(2)=0$, $90<g(9)<100$, $120<g(10)<130$, $7000k<g(150)<7000(k+1)$ for some integer $k$. What is $k$? | k=6 | 1/8 |
How many solutions in natural numbers \( x, y \) does the inequality \( \frac{x}{76} + \frac{y}{71} < 1 \) have? | 2625 | 2/8 |
In the coordinate plane, the curve $xy = 1$ intersects a circle at four points, three of which are $\left( 2, \frac{1}{2} \right),$ $\left( -5, -\frac{1}{5} \right),$ and $\left( \frac{1}{3}, 3 \right).$ Find the fourth point of intersection. | \left( -\frac{3}{10}, -\frac{10}{3} \right) | 7/8 |
If $a$ and $b$ are additive inverses, $c$ and $d$ are multiplicative inverses, and the absolute value of $m$ is 1, find $(a+b)cd-2009m=$ \_\_\_\_\_\_. | 2009 | 1/8 |
For a quadratic function \( p(x) = ax^2 + bx + c \), there exists some integer \( n \) such that \( p(n) = p\left(n^2\right) \). Provide an example of a function \( p(x) \) for which the number of such integers \( n \) is the greatest. What is this greatest number of integers \( n \)? | 4 | 7/8 |
Find all the positive integers less than 1000 such that the cube of the sum of its digits is equal to the square of such integer. | 1 \text{ and } 27 | 1/8 |
A river flows at a constant speed. Piers A and B are located upstream and downstream respectively, with a distance of 200 kilometers between them. Two boats, A and B, depart simultaneously from piers A and B, traveling towards each other. After meeting, they continue to their respective destinations, immediately return, and meet again for the second time. If the time interval between the two meetings is 4 hours, and the still water speeds of boats A and B are 36 km/h and 64 km/h respectively, what is the speed of the current in km/h? | 14 | 2/8 |
Each of the $n$ students writes one of the numbers $1,2$ or $3$ on each of the $29$ boards. If any two students wrote different numbers on at least one of the boards and any three students wrote the same number on at least one of the boards, what is the maximum possible value of $n$ ? | 3^{28} | 1/8 |
Given the sequence $\{a_n\}$ satisfies $a_1=\frac{1}{2}$, $a_{n+1}=1-\frac{1}{a_n} (n\in N^*)$, find the maximum positive integer $k$ such that $a_1+a_2+\cdots +a_k < 100$. | 199 | 1/8 |
Given the ellipse $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with its right focus at $F(\sqrt{3}, 0)$, and point $M(-\sqrt{3}, \dfrac{1}{2})$ on ellipse $C$.
(Ⅰ) Find the standard equation of ellipse $C$;
(Ⅱ) Line $l$ passes through point $F$ and intersects ellipse $C$ at points $A$ and $B$. A perpendicular line from the origin $O$ to line $l$ meets at point $P$. If the area of $\triangle OAB$ is $\dfrac{\lambda|AB| + 4}{2|OP|}$ ($\lambda$ is a real number), find the value of $\lambda$. | -1 | 6/8 |
The equation
\[(x - \sqrt[3]{13})(x - \sqrt[3]{53})(x - \sqrt[3]{103}) = \frac{1}{3}\]has three distinct solutions $r,$ $s,$ and $t.$ Calculate the value of $r^3 + s^3 + t^3.$ | 170 | 7/8 |
In a bag, there are $5$ balls of the same size, including $3$ red balls and $2$ white balls.<br/>$(1)$ If one ball is drawn with replacement each time, and this process is repeated $3$ times, with the number of times a red ball is drawn denoted as $X$, find the probability distribution and expectation of the random variable $X$;<br/>$(2)$ If one ball is drawn without replacement each time, and the color is recorded before putting it back into the bag, the process continues until two red balls are drawn, with the number of draws denoted as $Y$, find the probability of $Y=4$. | \frac{108}{625} | 1/8 |
A sequence \( b_1, b_2, b_3, \dots \) is defined recursively by \( b_1 = 2, b_2 = 2, \) and for \( k \ge 3, \)
\[
b_k = \frac{1}{2} b_{k - 1} + \frac{1}{3} b_{k - 2}.
\]
Evaluate \( b_1 + b_2 + b_3 + \dotsb. \) | 18 | 5/8 |
John learned that Lisa scored exactly 85 on the American High School Mathematics Examination (AHSME). Due to this information, John was able to determine exactly how many problems Lisa solved correctly. If Lisa's score had been any lower but still over 85, John would not have been able to determine this. What was Lisa's score? Remember, the AHSME consists of 30 multiple choice questions, and the score, $s$, is given by $s = 30 + 4c - w$, where $c$ is the number of correct answers, and $w$ is the number of wrong answers (no penalty for unanswered questions). | 85 | 1/8 |
Let \(x_1, x_2, \ldots, x_n\) be positive real numbers, and let
\[ S = x_1 + x_2 + \cdots + x_n. \]
Prove that
\[ \left(1 + x_1\right)\left(1 + x_2\right) \cdots \left(1 + x_n\right) \leq 1 + S + \frac{S^2}{2!} + \frac{S^3}{3!} + \cdots + \frac{S^n}{n!}. \] | (1+x_{1})(1+x_{2})\cdots(1+x_{n})\le1+S+\frac{S^2}{2!}+\frac{S^3}{3!}+\cdots+\frac{S^n}{n!} | 3/8 |
On a plane with $2n$ points $(n > 1, n \in \mathbf{N})$, no three points are collinear. Connect any two points with a segment and color any $n^{2}+1$ of these segments red. Prove that there are at least $n$ triangles whose three sides are all red. | n | 5/8 |
In an isosceles triangle $ABC$ with base $AB$, the bisector of angle $B$ is perpendicular to the median of side $BC$. Find the cosine of angle $C$. | \frac{7}{8} | 6/8 |
Given point \( B \) and an equilateral triangle with side lengths of \( 1 \text{ cm} \). The distances from point \( B \) to two vertices of this triangle are \( 2 \text{ cm} \) and \( 3 \text{ cm} \).
Calculate the distance from point \( B \) to the third vertex of the triangle. | \sqrt{7}\, | 1/8 |
Given a sequence $\left\{a_{n}\right\}$ with the sum of the first $n$ terms $S_{n}$ related to $a_{n}$ by the equation $S_{n}=-b a_{n}+1-\frac{1}{(1+b)^{n}}$, where $b$ is a constant independent of $n$ and $b \neq -1$:
(1) Find the expression for $a_{n}$ in terms of $b$ and $n$.
(2) When $b=3$, from which term does the sequence $\left\{a_{n}\right\}$ satisfy $S_{n}-a_{n}>\frac{1}{2}$? | 4 | 4/8 |
Mike had a bag of candies, and all candies were whole pieces that cannot be divided. Initially, Mike ate $\frac{1}{4}$ of the candies. Then, he shared $\frac{1}{3}$ of the remaining candies with his sister, Linda. Next, both Mike and his father ate 12 candies each from the remaining candies Mike had. Later, Mike’s sister took between one to four candies, leaving Mike with five candies in the end. Calculate the number of candies Mike started with initially. | 64 | 3/8 |
Triangle $ABC$ is an obtuse, isosceles triangle. Angle $A$ measures 20 degrees. What is number of degrees in the measure of the largest interior angle of triangle $ABC$?
[asy]
draw((-20,0)--(0,8)--(20,0)--cycle);
label("$20^{\circ}$",(-13,-0.7),NE);
label("$A$",(-20,0),W);
label("$B$",(0,8),N);
label("$C$",(20,0),E);
[/asy] | 140 | 5/8 |
Emily cycles at a constant rate of 15 miles per hour, and Leo runs at a constant rate of 10 miles per hour. If Emily overtakes Leo when he is 0.75 miles ahead of her, and she can view him in her mirror until he is 0.6 miles behind her, calculate the time in minutes it takes for her to see him. | 16.2 | 7/8 |
At exactly noon, a truck left a village for the city, and at the same time, a car left the city for the village. If the truck had left 45 minutes earlier, they would have met 18 kilometers closer to the city. If the car had left 20 minutes earlier, they would have met $k$ kilometers closer to the village. Find $k$. | 8 | 7/8 |
Given the functions $f(x)=x^{2}+ax+3$, $g(x)=(6+a)\cdot 2^{x-1}$.
(I) If $f(1)=f(3)$, find the value of the real number $a$;
(II) Under the condition of (I), determine the monotonicity of the function $F(x)=\frac{2}{1+g(x)}$ and provide a proof;
(III) When $x \in [-2,2]$, $f(x) \geqslant a$, ($a \notin (-4,4)$) always holds, find the minimum value of the real number $a$. | -7 | 7/8 |
Inside the triangle \(ABC\), there are points \(P\) and \(Q\) such that point \(P\) is at distances 6, 7, and 12 from lines \(AB\), \(BC\), and \(CA\) respectively, and point \(Q\) is at distances 10, 9, and 4 from lines \(AB\), \(BC\), and \(CA\) respectively. Find the radius of the inscribed circle of triangle \(ABC\). | 8 | 6/8 |
Find all integers $A$ if it is known that $A^{6}$ is an eight-digit number composed of the digits $0, 1, 2, 2, 2, 3, 4, 4$. | 18 | 4/8 |
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