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Garfield and Odie are situated at $(0,0)$ and $(25,0)$ , respectively. Suddenly, Garfield and Odie dash in the direction of the point $(9, 12)$ at speeds of $7$ and $10$ units per minute, respectively. During this chase, the minimum distance between Garfield and Odie can be written as $\frac{m}{\sqrt{n}}$ for relatively prime positive integers $m$ and $n$ . Find $m+n$ .
*Proposed by **Th3Numb3rThr33*** | 159 | 6/8 |
In $\triangle ABC$ in the adjoining figure, $AD$ and $AE$ trisect $\angle BAC$. The lengths of $BD$, $DE$ and $EC$ are $2$, $3$, and $6$, respectively. The length of the shortest side of $\triangle ABC$ is | 2\sqrt{10} | 3/8 |
There are 288 externally identical coins weighing 7 and 8 grams (both types are present). 144 coins are placed on each pan of a balance scale such that the scale is in equilibrium. In one operation, you can take any two groups of the same number of coins from the pans and swap them. Prove that in no more than 11 operations it is possible to make the scale unbalanced. | 11 | 1/8 |
The 79 trainees of the Animath workshop each choose an activity for the free afternoon among 5 offered activities. It is known that:
- The swimming pool was at least as popular as soccer.
- The students went shopping in groups of 5.
- No more than 4 students played cards.
- At most one student stayed in their room.
We write down the number of students who participated in each activity. How many different lists could we have written? | 3240 | 4/8 |
Convert the following expression into the product of polynomials \( a \), \( b \), and \( c \):
$$
a(b-c)^{3}+b(c-a)^{3}+c(a-b)^{3}
$$ | (b)()()() | 4/8 |
In the 8th grade class "Я," there are quite a few failing students, but Vovochka is the worst. The pedagogical council decided that either Vovochka must improve his grades by the end of the term, or he will be expelled. If Vovochka improves his grades, then the class will have $24\%$ failing students, but if he is expelled, the percentage of failing students will become $25\%$. What is the current percentage of failing students in the 8th grade class "Я"? | 28 | 5/8 |
Given real numbers $x$ and $y$ , such that $$ x^4 y^2 + y^4 + 2 x^3 y + 6 x^2 y + x^2 + 8 \le 0 . $$ Prove that $x \ge - \frac16$ | x\ge-\frac{1}{6} | 1/8 |
Consider the set $S$ of permutations of $1, 2, \dots, 2022$ such that for all numbers $k$ in the
permutation, the number of numbers less than $k$ that follow $k$ is even.
For example, for $n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}$ If $|S| = (a!)^b$ where $a, b \in \mathbb{N}$ , then find the product $ab$ . | 2022 | 4/8 |
A trapezoid is inscribed in a circle. The larger base of the trapezoid forms an angle $\alpha$ with a lateral side and an angle $\beta$ with the diagonal. Find the ratio of the area of the circle to the area of the trapezoid. | \frac{\pi}{2\sin^2\alpha\sin2\beta} | 3/8 |
Alice needs to replace a light bulb located $10$ centimeters below the ceiling in her kitchen. The ceiling is $2.4$ meters above the floor. Alice is $1.5$ meters tall and can reach $46$ centimeters above the top of her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters?
$\textbf{(A)}\ 32 \qquad\textbf{(B)}\ 34\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 38\qquad\textbf{(E)}\ 40$ | \textbf{(B)}\34 | 1/8 |
Two football teams, A and B, play two friendly matches. For a win, a team earns 2 points. For a draw, a team earns 1 point. For a loss, a team earns 0 points. The probability of a draw in each match is the same and is equal to $p$.
The following year, a similar set of two friendlies took place. The teams played with the same lineup and were still of equal strength, but the probability $p$ of a draw increased. Can it be asserted that the probability of the teams acquiring an equal number of points has increased? | No | 2/8 |
David and Evan each repeatedly flip a fair coin. David will stop when he flips a tail, and Evan will stop once he flips 2 consecutive tails. Find the probability that David flips more total heads than Evan. | \frac{1}{5} | 1/8 |
Several stones are arranged in 5 piles. It is known that:
- The number of stones in the fifth pile is six times the number of stones in the third pile.
- The number of stones in the second pile is twice the total number of stones in the third and fifth piles combined.
- The number of stones in the first pile is three times less than the number in the fifth pile and 10 less than the number in the fourth pile.
- The number of stones in the fourth pile is half the number in the second pile.
How many stones are there in total in these five piles? | 60 | 7/8 |
Find the largest number $n$ such that $(2004!)!$ is divisible by $((n!)!)!$. | 6 | 5/8 |
Given circle $O$: $x^{2}+y^{2}=10$, a line $l$ passing through point $P(-3,-4)$ intersects with circle $O$ at points $A$ and $B$. If the area of triangle $AOB$ is $5$, find the slope of line $l$. | \frac{11}{2} | 7/8 |
Given that in triangle \( \triangle ABC \), \( a = 2b \), \( \cos B = \frac{2 \sqrt{2}}{3} \), find the value of \( \sin \frac{A-B}{2} + \sin \frac{C}{2} \). | \frac{\sqrt{10}}{3} | 4/8 |
Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$ . (Note: $n$ is written in the usual base ten notation.) | 9999 | 1/8 |
The number 123456789 is written on the board. Two adjacent digits are selected from the number, if neither of them is 0, 1 is subtracted from each digit, and the selected digits are swapped (for example, from 123456789, one operation can result in 123436789). What is the smallest number that can be obtained as a result of these operations? | 101010101 | 1/8 |
Let \(a_{1}, a_{2}, a_{3}, \ldots \) be the sequence of all positive integers that are relatively prime to 75, where \(a_{1}<a_{2}<a_{3}<\cdots\). (The first five terms of the sequence are: \(a_{1}=1, a_{2}=2, a_{3}=4, a_{4}=7, a_{5}=8\).) Find the value of \(a_{2008}\). | 3764 | 7/8 |
The distance between points A and B is 1200 meters. Dachen starts from point A, and 6 minutes later, Xiaogong starts from point B. After another 12 minutes, they meet. Dachen walks 20 meters more per minute than Xiaogong. How many meters does Xiaogong walk per minute? | 28 | 7/8 |
The founder of a noble family received a plot of land. Each man in the family, upon his death, divided the land he had equally among his sons. If he had no sons, the land was transferred to the state. No other family members acquired or lost land by any other means. In total, there were 150 people in the family. What is the smallest possible fraction of the original plot that could be received by a family member? | \frac{1}{2\cdot3^{49}} | 1/8 |
The mass of a vessel completely filled with kerosene is 31 kg. If the vessel is completely filled with water, its mass will be 33 kg. Determine the mass of the empty vessel. The density of water $\rho_{B}=1000 \, \mathrm{kg} / \mathrm{m}^{3}$, and the density of kerosene $\rho_{K}=800 \, \mathrm{kg} / \mathrm{m}^{3}$. | 23\, | 1/8 |
In a chess tournament, 12 people participated. After the tournament concluded, each participant created 12 lists. The first list includes only themselves, the second one includes themselves and those they defeated, the third includes everyone from the second list and those they defeated, and so on. The 12th list includes everyone from the 11th list and those they defeated. It is known that for any participant in the tournament, there is a person in their 12th list who was not in their 11th list. How many drawn games were played in the tournament? | 54 | 5/8 |
A trapezoid with side lengths \( a \) and \( b \) is circumscribed around a circle. Find the sum of the squares of the distances from the center of the circle to the vertices of the trapezoid. | ^2+b^2 | 1/8 |
Let $l$ be a line passing the origin on the coordinate plane and has a positive slope. Consider circles $C_1,\ C_2$ determined by the condition (i), (ii), (iii) as below.
(i) The circles $C_1,\ C_2$ are contained in the domain determined by the inequality $x\geq 0,\ y\geq 0.$ (ii) The circles $C_1,\ C_2$ touch the line $l$ at same point.
(iii) The circle $C_1$ touches the $x$ -axis at the point $(1,\ 0)$ and the circle $C_2$ touches the $y$ -axis.
Let $r_1,\ r_2$ be the radii of the circles $C_1,\ C_2$ , respectively. Find the equation of the line $l$ such that $8r_1+9r_2$ is minimized and the minimum value. | 7 | 3/8 |
Using the six digits 0, 1, 2, 3, 4, 5,
(1) How many distinct three-digit numbers can be formed?
(2) How many distinct three-digit odd numbers can be formed? | 48 | 6/8 |
How many 3-element subsets of the set $\{1,2,3, \ldots, 19\}$ have a sum of elements divisible by 4? | 244 | 5/8 |
$\triangle GHI$ is inscribed inside $\triangle XYZ$ such that $G, H, I$ lie on $YZ, XZ, XY$, respectively. The circumcircles of $\triangle GZC, \triangle HYD, \triangle IXF$ have centers $O_1, O_2, O_3$, respectively. Also, $XY = 26, YZ = 28, XZ = 27$, and $\stackrel{\frown}{YI} = \stackrel{\frown}{GZ},\ \stackrel{\frown}{XI} = \stackrel{\frown}{HZ},\ \stackrel{\frown}{XH} = \stackrel{\frown}{GY}$. The length of $GY$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$. | 29 | 1/8 |
On the side $AB$ of triangle $ABC$, a point $M$ is taken. It starts moving parallel to $BC$ until it intersects $AC$, then it moves parallel to $AB$ until it intersects $BC$, and so on. Is it true that after a certain number of such steps, the point $M$ will return to its original position? If true, what is the minimum number of steps required for the return? | 6 | 2/8 |
The number \( A \) in decimal notation has the form \( A = \overline{7a631b} \), where \( a \) and \( b \) are non-zero digits. The number \( B \) is obtained by summing all the six-digit, distinct numbers derived from \( A \) by cyclic permutations of its digits (the first digit moves to the second place, the second to the third, and so on, the last digit moves to the first place). How many such numbers \( A \) exist, for which \( B \) is divisible by 121? Find the largest such \( A \). | 796317 | 7/8 |
In trapezoid \(ABCD\), where \(\angle BAD = 45^\circ\) and \(\angle CDA = 60^\circ\), the base \(AD\) equals 15, the base \(BC\) equals 13. A perpendicular to side \(AB\), drawn from point \(M\) (the midpoint of side \(AB\)), intersects with a perpendicular to side \(CD\), drawn from point \(N\) (the midpoint of side \(CD\)), at some point \(L\). Find the ratio of the area of triangle \(MNL\) to the area of trapezoid \(ABCD\). | \frac{7\sqrt{3}}{6} | 1/8 |
Given the triangle \( ABC \) where \(\angle C = \angle B = 50^\circ\), \(\angle MAB = 50^\circ\), and \(\angle ABN = 30^\circ\), find \(\angle BNM\). | 40 | 2/8 |
How many such five-digit Shenma numbers exist, where the middle digit is the smallest, the digits increase as they move away from the middle, and all the digits are different? | 1512 | 5/8 |
The quadratic trinomial $y = ax^2 + bx + c$ has no roots and $a+b+c > 0$. Find the sign of the coefficient $c$. | 0 | 1/8 |
Given the function defined on the interval $0 \le x < 1$:
\[
f_{1}(x) = \begin{cases}
1, & \text{if } 0.5 \le x < 0.6 \\
0, & \text{otherwise}
\end{cases}
\]
determine the elements of the sequence of functions $f_{n}(x)$ step by step using the following procedure. Where $f_{n-1}(x) > 0$, let $f_{n}(x) = f_{n-1}(x)$; where $f_{n-1}(x) = 0$, let $f_{n}(x)$ be $q^{n-1}$ or $0$ according to whether the $n$-th decimal digit of $x$ is 5 or not, where $q$ is a positive number less than 1. Determine the limit of the sequence
\[
a_{n} = \int_{0}^{1} f_{n}(x) \, dx \quad (n = 1, 2, \ldots)
\] | \frac{1}{10-9q} | 3/8 |
Let the sequence \( a_{n} \) be defined as follows:
\[
a_{1}=1, \quad a_{2}=\sqrt{2+\sqrt{2}}, \quad \ldots, \quad a_{n}=\sqrt{n+\sqrt{n+\ldots+\sqrt{n}}}, \ldots
\]
- In \( a_{n} \), the number of square root signs is \( n \).
Prove that the sequence \( \frac{a_{n}}{\sqrt{n}} \) converges and determine its limit. | 1 | 2/8 |
The bases of a trapezoid are equal to \(a\) and \(b\). Find the segment of the line connecting the midpoints of its diagonals (\(a > b\)). | \frac{b}{2} | 7/8 |
2. Each of $n$ members of a club is given a different item of information. The members are allowed to share the information, but, for security reasons, only in the following way: A pair may communicate by telephone. During a telephone call only one member may speak. The member who speaks may tell the other member all the information (s)he knows. Determine the minimal number of phone calls that are required to convey all the information to each of the members.
Hi, from my sketches I'm thinking the answer is $2n-2$ but I dont know how to prove that this number of calls is the smallest. Can anyone enlighten me? Thanks | 2n-2 | 7/8 |
Four chess players - Ivanov, Petrov, Vasiliev, and Kuznetsov - played a round-robin tournament (each played one game against each of the others). A victory awards 1 point, a draw awards 0.5 points to each player. It was found that the player in first place scored 3 points, and the player in last place scored 0.5 points. How many possible distributions of points are there among the named chess players, if some of them could have scored the same number of points? (For example, the cases where Ivanov has 3 points and Petrov has 0.5 points, and where Petrov has 3 points and Ivanov has 0.5 points, are considered different!) | 36 | 3/8 |
Let $f(n)$ be the number of distinct prime divisors of $n$ less than 6. Compute $$\sum_{n=1}^{2020} f(n)^{2}$$ | 3431 | 7/8 |
Given $0 \leq a_k \leq 1$ for $k=1,2,\ldots,2020$, and defining $a_{2021}=a_1, a_{2022}=a_2$, find the maximum value of $\sum_{k=1}^{2020}\left(a_{k}-a_{k+1} a_{k+2}\right)$. | 1010 | 2/8 |
For the largest possible $n$, can you create two bi-infinite sequences $A$ and $B$ such that any segment of length $n$ from sequence $B$ is contained in $A$, where $A$ has a period of 1995, but $B$ does not have this property (it is either not periodic or has a different period)?
Comment: The sequences can consist of arbitrary symbols. The problem refers to the minimal period. | 1995 | 1/8 |
There are triangles \( A_{1} A_{2} A_{3} \), \( B_{1} B_{2} B_{3} \), and \( C_{1} C_{2} C_{3} \) in a plane. Find a point \( P \) in the plane such that
\[ \overline{PA_1}^2 + \overline{PA_2}^2 + \overline{PA_3}^2 = \overline{PB_1}^2 + \overline{PB_2}^2 + \overline{PB_3}^2 = \overline{PC_1}^2 + \overline{PC_2}^2 + \overline{PC_3}^2. \] | P | 4/8 |
Given the random variable $X \sim N(1, \sigma^{2})$, if $P(0 < x < 3)=0.5$, $P(0 < X < 1)=0.2$, then $P(X < 3)=$\_\_\_\_\_\_\_\_\_\_\_ | 0.8 | 5/8 |
What is the smallest positive integer that is neither prime nor a cube and that has an even number of prime factors, all greater than 60? | 3721 | 6/8 |
There are \(2n\) different numbers in a row. By one move we can interchange any two numbers or interchange any three numbers cyclically (choose \(a, b, c\) and place \(a\) instead of \(b\), \(b\) instead of \(c\), and \(c\) instead of \(a\)). What is the minimal number of moves that is always sufficient to arrange the numbers in increasing order? | n | 2/8 |
A paper equilateral triangle with area 2019 is folded over a line parallel to one of its sides. What is the greatest possible area of the overlap of folded and unfolded parts of the triangle? | 673 | 1/8 |
The side of a triangle is 48 cm, and the height drawn to this side is 8.5 cm. Find the distance from the center of the circle inscribed in the triangle to the vertex opposite the given side, if the radius of the inscribed circle is 4 cm. | 5 | 4/8 |
A fenced, rectangular field measures $24$ meters by $52$ meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plots into which the field can be partitioned using all or some of the 1994 meters of fence?
| 702 | 3/8 |
For \(0 \leq x \leq 1\) and positive integer \(n\), let \(f_0(x) = |1 - 2x|\) and \(f_n(x) = f_0(f_{n-1}(x))\). How many solutions are there to the equation \(f_{10}(x) = x\) in the range \(0 \leq x \leq 1\)? | 2048 | 5/8 |
In the pie chart shown, 168 students chose bananas as their favourite fruit. How many students chose apples as their favourite fruit?
(A) 42
(B) 56
(C) 48
(D) 60
(E) 38 | 56 | 1/8 |
If $i^2 = -1$, then the sum $\cos{45^\circ} + i\cos{135^\circ} + \cdots + i^n\cos{(45 + 90n)^\circ} + \cdots + i^{40}\cos{3645^\circ}$ equals | \frac{\sqrt{2}}{2}(21 - 20i) | 1/8 |
Given that all three vertices of \(\triangle ABC\) lie on the parabola defined by \(y = 4x^2\), with \(A\) at the origin and \(\overline{BC}\) parallel to the \(x\)-axis, calculate the length of \(BC\), given that the area of the triangle is 128. | 4\sqrt[3]{4} | 7/8 |
A regular $n$-gon is inscribed in a unit circle. Prove that:
a) The sum of the squares of the lengths of all sides and all diagonals is equal to $n^{2}$;
b) The sum of the lengths of all sides and all diagonals is equal to $n \cot \left( \pi / 2n \right)$;
c) The product of the lengths of all sides and all diagonals is equal to $n^{n / 2}$. | n^{n/2} | 3/8 |
In square \(ABCD\) with a side length of 10, points \(P\) and \(Q\) lie on the segment joining the midpoints of sides \(AD\) and \(BC\). Connecting \(PA\), \(PC\), \(QA\), and \(QC\) divides the square into three regions of equal area. Find the length of segment \(PQ\). | 20/3 | 1/8 |
Let \( S \) be a set of rational numbers that satisfies the following conditions:
1. If \( a \in S \) and \( b \in S \), then \( a + b \in S \) and \( a \cdot b \in S \).
2. For any rational number \( r \), exactly one of the following three statements is true: \( r \in S \), \(-r \in S \), \( r = 0 \).
Prove that \( S \) is the set of all positive rational numbers. | S | 1/8 |
At the CleverCat Academy, there are three skills that the cats can learn: jump, climb, and hunt. Out of the cats enrolled in the school:
- 40 cats can jump.
- 25 cats can climb.
- 30 cats can hunt.
- 10 cats can jump and climb.
- 15 cats can climb and hunt.
- 12 cats can jump and hunt.
- 5 cats can do all three skills.
- 6 cats cannot perform any of the skills.
How many cats are in the academy? | 69 | 6/8 |
For a positive integer $K$ , define a sequence, $\{a_n\}_n$ , as following $a_1=K$ , \[ a_{n+1} = \{ \begin{array} {cc} a_n-1 , & \mbox{ if } a_n \mbox{ is even} \frac{a_n-1}2 , & \mbox{ if } a_n \mbox{ is odd} \end{array}, \] for all $n\geq 1$ .
Find the smallest value of $K$ , which makes $a_{2005}$ the first term equal to 0. | 2^{1003}-2 | 1/8 |
An electronic clock displays time from 00:00:00 to 23:59:59. How much time throughout the day does the clock show a number that reads the same forward and backward? | 96 | 2/8 |
Given a quadrilateral \(ABCD\) inscribed in a circle, where \(AC = a\), \(BD = b\), and \(AB \perp CD\). Find the radius of the circle. | \frac{\sqrt{^2+b^2}}{2} | 2/8 |
Given the ellipse \(\frac{x^{2}}{6} + \frac{y^{2}}{2} = 1\) with the right focus \(F\). The line passing through \(F\) given by \(y = k(x-2)\) intersects the ellipse at points \(P\) and \(Q\) \((k \neq 0)\). If the midpoint of \(PQ\) is \(N\) and \(O\) is the origin, the line \(ON\) intersects the line \(x = 3\) at \(M\).
1. Find the angle \(\angle MFQ\).
2. Find the maximum value of \(\frac{|PQ|}{|MF|}\). | \sqrt{3} | 5/8 |
When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number? | 0.02 | 7/8 |
Prove that the fraction \(\frac{12 n + 1}{30 n + 2}\) is irreducible for any natural number \( n \). | 1 | 6/8 |
The polynomial $ P(x)\equal{}x^3\plus{}ax^2\plus{}bx\plus{}d$ has three distinct real roots. The polynomial $ P(Q(x))$ , where $ Q(x)\equal{}x^2\plus{}x\plus{}2001$ , has no real roots. Prove that $ P(2001)>\frac{1}{64}$ . | P(2001)>\frac{1}{64} | 2/8 |
The units of length include , and the conversion rate between two adjacent units is . | 10 | 5/8 |
A boat travels upstream from port $A$ and encounters a drifting raft at a distance of 4 kilometers from port $A$. The boat continues its journey to port $B$ upstream and immediately turns around, arriving at port $A$ simultaneously with the raft. Given that the current speed is 2 meters/second, and the boat's speed downstream is 8 meters/second, what is the distance between ports $A$ and $B$ in kilometers? | 8 | 3/8 |
There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \dots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \); circle \( C_{6} \) passes through exactly 6 points in \( M \); ..., circle \( C_{1} \) passes through exactly 1 point in \( M \). Determine the minimum number of points in set \( M \). | 12 | 1/8 |
In quadrilateral \(ABCD\), \(AB = BC\), \(\angle A = \angle B = 20^{\circ}\), \(\angle C = 30^{\circ}\). The extension of side \(AD\) intersects \(BC\) at point... | 30 | 1/8 |
We roll five dice, each a different color. In how many ways can the sum of the rolls be 11? | 205 | 7/8 |
A pile of stones has a total weight of 100 kg, with each stone weighing no more than 2 kg. By taking out some stones in various ways, find the difference between the total weight of these stones and 10 kg. Let the minimum absolute value of these differences be \( d \). Find the maximum value of \( d \). | \frac{10}{11} | 1/8 |
Let $n\ge 2,$ be a positive integer. Numbers $\{1,2,3, ...,n\}$ are written in a row in an arbitrary order. Determine the smalles positive integer $k$ with the property: everytime it is possible to delete $k$ numbers from those written on the table, such that the remained numbers are either in an increasing or decreasing order. | n-\lceil\sqrt{n}\rceil | 2/8 |
Suppose there are 100 cities in a country, interconnected by a network of roads. It is known that even if all roads of one city are closed, it is still possible to travel from any city to any other city via the remaining network. Prove that the country can be divided into two sovereign nations, each with 50 cities, such that within each nation, it is possible to travel from any city to any other city via the roads. | 2 | 1/8 |
Find the ratio of $AE:EC$ in $\triangle ABC$ given that $AB=6$, $BC=8$, $AC=10$, and $E$ is on $\overline{AC}$ with $BE=6$. | \frac{18}{7} | 7/8 |
There is a $6 \times 6$ grid of lights. There is a switch at the top of each column and on the left of each row. A light will only turn on if the switches corresponding to both its column and its row are in the "on" position. Compute the number of different configurations of lights. | 3970 | 1/8 |
The distance from home to work is $s = 6$ km. At the moment Ivan left work, his favorite dog dashed out of the house and ran to meet him. They met at a distance of one-third of the total route from work. The dog immediately turned back and ran home. Upon reaching home, the dog turned around instantly and ran back towards Ivan, and so on. Assuming Ivan and his dog move at constant speeds, determine the distance the dog will run by the time Ivan arrives home. | 12 | 5/8 |
Let \( S \) be the set of functions \( f \) defined on reals in the closed interval \([0, 1]\) with non-negative real values such that \( f(1) = 1 \) and \( f(x) + f(y) \leq f(x + y) \) for all \( x, y \) such that \( x + y \leq 1 \). What is the smallest \( k \) such that \( f(x) \leq kx \) for all \( f \) in \( S \) and all \( x \)? | 2 | 1/8 |
\(ABCD\) is a square and \(X\) is a point on the side \(DA\) such that the semicircle with diameter \(CX\) touches the side \(AB\). Find the ratio \(AX: XD\). | 1 : 3 | 6/8 |
In quadrilateral \(ABCD\), \(\angle DAB = \angle DBC = 90^\circ\). Additionally, \(DB = a\) and \(DC = b\).
Find the distance between the centers of two circles, one passing through points \(D, A, B\), and the other passing through points \(B, C, D\). | \frac{\sqrt{b^2-^2}}{2} | 4/8 |
A "stair-step" figure is made of alternating black and white squares in each row. Rows $1$ through $4$ are shown. All rows begin and end with a white square. The number of black squares in the $37\text{th}$ row is
[asy]
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(6,0)--(6,2)--(1,2)--cycle);
draw((2,0)--(5,0)--(5,3)--(2,3)--cycle);
draw((3,0)--(4,0)--(4,4)--(3,4)--cycle);
fill((1,0)--(2,0)--(2,1)--(1,1)--cycle,black);
fill((3,0)--(4,0)--(4,1)--(3,1)--cycle,black);
fill((5,0)--(6,0)--(6,1)--(5,1)--cycle,black);
fill((2,1)--(3,1)--(3,2)--(2,2)--cycle,black);
fill((4,1)--(5,1)--(5,2)--(4,2)--cycle,black);
fill((3,2)--(4,2)--(4,3)--(3,3)--cycle,black);
[/asy] | 36 | 5/8 |
Let diamond \( A_{1} A_{2} A_{3} A_{4} \) have side length \( 1 \) and \(\angle A_{1} A_{2} A_{3} = \frac{\pi}{6} \). Point \( P \) lies in the plane of diamond \( A_{1} A_{2} A_{3} A_{4} \). Determine the minimum value of \( \sum_{1 \leqslant i < j \leqslant 4} \overrightarrow{P A_{i}} \cdot \overrightarrow{P A_{j}} \). | -1 | 3/8 |
Extend each side of the quadrilateral $ABCD$ in the same direction beyond itself, such that the new points are $A_{1}, B_{1}, C_{1}$, and $D_{1}$. Connect these newly formed points to each other. Show that the area of the quadrilateral $A_{1}B_{1}C_{1}D_{1}$ is five times the area of the quadrilateral $ABCD$. | 5 | 2/8 |
The letter T is formed by placing a $2\:\text{inch}\!\times\!6\:\text{inch}$ rectangle vertically and a $3\:\text{inch}\!\times\!2\:\text{inch}$ rectangle horizontally on top of the vertical rectangle at its middle, as shown. What is the perimeter of this T, in inches?
```
[No graphic input required]
``` | 22 | 3/8 |
In a right triangle, one of the acute angles $\alpha$ satisfies
\[\tan \frac{\alpha}{2} = \frac{1}{\sqrt[3]{2}}.\]Let $\theta$ be the angle between the median and the angle bisector drawn from this acute angle. Find $\tan \theta.$ | \frac{1}{2} | 2/8 |
What is the probability that two people, A and B, randomly choosing their rooms among 6 different rooms in a family hotel, which has two rooms on each of the three floors, will stay in two rooms on the same floor? | \frac{1}{5} | 7/8 |
Workshop A and Workshop B together have 360 workers. The number of workers in Workshop A is three times that of Workshop B. How many workers are there in each workshop? | 270 | 1/8 |
Let $p=4k+1$ be a prime. $S$ is a set of all possible residues equal or smaller then $2k$ when $\frac{1}{2} \binom{2k}{k} n^k$ is divided by $p$ . Show that \[ \sum_{x \in S} x^2 =p \] | p | 1/8 |
Given that 8 balls are randomly and independently painted either red or blue with equal probability, find the probability that exactly 4 balls are red and exactly 4 balls are blue, and all red balls come before any blue balls in the order they were painted. | \frac{1}{256} | 7/8 |
In a regular triangular pyramid \(SABC\) with vertex \(S\), a height \(SD\) is dropped. Point \(K\) is taken on segment \(SD\) such that \(SK: KD = 1:2\). It is known that the dihedral angles between the base and the lateral faces are \(\frac{\pi}{6}\), and the distance from point \(K\) to the lateral edge is \(\frac{4}{\sqrt{13}}\). Find the volume of the pyramid. | 216 | 3/8 |
The decimal digits of a natural number $A$ form an increasing sequence (from left to right). Find the sum of the digits of $9A$ . | 9 | 4/8 |
Consider a string of $n$ $7$'s, $7777\cdots77,$ into which $+$ signs are inserted to produce an arithmetic expression. For example, $7+77+777+7+7=875$ could be obtained from eight $7$'s in this way. For how many values of $n$ is it possible to insert $+$ signs so that the resulting expression has value $7000$?
| 108 | 1/8 |
A fair six-sided die with uniform quality is rolled twice in succession. Let $a$ and $b$ denote the respective outcomes. Find the probability that the function $f(x) = \frac{1}{3}x^3 + \frac{1}{2}ax^2 + bx$ has an extreme value. | \frac{17}{36} | 4/8 |
Square $PQRS$ has a side length of $2$ units. Points $T$ and $U$ are on sides $PQ$ and $QR$, respectively, with $PT = QU$. When the square is folded along the lines $ST$ and $SU$, sides $PS$ and $RS$ coincide and lie along diagonal $RQ$. Exprress the length of segment $PT$ in the form $\sqrt{k} - m$ units. What is the integer value of $k+m$? | 10 | 1/8 |
Find the limit, when $n$ tends to the infinity, of $$ \frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k} $$ | \sqrt{3} | 6/8 |
The telephone company "Prosto-Telecom" in Prostokvashino uses three-digit phone numbers. The equipment is old, so there may be errors in individual digits of the transmitted subscriber number — each digit can be independently replaced by another random digit with a probability of $p=0.02$. To reduce the likelihood of incorrect connections, the company uses the following rule: the first digit of the number must always be the remainder when the sum of the other two digits is divided by 10. For example, the numbers 000 and 156 are possible, but the number 234 is not. If the check digit is incorrect during the connection, an error message is issued. Find the probability that, despite the measures taken, an incorrect connection will occur. | 0.000131 | 1/8 |
Adam and Simon start on bicycle trips from the same point at the same time. Adam travels north at 10 mph and Simon travels west at 12 mph. How many hours will it take for them to be 130 miles apart? | \frac{65}{\sqrt{61}} | 2/8 |
Given the sequence $\left\{a_{n}\right\}$ that satisfies
$$
a_{n+1} = -\frac{1}{2} a_{n} + \frac{1}{3^{n}}\quad (n \in \mathbf{Z}_{+}),
$$
find all values of $a_{1}$ such that the sequence $\left\{a_{n}\right\}$ is monotonic, i.e., $\left\{a_{n}\right\}$ is either increasing or decreasing. | \frac{2}{5} | 2/8 |
If the eccentricity of the conic section \(C\): \(x^{2}+my^{2}=1\) is \(2\), determine the value of \(m\). | -\dfrac {1}{3} | 7/8 |
A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x$. If $x=0$ is a root for $f(x)=0$, what is the least number of roots $f(x)=0$ must have in the interval $-1000\leq x \leq 1000$? | 401 | 6/8 |
There are 11 of the number 1, 22 of the number 2, 33 of the number 3, and 44 of the number 4 on the blackboard. The following operation is performed: each time, three different numbers are erased, and the fourth number, which is not erased, is written 2 extra times. For example, if 1 of 1, 1 of 2, and 1 of 3 are erased, then 2 more of 4 are written. After several operations, there are only 3 numbers left on the blackboard, and no further operations can be performed. What is the product of the last three remaining numbers? | 12 | 1/8 |
A father and son measured the length of the yard in steps during winter, starting from the same place and going in the same direction. In some places, the footprints of the father and son perfectly coincided. There were a total of 61 footprints on the snow along the measurement line. What is the length of the yard if the father's step length is $0.72$ meters and the son's step length is $0.54$ meters? | 21.6\, | 1/8 |
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