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There are two arithmetic sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$, with respective sums of the first $n$ terms denoted by $S_{n}$ and $T_{n}$. Given that $\dfrac{S_{n}}{T_{n}} = \dfrac{3n}{2n+1}$, find the value of $\dfrac{a_{1}+a_{2}+a_{14}+a_{19}}{b_{1}+b_{3}+b_{17}+b_{19}}$.
A) $\dfrac{27}{19}$
B) $\dfrac{18}{13}$
C) $\dfrac{10}{7}$
D) $\dfrac{17}{13}$ | \dfrac{17}{13} | 1/8 |
Assume that the probability of a certain athlete hitting the bullseye with a dart is $40\%$. Now, the probability that the athlete hits the bullseye exactly once in two dart throws is estimated using a random simulation method: first, a random integer value between $0$ and $9$ is generated by a calculator, where $1$, $2$, $3$, and $4$ represent hitting the bullseye, and $5$, $6$, $7$, $8$, $9$, $0$ represent missing the bullseye. Then, every two random numbers represent the results of two throws. A total of $20$ sets of random numbers were generated in the random simulation:<br/>
| $93$ | $28$ | $12$ | $45$ | $85$ | $69$ | $68$ | $34$ | $31$ | $25$ |
|------|------|------|------|------|------|------|------|------|------|
| $73$ | $93$ | $02$ | $75$ | $56$ | $48$ | $87$ | $30$ | $11$ | $35$ |
Based on this estimation, the probability that the athlete hits the bullseye exactly once in two dart throws is ______. | 0.5 | 7/8 |
Given a rectangle \(ABCD\). On two sides of the rectangle, different points are chosen: six points on \(AB\) and seven points on \(BC\). How many different triangles can be formed with vertices at the chosen points? | 231 | 7/8 |
For some positive integer \( n \), the number \( 150n^3 \) has \( 150 \) positive integer divisors, including \( 1 \) and the number \( 150n^3 \). How many positive integer divisors does the number \( 108n^5 \) have? | 432 | 2/8 |
Given that the last initial of Mr. and Mrs. Alpha's baby's monogram is 'A', determine the number of possible monograms in alphabetical order with no letter repeated. | 300 | 2/8 |
Add $175_{9} + 714_{9} + 61_9$. Express your answer in base $9$. | 1061_{9} | 7/8 |
What is the parameter of the parabola \( y^{2}=2px \) if the parabola touches the line \( ax + by + c^{2} = 0 \)? | \frac{2ac^2}{b^2} | 3/8 |
Given that the sum of the coefficients of the expansion of $(2x-1)^{n}$ is less than the sum of the binomial coefficients of the expansion of $(\sqrt{x}+\frac{1}{2\sqrt[4]{x}})^{2n}$ by $255$.
$(1)$ Find all the rational terms of $x$ in the expansion of $(\sqrt{x}+\frac{1}{2\sqrt[4]{x}})^{2n}$;
$(2)$ If $(2x-1)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\ldots+a_{n}x^{n}$, find the value of $(a_{0}+a_{2}+a_{4})^{2}-(a_{1}+a_{3})^{2}$. | 81 | 3/8 |
Consider a polynomial $P(x) \in \mathbb{R}[x]$ , with degree $2023$ , such that $P(\sin^2(x))+P(\cos^2(x)) =1$ for all $x \in \mathbb{R}$ . If the sum of all roots of $P$ is equal to $\dfrac{p}{q}$ with $p, q$ coprime, then what is the product $pq$ ? | 4046 | 5/8 |
A rectangular grazing area is to be fenced off on three sides using part of a $100$ meter rock wall as the fourth side. Fence posts are to be placed every $12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $36$ m by $60$ m? | 12 | 5/8 |
Given the equation of the parabola $y^{2}=-4x$, and the equation of the line $l$ as $2x+y-4=0$. There is a moving point $A$ on the parabola. The distance from point $A$ to the $y$-axis is $m$, and the distance from point $A$ to the line $l$ is $n$. Find the minimum value of $m+n$. | \frac{6 \sqrt{5}}{5}-1 | 1/8 |
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible committees that can be formed subject to these requirements.
| 88 | 7/8 |
Let $H$ be the orthocenter of the acute-angled triangle $ABC$, $O^{\prime}$ be the circumcenter of $\triangle BHC$, $N$ be the midpoint of the segment $AO^{\prime}$, and $D$ be the reflection of $N$ across the side $BC$. Prove that the points $A$, $B$, $D$, and $C$ are concyclic if and only if $b^{2}+c^{2}-a^{2}=3R^{2}$, where $a=BC, b=CA, c=AB$, and $R$ is the circumradius of $\triangle ABC$. | b^2+^2-^2=3R^2 | 1/8 |
Let $a_1, a_2, \cdots , a_n$ be real numbers, with $n\geq 3,$ such that $a_1 + a_2 +\cdots +a_n = 0$ and $$ 2a_k\leq a_{k-1} + a_{k+1} \ \ \ \text{for} \ \ \ k = 2, 3, \cdots , n-1. $$ Find the least number $\lambda(n),$ such that for all $k\in \{ 1, 2, \cdots, n\} $ it is satisfied that $|a_k|\leq \lambda (n)\cdot \max \{|a_1|, |a_n|\} .$ | \frac{n+1}{n-1} | 1/8 |
Given that $\{1, a, \frac{b}{a}\} = \{0, a^2, a+b\}$, find the value of $a^{2017} + b^{2017}$. | -1 | 6/8 |
Calculate the area of the crescent moon enclosed by the portion of the circle of radius 4 centered at (0,0) that lies in the first quadrant, the portion of the circle with radius 2 centered at (0,1) that lies in the first quadrant, and the line segment from (0,0) to (4,0). | 2\pi | 1/8 |
Given the equation of a circle $(x-1)^{2}+(y-1)^{2}=9$, point $P(2,2)$ lies inside the circle. The longest and shortest chords passing through point $P$ are $AC$ and $BD$ respectively. Determine the product $AC \cdot BD$. | 12\sqrt{7} | 7/8 |
The Fibonacci sequence is defined as follows: $F_{0}=0, F_{1}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for all integers $n \geq 2$. Find the smallest positive integer $m$ such that $F_{m} \equiv 0(\bmod 127)$ and $F_{m+1} \equiv 1(\bmod 127)$. | 256 | 2/8 |
Find all functions \( f: \mathbb{Q} \rightarrow \mathbb{R} \) such that \( f(x + y) = f(x) + f(y) + 2xy \) for all \( x, y \) in \( \mathbb{Q} \) (the rationals). | f(x)=x^2+kx | 2/8 |
In the arithmetic sequence \(\left(a_{n}\right)\) where \(a_{1}=1\) and \(d=4\),
Calculate
\[
A=\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\ldots+\frac{1}{\sqrt{a_{1579}}+\sqrt{a_{1580}}}
\]
Report the smallest integer greater than \(A\). | 20 | 7/8 |
Let $\frac{1}{1-x-x^{2}-x^{3}}=\sum_{i=0}^{\infty} a_{n} x^{n}$, for what positive integers $n$ does $a_{n-1}=n^{2}$ ? | 1, 9 | 1/8 |
Given an integer sequence \(\{a_i\}\) defined as follows:
\[ a_i = \begin{cases}
i, & \text{if } 1 \leq i \leq 5; \\
a_1 a_2 \cdots a_{i-1} - 1, & \text{if } i > 5.
\end{cases} \]
Find the value of \(\sum_{i=1}^{2019} a_i^2 - a_1 a_2 \cdots a_{2019}\). | 1949 | 6/8 |
Given that $a \in \{0,1,2\}$ and $b \in \{-1,1,3,5\}$, the probability that the function $f(x) = ax^2 - 2bx$ is an increasing function in the interval $(1, +\infty)$ is $(\quad\quad)$. | \frac{1}{3} | 1/8 |
A point \((x, y)\) is selected uniformly at random from the unit square \(S = \{ (x, y) \mid 0 \leq x \leq 1, 0 \leq y \leq 1 \}\). If the probability that \((3x + 2y, x + 4y)\) is in \(S\) is \(\frac{a}{b}\), where \(a\) and \(b\) are relatively prime positive integers, compute \(100a + b\). | 820 | 5/8 |
Let $f(x)=x^2+bx+1,$ where $b$ is a real number. Find the number of integer solutions to the inequality $f(f(x)+x)<0.$ | 2 | 1/8 |
In triangle $ABC$, a median $BM$ is drawn. It is given that $\angle ABM = 40^\circ$, and $\angle MBC = 70^\circ$. Find the ratio $AB:BM$. Justify your answer. | 2 | 3/8 |
How many positive integers less than $200$ are multiples of $5$, but not multiples of either $10$ or $6$? | 20 | 5/8 |
The function \( f: \mathbb{R} \rightarrow \mathbb{R} \) is continuous. For every real number \( x \), the equation \( f(x) \cdot f(f(x)) = 1 \) holds. It is known that \( f(1000) = 999 \). Find \( f(500) \). | \frac{1}{500} | 1/8 |
The natural numbers from 1951 to 1982 are arranged in a certain order one after another. A computer reads two consecutive numbers from left to right (i.e., the 1st and 2nd, the 2nd and 3rd, etc.) until the last two numbers. If the larger number is on the left, the computer swaps their positions. Then the computer reads in the same manner from right to left, following the same rules to change the positions of the two numbers. After reading, it is found that the number at the 100th position has not changed its position in either direction. Find this number. | 1982 | 1/8 |
A test has ten questions. Points are awarded as follows:
- Each correct answer is worth 3 points.
- Each unanswered question is worth 1 point.
- Each incorrect answer is worth 0 points.
A total score that is not possible is: | 29 | 7/8 |
How many multiples of 4 are between 70 and 300? | 57 | 5/8 |
If the function $$f(x)=(2m+3)x^{m^2-3}$$ is a power function, determine the value of $m$. | -1 | 1/8 |
Given sets $A=\{2,3,4\}$ and $B=\{a+2,a\}$, if $A \cap B = B$, find $A^cB$ ___. | \{3\} | 3/8 |
In a game, \( N \) people are in a room. Each of them simultaneously writes down an integer between 0 and 100 inclusive. A person wins the game if their number is exactly two-thirds of the average of all the numbers written down. There can be multiple winners or no winners in this game. Let \( m \) be the maximum possible number such that it is possible to win the game by writing down \( m \). Find the smallest possible value of \( N \) for which it is possible to win the game by writing down \( m \) in a room of \( N \) people. | 34 | 3/8 |
A triangle has sides \(a\), \(b\), \(c\). The radius of the inscribed circle is \(r\) and \(s = \frac{a + b + c}{2}\). Show that
\[
\frac{1}{(s - a)^2} + \frac{1}{(s - b)^2} + \frac{1}{(s - c)^2} \geq \frac{1}{r^2}.
\] | \frac{1}{(-)^2}+\frac{1}{(-b)^2}+\frac{1}{(-)^2}\ge\frac{1}{r^2} | 5/8 |
The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\ldots,r_{2007}$ . A polynomial $P$ of degree $2007$ has the property that $P\left(r_j+\dfrac{1}{r_j}\right)=0$ for $j=1,\ldots,2007$ . Determine the value of $P(1)/P(-1)$ . | \frac{289}{259} | 1/8 |
Given $\triangle ABC$ is an oblique triangle, with the lengths of the sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$, respectively. If $c\sin A= \sqrt {3}a\cos C$.
(Ⅰ) Find angle $C$;
(Ⅱ) If $c= \sqrt {21}$, and $\sin C+\sin (B-A)=5\sin 2A$, find the area of $\triangle ABC$. | \frac {5 \sqrt {3}}{4} | 6/8 |
A set of natural numbers is said to be "parfumé" (fragranced) if it contains at least two elements and each of its elements shares a prime factor with at least one of the other elements. Let \( P(n) = n^2 + n + 1 \). Determine the smallest positive integer \( b \) for which there exists a positive integer \( a \) such that the set
\[
\{P(a+1), P(a+2), \ldots, P(a+b)\}
\]
is parfumé. | 6 | 1/8 |
When one of two integers is increased by 1996 times, and the other is decreased by 96 times, their sum does not change. What can the ratio of these numbers be? | 2016 | 2/8 |
A student, Alex, is required to do a specified number of homework assignments to earn homework points using a different system: for the first four points, each point requires one homework assignment; for the next four points (points 5-8), each requires two homework assignments; then, for points 9-12, each requires three assignments, etc. Calculate the smallest number of homework assignments necessary for Alex to earn a total of 20 homework points. | 60 | 5/8 |
The diagonals of a convex quadrilateral $ABCD$ intersect at point $O$, such that triangles $BOC$ and $AOD$ are equilateral. Point $T$ is symmetric to point $O$ with respect to the midpoint of side $CD$.
a) Prove that triangle $ABT$ is equilateral.
b) Suppose additionally that $BC=2$ and $AD=4$. Find the ratio of the area of triangle $ABT$ to the area of quadrilateral $ABCD$. | \frac{7}{9} | 1/8 |
By permuting the digits of 20130518, how many different eight-digit positive odd numbers can be formed? | 3600 | 7/8 |
Let \( x, y, z \) be positive numbers that satisfy the following system of equations:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=12 \\
y^{2}+y z+z^{2}=16 \\
z^{2}+x z+x^{2}=28
\end{array}\right.
$$
Find the value of the expression \( x y + y z + x z \). | 16 | 7/8 |
The set of vectors $\mathbf{u}$ such that
\[\mathbf{u} \cdot \mathbf{u} = \mathbf{u} \cdot \begin{pmatrix} 6 \\ -28 \\ 12 \end{pmatrix}\] forms a solid in space. Find the volume of this solid. | \frac{4}{3} \pi \cdot 241^{3/2} | 1/8 |
Joe will randomly select two letters from the word CAMP, four letters from the word HERBS, and three letters from the word GLOW. What is the probability that he will have all of the letters from the word PROBLEM? Express your answer as a common fraction. | \frac{1}{30} | 7/8 |
Given an ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) where \( a > b > 0 \), with the left focus at \( F \). A tangent to the ellipse is drawn at a point \( A \) on the ellipse and intersects the \( y \)-axis at point \( Q \). Let \( O \) be the origin of the coordinate system. If \( \angle QFO = 45^\circ \) and \( \angle QFA = 30^\circ \), find the eccentricity of the ellipse. | \frac{\sqrt{6}}{3} | 3/8 |
Points A, B, C, and D lie along a line, in that order. If $AB:AC=1:5$, and $BC:CD=2:1$, what is the ratio $AB:CD$? | 1:2 | 7/8 |
Determine the least integer $k$ for which the following story could hold true:
In a chess tournament with $24$ players, every pair of players plays at least $2$ and at most $k$ games against each other. At the end of the tournament, it turns out that every player has played a different number of games. | 4 | 2/8 |
When evaluated, the sum of the digits of the integer equal to \(10^{2021} - 2021\) is: | 18185 | 6/8 |
Andrew buys 27 identical small cubes, each with two adjacent faces painted red. He then uses all of these cubes to build a large cube. What is the largest number of completely red faces that the large cube can have? | 4 | 1/8 |
In $\triangle ABC$, the ratio $AC:CB$ is $2:3$. The bisector of the exterior angle at $C$ intersects $BA$ extended at point $Q$ ($A$ is between $Q$ and $B$). Find the ratio $QA:AB$. | 2:1 | 7/8 |
Given that $(2-x)^{5}=a\_{0}+a\_{1}x+a\_{2}x^{2}+…+a\_{5}x^{5}$, find the value of $\frac{a\_0+a\_2+a\_4}{a\_1+a\_3}$. | -\frac{61}{60} | 5/8 |
How many different numbers are obtainable from five 5s by first concatenating some of the 5s, then multiplying them together? For example, we could do $5 \cdot 55 \cdot 55,555 \cdot 55$, or 55555, but not $5 \cdot 5$ or 2525. | 7 | 7/8 |
Find the least positive integer $n$ such that no matter how $10^{n}$ is expressed as the product of any two positive integers, at least one of these two integers contains the digit $0$. | 8 | 2/8 |
If Ravi shortens the length of one side of a $5 \times 7$ index card by $1$ inch, the card would have an area of $24$ square inches. What is the area of the card in square inches if instead he shortens the length of the other side by $1$ inch? | 18 | 1/8 |
Find the smallest integer $n$ such that each subset of $\{1,2,\ldots, 2004\}$ with $n$ elements has two distinct elements $a$ and $b$ for which $a^2-b^2$ is a multiple of $2004$. | 1003 | 1/8 |
Let $(a_{n})_{n\geq 1}$ be a sequence defined by $a_{n}=2^{n}+49$ . Find all values of $n$ such that $a_{n}=pg, a_{n+1}=rs$ , where $p,q,r,s$ are prime numbers with $p<q, r<s$ and $q-p=s-r$ . | 7 | 3/8 |
Given a set of data $(1)$, $(a)$, $(3)$, $(6)$, $(7)$, its average is $4$, what is its variance? | \frac{24}{5} | 4/8 |
In a new diagram showing the miles traveled by bikers Alberto, Bjorn, and Carlos over a period of 6 hours. The straight lines represent their paths on a coordinate plot where the y-axis represents miles and x-axis represents hours. Alberto's line passes through the points (0,0) and (6,90), Bjorn's line passes through (0,0) and (6,72), and Carlos’ line passes through (0,0) and (6,60). Determine how many more miles Alberto has traveled compared to Bjorn and Carlos individually after six hours. | 30 | 6/8 |
Let \( O \) be an interior point of \( \triangle ABC \). Extend \( AO \) to meet \( BC \) at \( D \). Similarly, extend \( BO \) and \( CO \) to meet \( CA \) and \( AB \) respectively at \( E \) and \( F \). Given \( AO = 30 \), \( FO = 20 \), \( BO = 60 \), \( DO = 10 \) and \( CO = 20 \), find \( EO \). | 20 | 3/8 |
Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers $a, b, c$ , respectively. They then compute the polynomial $f(x)=x^{3}+p x^{2}+q x+r$ with roots $a, b, c$ . If the expected value of the sum of the squares of the coefficients of $f(x)$ is $\frac{m}{n}$ for relatively prime positive integers $m, n$ , find the remainder when $m+n$ is divided by 1000 . | 551 | 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 $h$? | $2+2\sqrt 7$ | 6/8 |
In a caftan with an area of 1, there are 5 patches, each with an area of at least $1/2$. Prove that there exist two patches whose overlapping area is at least $1/5$. | \frac{1}{5} | 6/8 |
Let $\alpha$ be an arbitrary positive real number. Determine for this number $\alpha$ the greatest real number $C$ such that the inequality $$ \left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right) $$ is valid for all positive real numbers $x, y$ and $z$ satisfying $xy + yz + zx =\alpha.$ When does equality occur?
*(Proposed by Walther Janous)* | 16 | 3/8 |
A batch of disaster relief supplies is loaded into 26 trucks. The trucks travel at a constant speed of \( v \) kilometers per hour directly to the disaster area. If the distance between the two locations is 400 kilometers and the distance between every two trucks must be at least \( \left(\frac{v}{20}\right)^{2} \) kilometers, how many hours will it take to transport all the supplies to the disaster area? | 10 | 1/8 |
A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot?
$\textbf{(A)}\ 52 \qquad\textbf{(B)}\ 58 \qquad\textbf{(C)}\ 62 \qquad\textbf{(D)}\ 68 \qquad\textbf{(E)}\ 70$ | \textbf{(C)}\62 | 1/8 |
In an $n \times n$ grid $C$ with numbers $1, 2, \ldots, n^{2}$ written such that each cell contains one number, adjacent cells are defined as those sharing a common edge. Now, we calculate the absolute difference between the numbers in any two adjacent cells, and denote the maximum of these differences as $g$. Find the smallest possible value of $g$. | n | 6/8 |
If \( S = \sum_{k=1}^{99} \frac{(-1)^{k+1}}{\sqrt{k(k+1)}(\sqrt{k+1}-\sqrt{k})} \), find the value of \( 1000 S \). | 1100 | 6/8 |
Three fair six-sided dice are rolled. The expected value of the median of the numbers rolled can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime integers. Find $m+n$ .
*Proposed by **AOPS12142015*** | 9 | 3/8 |
Given the function $f(x)=e^{x}(x^{3}-3x+3)-ae^{x}-x$, where $e$ is the base of the natural logarithm, find the minimum value of the real number $a$ such that the inequality $f(x)\leqslant 0$ has solutions in the interval $x\in\[-2,+\infty)$. | 1-\frac{1}{e} | 6/8 |
Given unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}+\overrightarrow{b}|+2\overrightarrow{a}\cdot\overrightarrow{b}=0$, determine the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 1/8 |
While one lion cub, who was 5 minutes away from the watering hole, went to get water, the second cub, who had already quenched his thirst, headed back along the same road 1.5 times faster than the first. At the same time, a turtle on the same road started heading to the watering hole from a distance of 30 minutes away. All three met at some point and then continued their journeys. How many minutes after the meeting did the turtle reach the watering hole, given that all three traveled at constant speeds? | 28 | 6/8 |
In the rectangular coordinate system $xOy$, the coordinates of point $A\left(x_{1}, y_{1}\right)$ and point $B\left(x_{2}, y_{2}\right)$ are both positive integers. The angle between $OA$ and the positive direction of the x-axis is greater than $45^{\circ}$, and the angle between $OB$ and the positive direction of the x-axis is less than $45^{\circ}$. The projection of $B$ on the x-axis is $B^{\prime}$, and the projection of $A$ on the y-axis is $A^{\prime}$. The area of $\triangle OB^{\prime} B$ is 33.5 larger than the area of $\triangle OA^{\prime} A$. The four-digit number formed by $x_{1}, y_{1}, x_{2}, y_{2}$ is $\overline{x_{1} x_{2} y_{2} y_{1}}=x_{1} \cdot 10^{3}+x_{2} \cdot 10^{2}+y_{2} \cdot 10+y_{1}$. Find all such four-digit numbers and describe the solution process. | 1985 | 4/8 |
Find all positive integers $m$ and $n$ such that the inequality
$$
[(m+n) \alpha]+[(m+n) \beta] \geqslant[m \alpha]+[m \beta]+[n(\alpha+\beta)]
$$
holds for any real numbers $\alpha$ and $\beta$. | n | 1/8 |
What is "Romney"? If
$$
\frac{N}{O}=. \text{Romney Romney Romney} \ldots
$$
is the decimal representation of a certain proper fraction, where each letter represents some decimal digit, find the value of the word Romney (the letters $N$ and $n$ represent the same digit; the same applies to $O$ and $o$). | 571428 | 4/8 |
The number $2027$ is prime. Let $T = \sum \limits_{k=0}^{72} \binom{2024}{k}$. What is the remainder when $T$ is divided by $2027$? | 1369 | 4/8 |
We are given a combination lock consisting of $6$ rotating discs. Each disc consists of digits $0, 1, 2,\ldots , 9$ in that order (after digit $9$ comes $0$ ). Lock is opened by exactly one combination. A move consists of turning one of the discs one digit in any direction and the lock opens instantly if the current combination is correct. Discs are initially put in the position $000000$ , and we know that this combination is not correct.
[list]
a) What is the least number of moves necessary to ensure that we have found the correct combination?
b) What is the least number of moves necessary to ensure that we have found the correct combination, if we
know that none of the combinations $000000, 111111, 222222, \ldots , 999999$ is correct?[/list]
*Proposed by Ognjen Stipetić and Grgur Valentić* | 999,999 | 1/8 |
In the quadrilateral pyramid \( S A B C D \):
- The lateral faces \( S A B \), \( S B C \), \( S C D \), and \( S D A \) have areas 9, 9, 27, 27 respectively;
- The dihedral angles at the edges \( A B \), \( B C \), \( C D \), \( D A \) are equal;
- The quadrilateral \( A B C D \) is inscribed in a circle, and its area is 36.
Find the volume of the pyramid \( S A B C D \). | 54 | 4/8 |
All vertices of a regular tetrahedron \( A B C D \) are located on one side of the plane \( \alpha \). It turns out that the projections of the vertices of the tetrahedron onto the plane \( \alpha \) are the vertices of a certain square. Find the value of \(A B^{2}\), given that the distances from points \( A \) and \( B \) to the plane \( \alpha \) are 17 and 21, respectively. | 32 | 4/8 |
Prove that if the sides of a triangle form an arithmetic sequence, then
$$
3 \operatorname{tg} \frac{\alpha}{2} \operatorname{tg} \frac{\gamma}{2}=1
$$
where $\alpha$ is the smallest angle and $\gamma$ is the largest angle of the triangle. | 1 | 6/8 |
Given \( f(x) = x^2 + a x + b \cos x \), and the sets \( \{x \mid f(x) = 0, x \in \mathbb{R}\} \) and \( \{x \mid f(f(x)) = 0, x \in \mathbb{R}\} \) are equal and non-empty, determine the range of values for \(a + b\). | [0,4) | 4/8 |
Let \( G \) be the group \( \{ (m, n) : m, n \text{ are integers} \} \) with the operation \( (a, b) + (c, d) = (a + c, b + d) \). Let \( H \) be the smallest subgroup containing \( (3, 8) \), \( (4, -1) \) and \( (5, 4) \). Let \( H_{ab} \) be the smallest subgroup containing \( (0, a) \) and \( (1, b) \). Find \( a > 0 \) such that \( H_{ab} = H \). | 7 | 1/8 |
Can two parallel non-coinciding lines be considered an image of skew lines? | Yes | 7/8 |
Given that the sequence $\{a_n\}$ is a geometric sequence, and the sequence $\{b_n\}$ is an arithmetic sequence. If $a_1-a_6-a_{11}=-3\sqrt{3}$ and $b_1+b_6+b_{11}=7\pi$, then the value of $\tan \frac{b_3+b_9}{1-a_4-a_3}$ is ______. | -\sqrt{3} | 3/8 |
Find the sum of the digits of all counting numbers less than 1000. | 13500 | 5/8 |
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?
$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$ | \textbf{(B)}\3 | 1/8 |
Calculate \(\sin (\alpha-\beta)\) if \(\sin \alpha - \cos \beta = \frac{3}{4}\) and \(\cos \alpha + \sin \beta = -\frac{2}{5}\). | \frac{511}{800} | 7/8 |
Two people are flipping a coin: one flipped it 10 times, and the other 11 times.
What is the probability that the second person gets more heads than the first person? | \frac{1}{2} | 5/8 |
A frog is positioned at the origin of the coordinate plane. From the point $(x, y)$, the frog can jump to any of the points $(x + 1, y)$, $(x + 2, y)$, $(x, y + 1)$, or $(x, y + 2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0, 0)$ and ends at $(4, 4)$. | 556 | 7/8 |
Prove that if in a standard shuffled deck of cards, there are more red cards among the top 26 cards than there are black cards among the bottom 26 cards, then there are at least 3 cards of the same color in a row in this deck. | 3 | 2/8 |
In triangle $A_{1} A_{2} A_{3}$, the point $B_{i}$ is the quarter-point on the side $A_{i} A_{i+1}$ closest to $A_{i}$, and the point $C_{i}$ is the quarter-point on the side $A_{i} A_{i+1}$ closest to $A_{i+1}$ $(i=1,2,3, A_{4} \equiv A_{1})$. What fraction of the area of triangle $A_{1} A_{2} A_{3}$ is the area of the common region of triangles $B_{1} B_{2} B_{3}$ and $C_{1} C_{2} C_{3}$? | \frac{49}{160} | 1/8 |
One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that \[133^5+110^5+84^5+27^5=n^{5}.\] Find the value of $n$. | 144 | 7/8 |
Given an ellipse \(\frac{x^{2}}{a_{1}^{2}}+\frac{y^{2}}{b_{1}^{2}}=1\) \((a_{1}>b_{1}>0)\) and a hyperbola \(\frac{x^{2}}{a_{2}^{2}}-\frac{y^{2}}{b_{2}^{2}}=1\) \((a_{2}>0, b_{2}>0)\) with the same foci, let \(P\) be an intersection point of the two curves. The slopes of the tangents to the ellipse and hyperbola at point \(P\) are \(k_{1}\) and \(k_{2}\) respectively. What is the value of \(k_{1} k_{2}\)? | -1 | 7/8 |
The base of a pyramid is a right-angled triangle. The lateral edges of the pyramid are equal, and the lateral faces passing through the legs of the triangle make angles of $30^\circ$ and $60^\circ$ with the plane of the base. Find the volume of the cone circumscribed around the pyramid if the height of the pyramid is $h$. | \frac{10\pi^3}{9} | 1/8 |
Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 8.\] | \frac{89}{9} | 4/8 |
Call a lattice point visible if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer \(k\), denote by \(S_{k}\) the set of all visible lattice points \((x, y)\) such that \(x^{2}+y^{2}=k^{2}\). Let \(D\) denote the set of all positive divisors of \(2021 \cdot 2025\). Compute the sum
$$
\sum_{d \in D}\left|S_{d}\right|
$$
Here, a lattice point is a point \((x, y)\) on the plane where both \(x\) and \(y\) are integers, and \(|A|\) denotes the number of elements of the set \(A\). | 20 | 3/8 |
The incircle of triangle $ABC$ touches the sides $[AC]$ and $ [AB]$ at $E$ and $F$. Lines $(BE)$ and $(CF)$ intersect at $G$. Points $R$ and $S$ are such that $BCER$ and $BCSF$ are parallelograms. Show that $GR = GS$. | GR=GS | 1/8 |
You have a length of string and 7 beads in the 7 colors of the rainbow. You place the beads on the string as follows - you randomly pick a bead that you haven't used yet, then randomly add it to either the left end or the right end of the string. What is the probability that, at the end, the colors of the beads are the colors of the rainbow in order? (The string cannot be flipped, so the red bead must appear on the left side and the violet bead on the right side.) | \frac{1}{5040} | 1/8 |
Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$, $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$ | 676 | 6/8 |
A target consists of five zones: the center circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the score for hitting each zone is inversely proportional to the probability of hitting that zone, and hitting the bullseye is worth 315 points. How many points is hitting the blue (second-to-last) zone worth? | 45 | 7/8 |
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