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There are knights who always tell the truth and liars who always lie living on an island. One day, 30 residents of this island sat around a round table. Each of them said one of the following two phrases: "My neighbor on the left is a liar" or "My neighbor on the right is a liar." What is the minimum number of knights that can be seated at the table? | 10 | 2/8 |
If $x > 10$ , what is the greatest possible value of the expression
\[
{( \log x )}^{\log \log \log x} - {(\log \log x)}^{\log \log x} ?
\]
All the logarithms are base 10. | 0 | 2/8 |
A circle is located on a plane. What is the minimum number of lines that need to be drawn so that by symmetrically reflecting this circle with respect to these lines (in any order a finite number of times), it is possible to cover any given point on the plane? | 3 | 3/8 |
Given that square $ABCE$ has side lengths $AF = 3FE$ and $CD = 3DE$, calculate the ratio of the area of $\triangle AFD$ to the area of square $ABCE$. | \frac{3}{8} | 1/8 |
Out of 8 shots, 3 hit the target. The total number of ways in which exactly 2 hits are consecutive is: | 30 | 7/8 |
Three cones with apex $A$ and generator $\sqrt{8}$ are externally tangent to each other. For two of the cones, the angle between the generator and the axis of symmetry is $\frac{\pi}{6}$, and for the third cone, this angle is $\frac{\pi}{4}$. Find the volume of the pyramid $O_{1}O_{2}O_{3}A$, where $O_{1}, O_{2}, O_{3}$ are the centers of the bases of the cones. | \sqrt{\sqrt{3}+1} | 1/8 |
Bill draws two circles which intersect at $X,Y$ . Let $P$ be the intersection of the common tangents to the two circles and let $Q$ be a point on the line segment connecting the centers of the two circles such that lines $PX$ and $QX$ are perpendicular. Given that the radii of the two circles are $3,4$ and the distance between the centers of these two circles is $5$ , then the largest distance from $Q$ to any point on either of the circles can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $100m+n$ .
*Proposed by Tristan Shin* | 4807 | 7/8 |
In a tetrahedron \( PABC \), \(\angle APB = \angle BPC = \angle CPA = 90^\circ\). The dihedral angles formed between \(\triangle PBC, \triangle PCA, \triangle PAB\), and \(\triangle ABC\) are denoted as \(\alpha, \beta, \gamma\), respectively. Consider the following three propositions:
1. \(\cos \alpha \cdot \cos \beta + \cos \beta \cdot \cos \gamma + \cos \gamma \cdot \cos \alpha \geqslant \frac{3}{2}\)
2. \(\tan^2 \alpha + \tan^2 \beta + \tan^2 \gamma \geqslant 0\)
3. \(\sin \alpha \cdot \sin \beta \cdot \sin \gamma \leqslant \frac{1}{2}\)
Which of the following is correct? | (2) | 2/8 |
Given a sequence $\{a_n\}$ where $a_n = n$, for each positive integer $k$, in between $a_k$ and $a_{k+1}$, insert $3^{k-1}$ twos (for example, between $a_1$ and $a_2$, insert three twos, between $a_2$ and $a_3$, insert $3^1$ twos, between $a_3$ and $a_4$, insert $3^2$ twos, etc.), to form a new sequence $\{d_n\}$. Let $S_n$ denote the sum of the first $n$ terms of the sequence $\{d_n\}$. Find the value of $S_{120}$. | 245 | 4/8 |
Sector $OAB$ is a quarter of a circle with radius 5 cm. Inside this sector, a circle is inscribed, tangent at three points. Find the radius of the inscribed circle in simplest radical form. | 5\sqrt{2} - 5 | 2/8 |
Find the largest integer that divides $m^{5}-5 m^{3}+4 m$ for all $m \geq 5$. | 120 | 7/8 |
Find all real numbers $a$ for which there exists a function $f$ defined on the set of all real numbers which takes as its values all real numbers exactly once and satisfies the equality $$ f(f(x))=x^2f(x)+ax^2 $$ for all real $x$ . | 0 | 2/8 |
From the first 2005 natural numbers, \( k \) of them are arbitrarily chosen. What is the least value of \( k \) to ensure that there is at least one pair of numbers such that one of them is divisible by the other? | 1004 | 7/8 |
Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together? | 120 | 1/8 |
Determine how much money the Romanov family will save by using a multi-tariff meter over three years.
The cost of the meter is 3500 rubles. The installation cost is 1100 rubles. On average, the family's electricity consumption is 300 kWh per month, with 230 kWh used from 23:00 to 07:00.
Electricity rates with a multi-tariff meter: from 07:00 to 23:00 - 5.2 rubles per kWh, from 23:00 to 07:00 - 3.4 rubles per kWh.
Electricity rate with a standard meter: 4.6 rubles per kWh. | 3824 | 4/8 |
Each edge of a cube is divided into three congruent parts. Prove that the twenty-four division points obtained lie on a single sphere. Calculate the surface area of this sphere if the edge length of the cube is $a$. | \frac{19\pi^2}{9} | 3/8 |
If $a$, $b$, $c$, and $d$ are the solutions of the equation $x^4 - bx - 3 = 0$, then an equation whose solutions are
\[\dfrac {a + b + c}{d^2}, \dfrac {a + b + d}{c^2}, \dfrac {a + c + d}{b^2}, \dfrac {b + c + d}{a^2}\]is
$\textbf{(A)}\ 3x^4 + bx + 1 = 0\qquad \textbf{(B)}\ 3x^4 - bx + 1 = 0\qquad \textbf{(C)}\ 3x^4 + bx^3 - 1 = 0\qquad \\\textbf{(D)}\ 3x^4 - bx^3 - 1 = 0\qquad \textbf{(E)}\ \text{none of these}$ | \textbf{(D)}\3x^4-bx^3-1=0 | 1/8 |
A circle is covered by several arcs. These arcs may overlap with each other, but none of them covers the entire circle on its own. Prove that it is possible to select some of these arcs such that they also cover the entire circle and together have a total arc measure of no more than $720^{\circ}$. | 720 | 2/8 |
The faces of a parallelepiped with dimensions \( a \times b \times c \) are divided into unit squares. Additionally, there are a large number of five-cell strips that can be folded along the boundaries of the unit squares. For which values of \( a \), \( b \), and \( c \) can the three faces of the parallelepiped that share a common vertex be completely covered with these strips without overlaps or gaps, so that the unit squares of the faces align perfectly with those of the strips? | 2 | 1/8 |
Given a convex pentagon \(ABCDE\), show that the centroids of the four triangles \(ABE\), \(BCE\), \(CDE\), and \(DAE\) form a parallelogram whose area is \(\frac{2}{9}\) of the area of quadrilateral \(ABCD\). | \frac{2}{9} | 4/8 |
Given that $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $C: \frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, point $P$ is on the hyperbola $C$, $PF_{2}$ is perpendicular to the x-axis, and $\sin \angle PF_{1}F_{2} = \frac {1}{3}$, determine the eccentricity of the hyperbola $C$. | \sqrt{2} | 7/8 |
Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ are perpendicular. If $DP= 15$ and $EQ = 20$, then what is ${DF}$? | \frac{20\sqrt{13}}{3} | 7/8 |
Mario is once again on a quest to save Princess Peach. Mario enters Peach's castle and finds himself in a room with 4 doors. This room is the first in a sequence of 6 indistinguishable rooms. In each room, 1 door leads to the next room in the sequence (or, for the last room, Bowser's level), while the other 3 doors lead to the first room. Now what is the expected number of doors through which Mario will pass before he reaches Bowser's level? | 5460 | 1/8 |
Fill the nine numbers $1, 2, \cdots, 9$ into a $3 \times 3$ grid, placing one number in each cell, such that the numbers in each row increase from left to right and the numbers in each column decrease from top to bottom. How many different ways are there to achieve this arrangement? | 42 | 7/8 |
A container holds one liter of wine, and another holds one liter of water. From the first container, we pour one deciliter into the second container and mix thoroughly. Then, we pour one deciliter of the mixture back into the first container. Calculate the limit of the amount of wine in the first container if this process is repeated infinitely many times, assuming perfect mixing with each pour and no loss of liquid. | \frac{1}{2} | 2/8 |
Five students, A, B, C, D, and E, participated in a labor skills competition. A and B asked about the results. The respondent told A, "Unfortunately, neither you nor B got first place." To B, the respondent said, "You certainly are not the worst." Determine the number of different possible rankings the five students could have. | 54 | 5/8 |
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all real numbers \( x \) and \( y \),
\[ f(f(x)+2y) = 6x + f(f(y)-x). \] | f(x)=2x+ | 5/8 |
We usually write the date in the format day, month, and year (for example, 17.12.2021). In the USA, it is customary to write the month number, day number, and year sequentially (for example, 12.17.2021). How many days in a year cannot be uniquely determined by this notation? | 132 | 4/8 |
In the diagram, \( Z \) lies on \( XY \) and the three circles have diameters \( XZ \), \( ZY \), and \( XY \). If \( XZ = 12 \) and \( ZY = 8 \), then the ratio of the area of the shaded region to the area of the unshaded region is | 12:13 | 1/8 |
Define \( f(n) \) as the number of zeros in the base 3 representation of the positive integer \( n \). For which positive real \( x \) does the series
\[ F(x) = \sum_{n=1}^{\infty} \frac{x^{f(n)}}{n^3} \]
converge? | x<25 | 1/8 |
Find the smallest three-digit number with the following property: if you append a number that is 1 greater to its right, the resulting six-digit number will be a perfect square. | 183 | 4/8 |
The price of the book "Nové hádanky" was reduced by 62.5%. Matěj found out that both prices (before and after the reduction) are two-digit numbers and can be expressed with the same digits, just in different orders. By how many Kč was the book discounted? | 45\, | 1/8 |
Let the polynomial \( p(x)=\left(x-a_{1}\right) \cdot\left(x-a_{2}\right) \cdots\left(x-a_{n}\right)-1 \), where \( a_{i} \) \((i=1,2, \cdots, n)\) are \( n \) distinct integers. Prove that \( p(x) \) cannot be factored into the product of two polynomials with integer coefficients, each having a degree greater than zero.
(1984 Shanghai Mathematical Competition Problem) | p(x) | 3/8 |
Among 150 schoolchildren, only boys collect stamps. 67 people collect USSR stamps, 48 people collect African stamps, and 32 people collect American stamps. 11 people collect only USSR stamps, 7 people collect only African stamps, 4 people collect only American stamps, and only Ivanov collects stamps from the USSR, Africa, and America. Find the maximum number of girls. | 66 | 7/8 |
Given that the plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and for all $t\in \mathbb{R}$, $|\overrightarrow{b}+t\overrightarrow{a}| \geq |\overrightarrow{b}-\overrightarrow{a}|$ always holds, determine the angle between $2\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 2/8 |
For the power function $f(x) = (m^2 - m - 1)x^{m^2 + m - 3}$ to be a decreasing function on the interval $(0, +\infty)$, then $m = \boxed{\text{answer}}$. | -1 | 1/8 |
Find the number of integer points that satisfy the system of inequalities:
\[
\begin{cases}
y \leqslant 3x \\
y \geqslant \frac{1}{3}x \\
x + y \leqslant 100
\end{cases}
\] | 2551 | 1/8 |
Through a point \( O \) inside \( \triangle ABC \), draw lines parallel to the three sides: \( DE \parallel BC \), \( FG \parallel CA \), and \( HI \parallel AB \). Points \( D, E, F, G, H, I \) all lie on the sides of \( \triangle ABC \). Let \( S_1 \) represent the area of the hexagon \( DGHEDF \), and let \( S_2 \) represent the area of \( \triangle ABC \). Prove that \( S_1 \geq \frac{2}{3} S_2 \). | S_{1}\ge\frac{2}{3}S_{2} | 1/8 |
Point $C_{1}$ of hypothenuse $AC$ of a right-angled triangle $ABC$ is such that $BC = CC_{1}$ . Point $C_{2}$ on cathetus $AB$ is such that $AC_{2} = AC_{1}$ ; point $A_{2}$ is defined similarly. Find angle $AMC$ , where $M$ is the midpoint of $A_{2}C_{2}$ . | 135 | 1/8 |
Given that the equation about $x$, $x^{2}-2a\ln x-2ax=0$ has a unique solution, find the value of the real number $a$. | \frac{1}{2} | 7/8 |
Given a non-empty set of numbers, the sum of its maximum and minimum elements is called the "characteristic value" of the set. $A\_1$, $A\_2$, $A\_3$, $A\_4$, $A\_5$ each contain $20$ elements, and $A\_1∪A\_2∪A\_3∪A\_4∪A\_5={x∈N^⁎|x≤slant 100}$, find the minimum value of the sum of the "characteristic values" of $A\_1$, $A\_2$, $A\_3$, $A\_4$, $A\_5$. | 325 | 1/8 |
In quadrilateral \(ABCD\), \(BC \parallel AD\), \(BC = 26\), \(AD = 5\), \(AB = 10\), and \(CD = 17\). If the angle bisectors of \(\angle A\) and \(\angle B\) intersect at point \(M\), and the angle bisectors of \(\angle C\) and \(\angle D\) intersect at point \(N\), what is the length of \(MN\)? | 2 | 7/8 |
We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number? | 606 | 2/8 |
$J K L M$ is a square. Points $P$ and $Q$ are outside the square such that triangles $J M P$ and $M L Q$ are both equilateral. The size, in degrees, of angle $P Q M$ is | 15 | 7/8 |
The graph of \(y^2 + 2xy + 60|x| = 900\) partitions the plane into several regions. What is the area of the bounded region? | 1800 | 6/8 |
Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color. | 33 | 1/8 |
Jeffrey rolls fair three six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $m+n$ .
*Proposed by Nathan Xiong* | 101 | 6/8 |
Given \(\theta \in \left(0, \frac{\pi}{2}\right)\), find the maximum value of \(\frac{2 \sin \theta \cos \theta}{(\sin \theta + 1)(\cos \theta + 1)}\). | 6-4\sqrt{2} | 6/8 |
A cryptographer designed the following method to encode natural numbers: first, represent the natural number in base 5, then map the digits in the base 5 representation to the elements of the set $\{V, W, X, Y, Z\}$ in a one-to-one correspondence. Using this correspondence, he found that three consecutive increasing natural numbers were encoded as $V Y Z, V Y X, V V W$. What is the decimal representation of the number encoded as $X Y Z$?
(38th American High School Mathematics Examination, 1987) | 108 | 7/8 |
If natural numbers \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \) satisfy \( x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = x_{1} x_{2} x_{3} x_{4} x_{5} \),
then the maximum value of \( x_{5} \) is | 5 | 4/8 |
In an $n \times 6$ rectangular grid with $6n$ unit squares, each unit square is filled with either a 0 or a 1. A filling method is called an "N-type filling" if it ensures that there is no rectangular subgrid whose four corners have the same number. Otherwise, the filling method is called a "Y-type filling." Find the smallest positive integer $n$ such that no matter how the numbers are filled, the filling method will always be a "Y-type filling." | 5 | 2/8 |
Let $C_1$ and $C_2$ be circles defined by $$
(x-10)^2+y^2=36
$$and $$
(x+15)^2+y^2=81,
$$respectively. What is the length of the shortest line segment $\overline{PQ}$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$? | 20 | 2/8 |
Let \(a_{1}, a_{2}, \ldots, a_{n}\) be positive real numbers such that \(\sum_{i=1}^{n} a_{i}^{3}=3\) and \(\sum_{i=1}^{n} a_{i}^{5}=5\). Prove that \(\sum_{i=1}^{n} a_{i}>\frac{3}{2}\). | \sum_{i=1}^{n}a_i>\frac{3}{2} | 1/8 |
The lengths of the three sides of a triangle are \( 10 \), \( y+5 \), and \( 3y-2 \). The perimeter of the triangle is \( 50 \). What is the length of the longest side of the triangle? | 25.75 | 4/8 |
Place the numbers 1, 2, 3, 4, 5, 6, 7, and 8 on the eight vertices of a cube such that the sum of any three numbers on a face is at least 10. Find the minimum sum of the four numbers on any face. | 16 | 2/8 |
The reform pilot of basic discipline enrollment, also known as the Strong Foundation Plan, is an enrollment reform project initiated by the Ministry of Education, mainly to select and cultivate students who are willing to serve the country's major strategic needs and have excellent comprehensive quality or outstanding basic subject knowledge. The school exam of the Strong Foundation Plan is independently formulated by pilot universities. In the school exam process of a certain pilot university, students can only enter the interview stage after passing the written test. The written test score $X$ of students applying to this pilot university in 2022 approximately follows a normal distribution $N(\mu, \sigma^2)$. Here, $\mu$ is approximated by the sample mean, and $\sigma^2$ is approximated by the sample variance $s^2$. It is known that the approximate value of $\mu$ is $76.5$ and the approximate value of $s$ is $5.5$. Consider the sample to estimate the population.
$(1)$ Assuming that $84.135\%$ of students scored higher than the expected average score of the university, what is the approximate expected average score of the university?
$(2)$ If the written test score is above $76.5$ points to enter the interview, and $10$ students are randomly selected from those who applied to this pilot university, with the number of students entering the interview denoted as $\xi$, find the expected value of the random variable $\xi$.
$(3)$ Four students, named A, B, C, and D, have entered the interview, and their probabilities of passing the interview are $\frac{1}{3}$, $\frac{1}{3}$, $\frac{1}{2}$, and $\frac{1}{2}$, respectively. Let $X$ be the number of students among these four who pass the interview. Find the probability distribution and the mathematical expectation of the random variable $X$.
Reference data: If $X \sim N(\mu, \sigma^2)$, then: $P(\mu - \sigma < X \leq \mu + \sigma) \approx 0.6827$, $P(\mu - 2\sigma < X \leq \mu + 2\sigma) \approx 0.9545$, $P(\mu - 3\sigma < X \leq \mu + 3\sigma) \approx 0.9973$. | \frac{5}{3} | 4/8 |
A and B play a number-filling game on a $5 \times 5$ grid where A starts and they take turns filling empty squares. Each time A fills a square with a 1, and each time B fills a square with a 0. After the grid is completely filled, the sum of the numbers in each $3 \times 3$ square is calculated, and the highest sum among these is recorded as $A$. A aims to maximize $A$, while B aims to minimize $A$. What is the maximum value of $A$ that A can achieve? | 6 | 1/8 |
Three points $A,B,C$ are such that $B\in AC$ . On one side of $AC$ , draw the three semicircles with diameters $AB,BC,CA$ . The common interior tangent at $B$ to the first two semicircles meets the third circle $E$ . Let $U,V$ be the points of contact of the common exterior tangent to the first two semicircles.
Evaluate the ratio $R=\frac{[EUV]}{[EAC]}$ as a function of $r_{1} = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$ , where $[X]$ denotes the area of polygon $X$ . | \frac{r_1r_2}{(r_1+r_2)^2} | 1/8 |
Abe holds 1 green and 1 red jelly bean in his hand. Bob holds 1 green, 1 yellow, and 2 red jelly beans in his hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?
$\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac23$ | \textbf{(C)}\\frac{3}{8} | 1/8 |
What is the value of $\frac{(2112-2021)^2}{169}$?
$\textbf{(A) } 7 \qquad\textbf{(B) } 21 \qquad\textbf{(C) } 49 \qquad\textbf{(D) } 64 \qquad\textbf{(E) } 91$ | \textbf{(C)}49 | 1/8 |
Yannick is playing a game with 100 rounds, starting with 1 coin. During each round, there is an \( n \% \) chance that he gains an extra coin, where \( n \) is the number of coins he has at the beginning of the round. What is the expected number of coins he will have at the end of the game? | 1.01^{100} | 7/8 |
The city administration of a certain city launched a campaign that allows the exchange of four empty 1-liter bottles for one full 1-liter bottle of milk. How many liters of milk can a person obtain if they have 43 empty 1-liter bottles by making several of these exchanges? | 14 | 7/8 |
The base of the quadrilateral pyramid \( MABCD \) is the parallelogram \( ABCD \). Point \( K \) bisects edge \( DM \). Point \( P \) lies on edge \( BM \) such that \( BP: PM = 1: 3 \). The plane \( APK \) intersects edge \( MC \) at point \( X \). Find the ratio of segments \( MX \) and \( XC \). | 3:4 | 6/8 |
Find all three-digit integers \( abc = n \) such that \( \frac{2n}{3} = a! \cdot b! \cdot c! \). | 432 | 1/8 |
The value $b^n$ has both $b$ and $n$ as positive integers less than or equal to 15. What is the greatest number of positive factors $b^n$ can have? | 496 | 5/8 |
People are standing in a circle - there are liars, who always lie, and knights, who always tell the truth. Each of them said that among the people standing next to them, there is an equal number of liars and knights. How many people are there in total if there are 48 knights? | 72 | 5/8 |
Given the ellipse $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, its minor axis length is $2$, and the eccentricity is $\frac{\sqrt{6}}{3}$. The line $l$ passes through the point $(-1,0)$ and intersects the ellipse $E$ at points $A$ and $B$. $O$ is the coordinate origin.
(1) Find the equation of the ellipse $E$;
(2) Find the maximum area of $\triangle OAB$. | \frac{\sqrt{6}}{3} | 4/8 |
Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2010$, and $a^2 - b^2 + c^2 - d^2 = 2010$. Find the number of possible values of $a$. | 501 | 5/8 |
In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?
$\begin{array}{cccccc}&A&B&B&C&B\ +&B&C&A&D&A\ \hline &D&B&D&D&D\end{array}$ | 7 | 5/8 |
Let \( X = \{1,2, \cdots, 100\} \). For any non-empty subset \( M \) of \( X \), define the characteristic of \( M \), denoted as \( m(M) \), as the sum of the maximum and minimum elements of \( M \). Find the average value of the characteristics of all non-empty subsets of \( X \). | 101 | 5/8 |
Determine all positive integers $ n$ such that $f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx$ divides $g_n(x,y, z) = (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}$ , as polynomials in $x, y, z$ with integer coefficients. | 1 | 1/8 |
In the triangular prism \( ABC-A_1B_1C_1 \), the following are given: \( AC = \sqrt{2} \), \( AA_1 = A_1C = 1 \), \( AB = \frac{\sqrt{6}}{3} \), \( A_1B = \frac{\sqrt{3}}{3} \), \( BC = \frac{2\sqrt{3}}{3} \). Find the angle \(\angle B_1C_1C \). | \arccos\frac{\sqrt{3}}{6} | 1/8 |
Find the greatest common divisor of the numbers $2002+2,2002^{2}+2,2002^{3}+2, \ldots$. | 6 | 7/8 |
Let $p = 101$ and let $S$ be the set of $p$ -tuples $(a_1, a_2, \dots, a_p) \in \mathbb{Z}^p$ of integers. Let $N$ denote the number of functions $f: S \to \{0, 1, \dots, p-1\}$ such that
- $f(a + b) + f(a - b) \equiv 2\big(f(a) + f(b)\big) \pmod{p}$ for all $a, b \in S$ , and
- $f(a) = f(b)$ whenever all components of $a-b$ are divisible by $p$ .
Compute the number of positive integer divisors of $N$ . (Here addition and subtraction in $\mathbb{Z}^p$ are done component-wise.)
*Proposed by Ankan Bhattacharya* | 5152 | 5/8 |
Two positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers? | 156 | 5/8 |
Let \(\mathrm{AB}\) be the major axis of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{6}=1\). The chord \(\mathrm{PQ}\) of this ellipse passes through point \(\mathrm{C}(2,0)\) but does not pass through the origin. Line \(\mathrm{AP}\) intersects \(\mathrm{QB}\) at point \(\mathrm{M}\), and \(\mathrm{PB}\) intersects \(\mathrm{AQ}\) at point \(\mathrm{N}\). Find the equation of the line \(\mathrm{MN}\). | 8 | 1/8 |
The value of x that satisfies the equation \( x^{x^x} = 2 \) is calculated. | \sqrt{2} | 1/8 |
In the subtraction shown, \( K, L, M \), and \( N \) are digits. What is the value of \( K+L+M+N \)?
\[
\begin{array}{llll}
5 & K & 3 & L \\
\end{array}
\]
\[
\begin{array}{r}
M & 4 & N & 1 \\
\hline
4 & 4 & 5 & 1 \\
\end{array}
\] | 20 | 7/8 |
Given the ratio of length $AD$ to width $AB$ of the rectangle is $4:3$ and $AB$ is 40 inches, determine the ratio of the area of the rectangle to the combined area of the semicircles. | \frac{16}{3\pi} | 6/8 |
Given a right square prism \(A B C D-A_{1} B_{1} C_{1} D_{1}\) with a base \(A B C D\) that is a unit square, if the dihedral angle \(A_{1}-B D-C_{1}\) is \(\frac{\pi}{3}\), find the length of \(A A_{1}\). | \frac{\sqrt{6}}{2} | 7/8 |
About $5$ years ago, Joydip was researching on the number $2017$ . He understood that $2017$ is a prime number. Then he took two integers $a,b$ such that $0<a,b <2017$ and $a+b\neq 2017.$ He created two sequences $A_1,A_2,\dots ,A_{2016}$ and $B_1,B_2,\dots, B_{2016}$ where $A_k$ is the remainder upon dividing $ak$ by $2017$ , and $B_k$ is the remainder upon dividing $bk$ by $2017.$ Among the numbers $A_1+B_1,A_2+B_2,\dots A_{2016}+B_{2016}$ count of those that are greater than $2017$ is $N$ . Prove that $N=1008.$ | 1008 | 5/8 |
A blank \( n \times n \) (where \( n \geq 4 \)) grid is filled with either +1 or -1 in each cell. Define a basic term as the product of \( n \) numbers chosen from the grid such that no two numbers are from the same row or column. Prove that the sum of all basic terms in such a filled grid is always divisible by 4. | 4 | 1/8 |
If a function $f(x)$ satisfies both (1) for any $x$ in the domain, $f(x) + f(-x) = 0$ always holds; and (2) for any $x_1, x_2$ in the domain where $x_1 \neq x_2$, the inequality $\frac{f(x_1) - f(x_2)}{x_1 - x_2} < 0$ always holds, then the function $f(x)$ is called an "ideal function." Among the following three functions: (1) $f(x) = \frac{1}{x}$; (2) $f(x) = x + 1$; (3) $f(x) = \begin{cases} -x^2 & \text{if}\ x \geq 0 \\ x^2 & \text{if}\ x < 0 \end{cases}$; identify which can be called an "ideal function" by their respective sequence numbers. | (3) | 7/8 |
A rectangular board of 8 columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result? | 120 | 6/8 |
In the ellipse \( x^2 + 4y^2 = 8 \), \( AB \) is a chord of length \( \frac{5}{2} \), and \( O \) is the origin. Find the range of the area of \(\triangle AOB\). | [\frac{5\sqrt{103}}{32},2] | 1/8 |
The sum of three positive integers is 15, and the sum of their reciprocals is $\frac{71}{105}$. Determine the numbers! | 3,5,7 | 1/8 |
Given a cyclic quadrilateral \(ABCD\), it is known that \(\angle ADB = 48^{\circ}\) and \(\angle BDC = 56^{\circ}\). Inside the triangle \(ABC\), a point \(X\) is marked such that \(\angle BCX = 24^{\circ}\), and the ray \(AX\) is the angle bisector of \(\angle BAC\). Find the angle \(CBX\). | 38 | 5/8 |
Find a number that, when added to 13600, results in a perfect square. To solve this geometrically, consider a rectangle with an area of 13600 with one side equal to 136. Divide it into a square \( A \) and a rectangle \( B \). Cut the rectangle \( B \) into two equal rectangles, both denoted by \( C \). Position the rectangles \( C \) on two consecutive sides of the square \( A \).
(a) What is the area of rectangle \( C \)?
(b) Notice that if we add square \( X \), we complete a larger square. What should be the side length of square \( X \)?
(c) After answering the two previous questions, determine a number that, when added to 13600, results in a perfect square and find the square root of this perfect square. | 118 | 6/8 |
How many 11 step paths are there from point A to point D, which pass through points B and C in that order? Assume the grid layout permits only right and down steps, where B is 2 right and 2 down steps from A, and C is 1 right and 3 down steps from B, and finally, D is 3 right and 1 down step from C. | 96 | 3/8 |
In the rectangular coordinate system \( xOy \), find the area of the graph formed by all points \( (x, y) \) that satisfy \( \lfloor x \rfloor \cdot \lfloor y \rfloor = 2013 \), where \( \lfloor x \rfloor \) represents the greatest integer less than or equal to the real number \( x \). | 16 | 7/8 |
Let \( a \) and \( b \) be two positive integers whose least common multiple is 232848. How many such ordered pairs \( (a, b) \) are there? | 945 | 7/8 |
In $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a^{2}+c^{2}-b^{2}=ac$, $c=2$, and point $G$ satisfies $| \overrightarrow{BG}|= \frac { \sqrt {19}}{3}$ and $\overrightarrow{BG}= \frac {1}{3}( \overrightarrow{BA}+ \overrightarrow{BC})$, find the value of $\sin A$. | \frac {3 \sqrt {21}}{14} | 7/8 |
Let \( S = \{1, 2, \ldots, n\} \). Define \( A \) as an arithmetic sequence (with a positive difference) containing at least two terms, where all the terms are elements of \( S \). Moreover, adding any other element from \( S \) to \( A \) must not create a new arithmetic sequence with the same difference as \( A \). Find the number of such sequences \( A \). Note that a sequence with only two terms is also considered an arithmetic sequence. | \lfloor\frac{n^2}{4}\rfloor | 2/8 |
Determine the distance in feet between the 5th red light and the 23rd red light, where the lights are hung on a string 8 inches apart in the pattern of 3 red lights followed by 4 green lights. Recall that 1 foot is equal to 12 inches. | 28 | 7/8 |
For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? | 19 | 7/8 |
Given nine cards with the numbers \(5, 5, 6, 6, 6, 7, 8, 8, 9\) written on them, three three-digit numbers \(A\), \(B\), and \(C\) are formed using these cards such that each of these numbers has three different digits. What is the smallest possible value of the expression \(A + B - C\)? | 149 | 1/8 |
Let $g(x) = 2x^7 - 3x^3 + 4x - 8.$ If $g(6) = 12,$ find $g(-6).$ | -28 | 4/8 |
There are $N{}$ points marked on the plane. Any three of them form a triangle, the values of the angles of which in are expressed in natural numbers (in degrees). What is the maximum $N{}$ for which this is possible?
*Proposed by E. Bakaev* | 180 | 7/8 |
Find the smallest positive integer $m$ such that for all positive integers $n \geq m$, there exists a positive integer $l$ satisfying
$$
n < l^2 < \left(1+\frac{1}{2009}\right)n.
$$ | 16144325 | 1/8 |
A shuffle of a deck of $n$ cards is defined as the following operation: the deck is divided into some (arbitrary) number of parts, which are then rearranged in reverse order without changing the positions of the cards within each part. Prove that a deck of 1000 cards can be transformed from any arrangement into any other arrangement in no more than 56 shuffles. | 56 | 1/8 |
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