problem
stringlengths 10
7.44k
| answer
stringlengths 1
270
| difficulty
stringclasses 8
values |
|---|---|---|
For positive integer $n$ we define $f(n)$ as sum of all of its positive integer divisors (including $1$ and $n$ ). Find all positive integers $c$ such that there exists strictly increasing infinite sequence of positive integers $n_1, n_2,n_3,...$ such that for all $i \in \mathbb{N}$ holds $f(n_i)-n_i=c$ | 1 | 1/8 |
The solution set of $6x^2+5x<4$ is the set of all values of $x$ such that | -\frac{4}{3}<x<\frac{1}{2} | 1/8 |
Given the broadcast time of the "Midday News" program is from 12:00 to 12:30 and the news report lasts 5 minutes, calculate the probability that Xiao Zhang can watch the entire news report if he turns on the TV at 12:20. | \frac{1}{6} | 4/8 |
Three fleas move on a grid forming a triangle $ABC$. Initially, $A$ is at $(0,0)$, $B$ is at $(1,0)$, and $C$ is at $(0,1)$. The fleas move successively in space following the rule: the flea that moves does so along the line parallel to the side opposite it, passing through its initial position and covering the desired distance.
The first flea to move is $A$, which covers a certain distance along the line parallel to $(BC)$ passing through $(0,0)$. Subsequently, it is $B$'s turn to move according to the rule.
Is there a sequence of movements that allows the fleas to occupy the positions $(1,0), (-1,0)$, and $(0,1)$? | No | 6/8 |
There are 2012 distinct points in the plane, each of which is to be coloured using one of \( n \) colours so that the number of points of each colour are distinct. A set of \( n \) points is said to be multi-coloured if their colours are distinct. Determine \( n \) that maximizes the number of multi-coloured sets. | 61 | 1/8 |
\(X\) is the unit \(n\)-cube, \([0, 1]^n\). Let
\[k_n = \int_X \cos^2 \left( \frac{\pi(x_1 + x_2 + \cdots + x_n)}{2n} \right) dx_1 \, dx_2 \, \cdots \, dx_n.\]
What is \(\lim_{n \to \infty} k_n\)? | \frac{1}{2} | 4/8 |
The lighting power increased by
\[ \Delta N = N_{\text {after}} - N_{\text {before}} = 300\, \text{BT} - 240\, \text{BT} = 60\, \text{BT} \] | 60 | 2/8 |
Given that in triangle $\triangle ABC$, the two medians $BD$ and $CE$ intersect at point $G$. Points $A$, $D$, $G$, and $E$ are concyclic, and $BC = 6$. Find the length of $AG$. | 2\sqrt{3} | 2/8 |
Let \( x, y, z \) be nonzero numbers. Prove that among the inequalities: \( x + y > 0 \), \( y + z > 0 \), \( z + x > 0 \), \( x + 2y < 0 \), \( y + 2z < 0 \), \( z + 2x < 0 \), at least two are incorrect. | 2 | 7/8 |
Find the minimum real $x$ that satisfies $$ \lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots $$ | \sqrt[3]{3} | 4/8 |
A transformation of the first [quadrant](https://artofproblemsolving.com/wiki/index.php/Quadrant) of the [coordinate plane](https://artofproblemsolving.com/wiki/index.php/Coordinate_plane) maps each point $(x,y)$ to the point $(\sqrt{x},\sqrt{y}).$ The [vertices](https://artofproblemsolving.com/wiki/index.php/Vertex) of [quadrilateral](https://artofproblemsolving.com/wiki/index.php/Quadrilateral) $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k_{}$ be the area of the region enclosed by the image of quadrilateral $ABCD.$ Find the greatest integer that does not exceed $k_{}.$ | 314 | 1/8 |
From the numbers \\(1, 2, \ldots, 100\\) totaling \\(100\\) numbers, three numbers \\(x, y, z\\) are chosen in sequence. The probability that these three numbers satisfy \\(x+z=2y\\) is __________. | \dfrac{1}{198} | 4/8 |
To celebrate 2019, Faraz gets four sandwiches shaped in the digits 2, 0, 1, and 9 at lunch. However, the four digits get reordered (but not flipped or rotated) on his plate and he notices that they form a 4-digit multiple of 7. What is the greatest possible number that could have been formed? | 1092 | 3/8 |
Given that point P is on the curve \( y = -x^{2} - 1 \) and point Q is on the curve \( x = 1 + y^{2} \), find the minimum value of \( |PQ| \). | \frac{3\sqrt{2}}{4} | 5/8 |
Angle $ABC$ of $\triangle ABC$ is a right angle. The sides of $\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\overline{AB}$ equals $8\pi$, and the arc of the semicircle on $\overline{AC}$ has length $8.5\pi$. What is the radius of the semicircle on $\overline{BC}$? | 7.5 | 1/8 |
Given that the circumradius of triangle \( \triangle ABC \) is \( R \), and
\[ 2R\left(\sin^2 A - \sin^2 C\right) = (\sqrt{2} a - b) \sin B, \]
where \( a \) and \( b \) are the sides opposite to \( \angle A \) and \( \angle B \) respectively. Determine the measure of \( \angle C \). | 45 | 7/8 |
Amanda has the list of even numbers $2, 4, 6, \dots 100$ and Billy has the list of odd numbers $1, 3, 5, \dots 99$ . Carlos creates a list by adding the square of each number in Amanda's list to the square of the corresponding number in Billy's list. Daisy creates a list by taking twice the product of corresponding numbers in Amanda's list and Billy's list. What is the positive difference between the sum of the numbers in Carlos's list and the sum of the numbers in Daisy's list?
*2016 CCA Math Bonanza Individual #3* | 50 | 7/8 |
A farmer contracted several acres of fruit trees. This year, he invested 13,800 yuan, and the total fruit yield was 18,000 kilograms. The fruit sells for a yuan per kilogram in the market and b yuan per kilogram when sold directly from the orchard (b < a). The farmer transports the fruit to the market for sale, selling an average of 1,000 kilograms per day, requiring the help of 2 people, paying each 100 yuan per day, and the transportation cost of the agricultural vehicle and other taxes and fees average 200 yuan per day.
(1) Use algebraic expressions involving a and b to represent the income from selling the fruit in both ways.
(2) If a = 4.5 yuan, b = 4 yuan, and all the fruit is sold out within the same period using both methods, calculate which method of selling is better.
(3) If the farmer strengthens orchard management, aiming for a net income of 72,000 yuan next year, and uses the better selling method from (2), what is the growth rate of the net income (Net income = Total income - Total expenses)? | 20\% | 6/8 |
Given the function $f(x)=|2x-a|+|x+ \frac {2}{a}|$
$(1)$ When $a=2$, solve the inequality $f(x)\geqslant 1$;
$(2)$ Find the minimum value of the function $g(x)=f(x)+f(-x)$. | 4 \sqrt {2} | 1/8 |
From the vertex $C$ of right triangle $ABC$, an altitude $CD$ is drawn to the hypotenuse $AB$. A circle is constructed with diameter $CD$, which intersects the leg $AC$ at point $E$ and the leg $BC$ at point $F$. Find the area of quadrilateral $CFDE$ if the leg $AC$ equals $b$ and the leg $BC$ equals $a$. | \frac{^3b^3}{(^2+b^2)^2} | 6/8 |
Given $f(x) = \sin \left( \frac{\pi}{3}x \right)$, and the set $A = \{1, 2, 3, 4, 5, 6, 7, 8\}$. Now, choose any two distinct elements $s$ and $t$ from set $A$. Find out the number of possible pairs $(s, t)$ such that $f(s)\cdot f(t) = 0$. | 13 | 1/8 |
Given positive integers \( x_1, x_2, x_3, x_4, x_5 \) satisfying
\[
x_1 + x_2 + x_3 + x_4 + x_5 = x_1 x_2 x_3 x_4 x_5,
\]
find the maximum value of \( x_5 \). | 5 | 3/8 |
The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 +...+ 10^{2013}$ If one writes down the number $S$ , how often does the digit ` $5$ ' occur in the result? | 4022 | 1/8 |
Find the number of real values of \( a \) such that for each \( a \), the cubic equation \( x^{3} = ax + a + 1 \) has an even root \( x \) with \( |x| < 1000 \). | 999 | 5/8 |
The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$. Suppose that the radius of the circle is $5$, that $BC=6$, and that $AD$ is bisected by $BC$. Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$. It follows that the sine of the central angle of minor arc $AB$ is a rational number. If this number is expressed as a fraction $\frac{m}{n}$ in lowest terms, what is the product $mn$?
[asy]size(100); defaultpen(linewidth(.8pt)+fontsize(11pt)); dotfactor=1; pair O1=(0,0); pair A=(-0.91,-0.41); pair B=(-0.99,0.13); pair C=(0.688,0.728); pair D=(-0.25,0.97); path C1=Circle(O1,1); draw(C1); label("$A$",A,W); label("$B$",B,W); label("$C$",C,NE); label("$D$",D,N); draw(A--D); draw(B--C); pair F=intersectionpoint(A--D,B--C); add(pathticks(A--F,1,0.5,0,3.5)); add(pathticks(F--D,1,0.5,0,3.5)); [/asy]
| 175 | 1/8 |
Show that \(\frac{(x + y + z)^2}{3} \geq x\sqrt{yz} + y\sqrt{zx} + z\sqrt{xy}\) for all non-negative reals \(x, y, z\). | \frac{(x+y+z)^2}{3}\gex\sqrt{yz}+y\sqrt{zx}+z\sqrt{xy} | 7/8 |
What is the value of $a^3 - b^3$ given that $a+b=12$ and $ab=20$? | 992 | 6/8 |
Mateo receives $20 every hour for one week, and Sydney receives $400 every day for one week. Calculate the difference between the total amounts of money that Mateo and Sydney receive over the one week period. | 560 | 7/8 |
For any natural number $n$ , expressed in base $10$ , let $S(n)$ denote the sum of all digits of $n$ . Find all positive integers $n$ such that $n^3 = 8S(n)^3+6S(n)n+1$ . | 17 | 3/8 |
Let $A\in \mathcal{M}_2(\mathbb{R})$ such that $\det(A)=d\neq 0$ and $\det(A+dA^*)=0$ . Prove that $\det(A-dA^*)=4$ .
*Daniel Jinga* | 4 | 5/8 |
\( \mathrm{n} \) is a positive integer not greater than 100 and not less than 10, and \( \mathrm{n} \) is a multiple of the sum of its digits. How many such \( \mathrm{n} \) are there? | 24 | 1/8 |
A function \( f(x_{1}, x_{2}, \ldots, x_{n}) \) is linear in each of the \( x_{i} \) and \( f(x_{1}, x_{2}, \ldots, x_{n}) = \frac{1}{x_{1} x_{2} \cdots x_{n}} \) when \( x_{i} \in \{3, 4\} \) for all \( i \). In terms of \( n \), what is \( f(5, 5, \ldots, 5) \)? | \frac{1}{6^{n}} | 3/8 |
It is easy to place the complete set of ships for the game "Battleship" on a $10 \times 10$ board (see illustration). What is the smallest square board on which this set can be placed? (Recall that according to the rules, ships must not touch each other, even at the corners.) | 7 \times 7 | 1/8 |
Let the equation \( x^n - x^{n-1} + a_2 x^{n-2} + \cdots + a_{n-1} x + a_n = 0 \) (with \( n \geq 2 \)) have \( n \) non-negative real roots. Prove that:
$$
0 \leq 2^2 a_2 + 2^3 a_3 + \cdots + 2^{n-1} a_{n-1} + 2^n a_n \leq \left( \frac{n-2}{n} \right)^n + 1.
$$ | 0\le2^{2}a_{2}+2^{3}a_{3}+\cdots+2^{n-1}a_{n-1}+2^{n}a_{n}\le(\frac{n-2}{n})^{n}+1 | 1/8 |
There are 100 points marked on a circle, painted either red or blue. Some points are connected by segments, with each segment having one blue end and one red end. It is known that no two red points are connected to the same number of segments. What is the maximum possible number of red points? | 50 | 3/8 |
Given the set $A=\{2,3,4,8,9,16\}$, if $a\in A$ and $b\in A$, the probability that the event "$\log_{a}b$ is not an integer but $\frac{b}{a}$ is an integer" occurs is $\_\_\_\_\_\_$. | \frac{1}{18} | 6/8 |
In the acute triangle $ABC$ the circle through $B$ touching the line $AC$ at $A$ has centre $P$ , the circle through $A$ touching the line $BC$ at $B$ has centre $Q$ . Let $R$ and $O$ be the circumradius and circumcentre of triangle $ABC$ , respectively. Show that $R^2 = OP \cdot OQ$ . | R^2=OP\cdotOQ | 4/8 |
There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball? | 4 | 1/8 |
Given the function \( f(x) = \ln (x+1) + \frac{2}{x+1} + ax - 2 \) (where \( a > 0 \)).
(1) When \( a = 1 \), find the minimum value of \( f(x) \).
(2) If \( f(x) \geq 0 \) holds for \( x \in [0, 2] \), find the range of the real number \( a \). | [1,\infty) | 2/8 |
Find $PB$ given that $AP$ is a tangent to $\Omega$, $\angle PAB=\angle PCA$, and $\frac{PB}{PA}=\frac{4}{7}=\frac{PA}{PB+6}$. | \frac{32}{11} | 7/8 |
A $4\times 4\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$?
[asy] import graph3; import solids; real h=2+2*sqrt(7); currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2)); currentlight=light(4,-4,4); draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0)); draw(shift((1,3,1))*unitsphere,gray(0.85)); draw(shift((3,3,1))*unitsphere,gray(0.85)); draw(shift((3,1,1))*unitsphere,gray(0.85)); draw(shift((1,1,1))*unitsphere,gray(0.85)); draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85)); draw(shift((1,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,1,h-1))*unitsphere,gray(0.85)); draw(shift((1,1,h-1))*unitsphere,gray(0.85)); draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h)); [/asy]
$\textbf{(A) }2+2\sqrt 7\qquad \textbf{(B) }3+2\sqrt 5\qquad \textbf{(C) }4+2\sqrt 7\qquad \textbf{(D) }4\sqrt 5\qquad \textbf{(E) }4\sqrt 7\qquad$ | \textbf{(A)}\2+2\sqrt{7} | 1/8 |
Find the minimum value of the expression
$$
\frac{|a-3b-2| + |3a-b|}{\sqrt{a^2 + (b+1)^2}}
$$
for \(a, b \geq 0\). | 2 | 1/8 |
In the tetrahedron \(A B C D\), it is known that \(DA \perp\) the base \(ABC\), the face \(ABD \perp\) the face \(BCD\), and \(BD = BC = 2\). The sum of the squares of the areas of the three faces \(DAB\), \(DBC\), and \(DCA\) is 8. Find \(\angle ADB\). | \frac{\pi}{4} | 1/8 |
Let $\sigma(n)$ be the sum of all positive divisors of a positive integer $n$. Evaluate $\left\lfloor\sqrt[3]{\sum_{n=1}^{2020} \frac{\sigma(n)}{n}}\right\rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. | 14 | 1/8 |
The military kitchen needs 1000 jin of rice and 200 jin of millet for dinner. Upon arriving at the rice store, the quartermaster finds a promotion: "Rice is 1 yuan per jin, with 1 jin of millet given for every 10 jin purchased (fractions of 10 jins do not count); Millet is 2 yuan per jin, with 2 jins of rice given for every 5 jin purchased (fractions of 5 jins do not count)." How much money does the quartermaster need to spend to buy enough rice and millet for dinner? | 1200 | 3/8 |
Given that the real numbers \( x, y \) and \( z \) satisfy the condition \( x + y + z = 3 \), find the maximum value of \( f(x, y, z) = \sqrt{2x + 13} + \sqrt[3]{3y + 5} + \sqrt[4]{8z + 12} \). | 8 | 5/8 |
Let $ n $ $(n\geq2)$ be an integer. Find the greatest possible value of the expression $$ E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2} $$ if the positive real numbers $a_1,a_2,\ldots,a_n$ satisfy $a_1+a_2+\ldots+a_n=\frac{n}{2}.$ What are the values of $a_1,a_2,\ldots,a_n$ when the greatest value is achieved? | \frac{2n}{5} | 4/8 |
How many 5-digit positive numbers contain only odd numbers and have at least one pair of consecutive digits whose sum is 10? | 1845 | 7/8 |
In an isosceles right-angled triangle AOB, points P; Q and S are chosen on sides OB, OA, and AB respectively such that a square PQRS is formed as shown. If the lengths of OP and OQ are a and b respectively, and the area of PQRS is 2 5 that of triangle AOB, determine a : b.
[asy]
pair A = (0,3);
pair B = (0,0);
pair C = (3,0);
pair D = (0,1.5);
pair E = (0.35,0);
pair F = (1.2,1.8);
pair J = (0.17,0);
pair Y = (0.17,0.75);
pair Z = (1.6,0.2);
draw(A--B);
draw(B--C);
draw(C--A);
draw(D--F--Z--E--D);
draw(" $O$ ", B, dir(180));
draw(" $B$ ", A, dir(45));
draw(" $A$ ", C, dir(45));
draw(" $Q$ ", E, dir(45));
draw(" $P$ ", D, dir(45));
draw(" $R$ ", Z, dir(45));
draw(" $S$ ", F, dir(45));
draw(" $a$ ", Y, dir(210));
draw(" $b$ ", J, dir(100));
[/asy] | 2 : 1 | 4/8 |
What is the smallest number of weights in a set that can be divided into 3, 4, and 5 piles of equal mass? | 9 | 1/8 |
In the rectangular table shown below, the number $1$ is written in the upper-left hand corner, and every number is the sum of the any numbers directly to its left and above. The table extends infinitely downwards and to the right.
\[
\begin{array}{cccccc}
1 & 1 & 1 & 1 & 1 & \cdots
1 & 2 & 3 & 4 & 5 & \cdots
1 & 3 & 6 & 10 & 15 & \cdots
1 & 4 & 10 & 20 & 35 & \cdots
1 & 5 & 15 & 35 & 70 & \cdots
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots
\end{array}
\]
Wanda the Worm, who is on a diet after a feast two years ago, wants to eat $n$ numbers (not necessarily distinct in value) from the table such that the sum of the numbers is less than one million. However, she cannot eat two numbers in the same row or column (or both). What is the largest possible value of $n$ ?
*Proposed by Evan Chen* | 19 | 1/8 |
Any type of nature use affects at least one of the natural resources, including lithogenic base, soil, water, air, plant world, and animal world. Types that affect the same set of resources belong to the same type. Research has shown that types of nature use developed in the last 700 years can be divided into 23 types. How many types remain unused? | 40 | 7/8 |
Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7. | 27 | 2/8 |
Let \( A B C \) be a triangle with \( AB = 8 \), \( AC = 12 \), and \( BC = 5 \). Let \( M \) be the second intersection of the internal angle bisector of \( \angle B A C \) with the circumcircle of \( A B C \). Let \( \omega \) be the circle centered at \( M \) tangent to \( AB \) and \( AC \). The tangents to \( \omega \) from \( B \) and \( C \), other than \( AB \) and \( AC \) respectively, intersect at a point \( D \). Compute \( AD \). | 16 | 1/8 |
Define \( P = \{\mathrm{S}, \mathrm{T}\} \) and let \( \mathcal{P} \) be the set of all proper subsets of \( P \). (A proper subset is a subset that is not the set itself.) How many ordered pairs \( (\mathcal{S}, \mathcal{T}) \) of proper subsets of \( \mathcal{P} \) are there such that
(a) \( \mathcal{S} \) is not a proper subset of \( \mathcal{T} \) and \( \mathcal{T} \) is not a proper subset of \( \mathcal{S} \); and
(b) for any sets \( S \in \mathcal{S} \) and \( T \in \mathcal{T} \), \( S \) is not a proper subset of \( T \) and \( T \) is not a proper subset of \( S \)? | 7 | 1/8 |
Among the natural numbers from 1 to 1200, 372 different numbers were chosen such that no two of them differ by 4, 5, or 9. Prove that the number 600 is one of the chosen numbers. | 600 | 1/8 |
Two circles with centers $A$ and $B$ intersect at points $X$ and $Y$ . The minor arc $\angle{XY}=120$ degrees with respect to circle $A$ , and $\angle{XY}=60$ degrees with respect to circle $B$ . If $XY=2$ , find the area shared by the two circles. | \frac{10\pi - 12\sqrt{3}}{9} | 2/8 |
Determine all real values of \(a\) for which there exist five non-negative real numbers \(x_{1}, \ldots, x_{5}\) satisfying the following equations:
\[
\sum_{k=1}^{5} k x_{k} = a
\]
\[
\sum_{k=1}^{5} k^{3} x_{k} = a^{2}
\]
\[
\sum_{k=1}^{5} k^{5} x_{k} = a^{3}
\] | 0,1,4,9,16,25 | 1/8 |
Let $C$ be a circle with two diameters intersecting at an angle of 30 degrees. A circle $S$ is tangent to both diameters and to $C$, and has radius 1. Find the largest possible radius of $C$. | 1+\sqrt{2}+\sqrt{6} | 6/8 |
Given the planar vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $\overrightarrow{a} \cdot (\overrightarrow{a} + \overrightarrow{b}) = 3$, and $|\overrightarrow{a}| = 2$, $|\overrightarrow{b}| = 1$, find the angle between the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 1/8 |
In the rhombus \(ABCD\), a height \(DE\) is dropped on side \(BC\). The diagonal \(AC\) of the rhombus intersects the height \(DE\) at point \(F\), where \(DF : FE = 5\).
Find the side length of the rhombus, given that \(AE = 5\). | \frac{25}{7} | 3/8 |
Find all integers \( n \) for which \( n^2 + 20n + 11 \) is a perfect square. | 35 | 7/8 |
Solve the system of equations \(\left\{\begin{array}{l}x^{2} y+x y^{2}-2 x-2 y+10=0, \\ x^{3} y-x y^{3}-2 x^{2}+2 y^{2}-30=0 .\end{array}\right.\) | (-4,-1) | 7/8 |
A metallic weight has a mass of 25 kg and is an alloy of four metals. The first metal in this alloy is one and a half times more than the second; the mass of the second metal is related to the mass of the third as \(3: 4\), and the mass of the third metal to the mass of the fourth as \(5: 6\). Determine the mass of the fourth metal. Give the answer in kilograms, rounding to the nearest hundredth if necessary. | 7.36 | 3/8 |
Find the number of ordered triples $(x,y,z)$ of non-negative integers satisfying
(i) $x \leq y \leq z$
(ii) $x + y + z \leq 100.$ | 30787 | 2/8 |
Find $\frac{a^{8}-6561}{81 a^{4}} \cdot \frac{3 a}{a^{2}+9}$, given that $\frac{a}{3}-\frac{3}{a}=4$. | 72 | 5/8 |
In the plane $xOy$, the lines $y = 3x - 3$ and $x = -1$ intersect at point $\mathrm{B}$. A line passing through the point $M(1, 2)$ intersects the given lines at points $\mathrm{A}$ and $\mathrm{C}$, respectively. For which positive value of the $x$-coordinate of point $\mathrm{A}$ will the area of triangle $\mathrm{ABC}$ be minimized?
(12 points) | 3 | 2/8 |
Given the ellipse $\frac{x^2}{4} + y^2 = 1$ with points A and B symmetric about the line $4x - 2y - 3 = 0$, find the magnitude of the vector sum of $\overrightarrow{OA}$ and $\overrightarrow{OB}$. | \sqrt {5} | 5/8 |
In a convex polygon, all its diagonals are drawn. These diagonals divide the polygon into several smaller polygons. What is the maximum number of sides that a polygon in the subdivision can have if the original polygon has:
a) 13 sides;
b) 1950 sides? | 1950 | 7/8 |
In each square of an $11\times 11$ board, we are to write one of the numbers $-1$ , $0$ , or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer. | 11 | 3/8 |
The polynomial \( f(x) \) satisfies the equation \( f(x) - f(x-2) = (2x-1)^{2} \) for all \( x \). Find the sum of the coefficients of \( x^{2} \) and \( x \) in \( f(x) \). | \frac{5}{6} | 5/8 |
Let $M$ be the maximum possible value of $x_1x_2+x_2x_3+\cdots +x_5x_1$ where $x_1, x_2, \dots, x_5$ is a permutation of $(1,2,3,4,5)$ and let $N$ be the number of permutations for which this maximum is attained. Evaluate $M+N$. | 58 | 1/8 |
In the cyclic quadrilateral \(ABCD\), there is a point \(X\) on side \(AB\) such that the diagonal \(BD\) bisects \(CX\) and \(AC\) bisects \(DX\). Find the minimum value of \(\frac{AB}{CD}\). | 2 | 1/8 |
Given the function \( f(x) = \log_a \frac{x-3}{x+3} \) with the domain \( \alpha \leq x < \beta \) and range \( \log_a[a(\beta-1)] < f(x) \leq \log_a[a(\alpha-1)] \) (where \( a \) is a positive number less than 1).
1. Prove that \( \alpha > 3 \).
2. Discuss the monotonicity of \( f(x) \).
3. Find the range of values for the positive number \( a \). | 0<\frac{2-\sqrt{3}}{4} | 1/8 |
Let \( a_{0} > 0 \) be a real number, and let
\[
a_{n}=\frac{a_{n-1}}{\sqrt{1+2020 \cdot a_{n-1}^{2}}}, \quad \text{for} \quad n=1,2,\ldots, 2020
\]
Show that \( a_{2020} < \frac{1}{2020} \). | a_{2020}<\frac{1}{2020} | 3/8 |
Given a trapezoid \(ABCD\) with bases \(AD = 3\) and \(BC = 18\). Point \(M\) is located on the diagonal \(AC\), with the ratio \(AM : MC = 1 : 2\). A line passing through point \(M\) parallel to the bases of the trapezoid intersects the diagonal \(BD\) at point \(N\). Find the length of \(MN\). | 4 | 5/8 |
Car A and Car B are traveling from point A to point B. Car A departs 6 hours later than Car B. The speed ratio of Car A to Car B is 4:3. 6 hours after Car A departs, its speed doubles, and both cars arrive at point B simultaneously. How many hours in total did Car A take to travel from A to B? | 8.4 | 1/8 |
Parallelogram $ABCD$ has vertices $A(3,4)$, $B(-2,1)$, $C(-5,-2)$, and $D(0,1)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is left of the $y$-axis? Express your answer as a common fraction. | \frac{1}{2} | 1/8 |
Consider the line \( l: y = kx + m \) (where \( k \) and \( m \) are integers) which intersects the ellipse \( \frac{x^2}{16} + \frac{y^2}{12} = 1 \) at two distinct points \( A \) and \( B \), and the hyperbola \( \frac{x^2}{4} - \frac{y^2}{12} = 1 \) at two distinct points \( C \) and \( D \). Determine whether there exists a line \( l \) such that the vector \( \overrightarrow{AC} + \overrightarrow{BD} = 0 \). If such a line exists, how many such lines are there? If not, provide an explanation. | 9 | 2/8 |
Let $A B C$ be a triangle with $A B=A C=5$ and $B C=6$. Denote by $\omega$ the circumcircle of $A B C$. We draw a circle $\Omega$ which is externally tangent to $\omega$ as well as to the lines $A B$ and $A C$ (such a circle is called an $A$-mixtilinear excircle). Find the radius of $\Omega$. | \frac{75}{8} | 1/8 |
Determine the value of
\[2002 + \frac{1}{2} \left( 2001 + \frac{1}{2} \left( 2000 + \dots + \frac{1}{2} \left( 3 + \frac{1}{2} \cdot 2 \right) \right) \dotsb \right).\] | 4002 | 6/8 |
Find the smallest positive integer \( n \) such that \( n(n+1)(n+2) \) is divisible by 247. | 37 | 7/8 |
How many nonzero complex numbers $z$ have the property that $0, z,$ and $z^3,$ when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle? | 4 | 3/8 |
Daniel worked for 50 hours per week for 10 weeks during the summer, earning \$6000. If he wishes to earn an additional \$8000 during the school year which lasts for 40 weeks, how many fewer hours per week must he work compared to the summer if he receives the same hourly wage? | 33.33 | 1/8 |
Let the three sides of a triangle be $\ell, m, n$ , respectively, satisfying $\ell>m>n$ and $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$ , where $\{x\}=x-\lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$ . Find the minimum perimeter of such a triangle. | 3003 | 6/8 |
Given that a square $S_1$ has an area of $25$, the area of the square $S_3$ constructed by bisecting the sides of $S_2$ is formed by the points of bisection of $S_2$. | 6.25 | 1/8 |
We call the pair $(m, n)$ of positive integers a happy pair if the greatest common divisor of $m$ and $n$ is a perfect square. For example, $(20, 24)$ is a happy pair because the greatest common divisor of 20 and 24 is 4. Suppose that $k$ is a positive integer such that $(205800, 35k)$ is a happy pair. What is the number of possible values of $k$ with $k \leq 2940$? | 30 | 1/8 |
Set \( S \) satisfies the following conditions:
1. The elements of \( S \) are positive integers not exceeding 100.
2. For any \( a, b \in S \) where \( a \neq b \), there exists \( c \in S \) different from \( a \) and \( b \) such that \(\gcd(a + b, c) = 1\).
3. For any \( a, b \in S \) where \( a \neq b \), there exists \( c \in S \) different from \( a \) and \( b \) such that \(\gcd(a + b, c) > 1\).
Determine the maximum value of \( |S| \). | 50 | 1/8 |
Exactly three faces of a \(2 \times 2 \times 2\) cube are partially shaded. Calculate the fraction of the total surface area of the cube that is shaded. | \frac{1}{4} | 5/8 |
14 people are participating in a round-robin Japanese chess tournament. Each person plays against the other 13 people with no draws. What is the maximum number of "triangular matches" (defined as a set of 3 players where each player wins one match and loses one match against the other two)? | 112 | 6/8 |
Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write
\[\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots , \]
where $r=r(p)$ ; for every $p \in S$ and every integer $k \geq 1$ define $f(k,p)$ by \[ f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)}\]
a) Prove that $S$ is infinite.
b) Find the highest value of $f(k,p)$ for $k \geq 1$ and $p \in S$ | 19 | 1/8 |
Let $\zeta=\cos \frac{2 \pi}{13}+i \sin \frac{2 \pi}{13}$. Suppose $a>b>c>d$ are positive integers satisfying $$\left|\zeta^{a}+\zeta^{b}+\zeta^{c}+\zeta^{d}\right|=\sqrt{3}$$ Compute the smallest possible value of $1000 a+100 b+10 c+d$. | 7521 | 1/8 |
Find the number of eight-digit numbers where the product of the digits is 64827. Provide the answer as an integer. | 1120 | 7/8 |
Let \( ABC \) be an isosceles triangle at \( A \) with \( \angle CAB = 20^\circ \). Let \( D \) be a point on the segment \( [AC] \) such that \( AD = BC \). Calculate the angle \( \angle BDC \). | 30 | 5/8 |
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$? | 385 | 5/8 |
Given that the two lines $ax+2y+6=0$ and $x+(a-1)y+(a^{2}-1)=0$ are parallel, determine the set of possible values for $a$. | \{-1\} | 2/8 |
Note that there are exactly three ways to write the integer $4$ as a sum of positive odd integers where the order of the summands matters:
\begin{align*}
1+1+1+1&=4,
1+3&=4,
3+1&=4.
\end{align*}
Let $f(n)$ be the number of ways to write a natural number $n$ as a sum of positive odd integers where the order of the summands matters. Find the remainder when $f(2008)$ is divided by $100$ . | 71 | 4/8 |
Let $S_{7}$ denote all the permutations of $1,2, \ldots, 7$. For any \pi \in S_{7}$, let $f(\pi)$ be the smallest positive integer $i$ such that \pi(1), \pi(2), \ldots, \pi(i)$ is a permutation of $1,2, \ldots, i$. Compute \sum_{\pi \in S_{7}} f(\pi)$. | 29093 | 2/8 |
Find the number of ordered pairs \((A, B)\) such that the following conditions hold:
- \(A\) and \(B\) are disjoint subsets of \(\{1,2, \ldots, 50\}\).
- \(|A|=|B|=25\)
- The median of \(B\) is 1 more than the median of \(A\). | (\binom{24}{12})^2 | 1/8 |
In the plane, a square with consecutively located vertices \( A, B, C, D \) and a point \( O \), lying outside the square, are given. It is known that \( AO = OB = 5 \) and \( OD = \sqrt{13} \). Find the area of the square. | 2 | 7/8 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.