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In an abandoned chemistry lab, Gerome found a two-pan balance scale and three 1-gram weights, three 5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile. Find the number of different positive weights of chemicals that Gerome could measure.
|
129
|
3/8
|
For integers $0 \le m,n \le 2^{2017}-1$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor \frac{m}{2^k} \right\rfloor$ and $\left\lfloor \frac{n}{2^k} \right\rfloor$ are both odd integers. Consider a $2^{2017} \times 2^{2017}$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 2^{2017}$) is $(-1)^{\alpha(i-1, j-1)}$. For $1 \le i, j \le 2^{2017}$, let $M_{i,j}$ be the matrix with the same entries as $M$ except for the $(i,j)$th entry, denoted by $a_{i,j}$, and such that $\det M_{i,j}=0$. Suppose that $A$ is the $2^{2017} \times 2^{2017}$ matrix whose $(i,j)$th entry is $a_{i,j}$ for all $1 \le i, j \le 2^{2017}$. Compute the remainder when $\det A$ is divided by $2017$.
|
1382
|
3/8
|
Define a \textit{subsequence} of a string \( \mathcal{S} \) of letters to be a positive-length string using any number of the letters in \( \mathcal{S} \) in order. For example, a subsequence of \( HARRISON \) is \( ARRON \). Compute the number of subsequences in \( HARRISON \).
|
255
|
2/8
|
Let $O$ be the center and let $F$ be one of the foci of the ellipse $25x^2 +16 y^2 = 400$. A second ellipse, lying inside and tangent to the first ellipse, has its foci at $O$ and $F$. What is the length of the minor axis of this second ellipse?The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
|
12
|
2/8
|
The parabola $y = x^2$ is tangent to the graph of $y = x^4 + ax^3 + x^2 + bx + 1$ at two points. Find the positive difference between the $x$-coordinates of the points of tangency.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
|
7
|
3/8
|
Let $\tau = \frac{1 + \sqrt{5}}{2}.$ Find
\[\sum_{n = 0}^\infty \frac{\lfloor \tau^n \rceil}{2^n}.\]Note: For a real number $x,$ $\lfloor x \rceil$ denotes the integer closest to $x.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
|
13
|
2/8
|
4. Let $M$ be a set of six mutually different positive integers which sum up to 60 . We write these numbers on faces of a cube (on each face one). In a move we choose three faces with a common vertex and we increase each number on these faces by one. Find the number of all sets $M$, whose elements (numbers) can be written on the faces of the cube in such a way that we can even up the numbers on the faces in finitely many moves.
(Peter Novotný)
|
84
|
1/8
|
10.5. An increasing geometric progression consists of four different positive numbers, three of which form an arithmetic progression. What can the denominator of this progression be? Provide all possible answers and prove that there are no others.
|
\dfrac{1 + \sqrt{5}}{2}
|
0/8
|
G2.3 In Figure 1 , the square $P Q R S$ is inscribed in $\triangle A B C$. The areas of $\triangle A P Q, \triangle P B S$ and $\triangle Q R C$ are 4,4 and 12 respectively. If the area of the square is $c$, find the value of $c$.
|
16
|
0/8
|
2. Simplify $\sqrt[3]{10+7 \sqrt{2}}+\sqrt[3]{10-7 \sqrt{2}}$ to get the result $\qquad$ .
|
\sqrt[3]{32}
|
1/8
|
There are 6 moments, $6: 30, 6: 31, 6: 32, 6: 33, 6: 34, 6: 35$. At $\qquad$ moment, the hour hand and the minute hand are closest to each other, and at $\qquad$ moment, the hour hand and the minute hand are farthest from each other.
|
6:33
|
0/8
|
Consider functions $f : [0, 1] \rightarrow \mathbb{R}$ which satisfy
(i)$f(x)\ge0$ for all $x$ in $[0, 1]$,
(ii)$f(1) = 1$,
(iii) $f(x) + f(y) \le f(x + y)$ whenever $x$, $y$, and $x + y$ are all in $[0, 1]$.
Find, with proof, the smallest constant $c$ such that
$f(x) \le cx$
for every function $f$ satisfying (i)-(iii) and every $x$ in $[0, 1]$.
|
2
|
1/8
|
A circle of radius $4$ is inscribed in a triangle $ABC$ . We call $D$ the touchpoint between the circle and side BC. Let $CD =8$ , $DB= 10$ . What is the length of the sides $AB$ and $AC$ ?
|
12.5
|
0/8
|
Every of $n$ guests invited to a dinner has got an invitation denoted by a number from $1$ to $n$ . The guests will be sitting around a round table with $n$ seats. The waiter has decided to derve them according to the following rule. At first, he selects one guest and serves him/her at any place. Thereafter, he selects the guests one by one: having chosen a guest, he goes around the table for the number of seats equal to the preceeding guest's invitation number (starting from the seat of the preceeding guest), and serves the guest there.
Find all $n$ for which he can select the guests in such an order to serve all the guests.
|
n
|
0/8
|
$n$ is a natural number. Given $3n \cdot 3n$ table, the unit cells are colored white and black such that starting from the left up corner diagonals are colored in pure white or black in ratio of 2:1 respectively. ( See the picture below). In one step any chosen $2 \cdot 2$ square's white cells are colored orange, orange are colored black and black are colored white. Find all $n$ such that with finite steps, all the white cells in the table turns to black, and all black cells in the table turns to white. ( From starting point)
|
n
|
1/8
|
For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$ , we define
\[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.\]
Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$ , $\Omega(P(k))$ is even. Show that $n$ is an even number.
|
n
|
4/8
|
We define a sequence $a_n$ so that $a_0=1$ and
\[a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, a_n + d & \textrm{ otherwise. } \end{cases} \]
for all postive integers $n$ .
Find all positive integers $d$ such that there is some positive integer $i$ for which $a_i=1$ .
|
d
|
1/8
|
A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers $a$ and $b$ . A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between $a$ and $b$ be for this to happen?
|
\frac{a}{b}
|
1/8
|
A town-planner has built an isolated city whose road network consists of $2N$ roundabouts, each connecting exactly three roads. A series of tunnels and bridges ensure that all roads in the town meet only at roundabouts. All roads are two-way, and each roundabout is oriented clockwise.
Vlad has recently passed his driving test, and is nervous about roundabouts. He starts driving from his house, and always takes the first edit at each roundabout he encounters. It turns out his journey incluldes every road in the town in both directions before he arrives back at the starting point in the starting direction. For what values of $N$ is this possible?
|
N
|
3/8
|
9.5. Let $M$ - be a finite set of numbers (distinct). It is known that among any three of its elements, there will be two whose sum belongs to $M$. What is the maximum number of elements that can be in $M$?
|
7
|
5/8
|
3. Given two sets of numbers, set $A$ is: $1,2, \cdots, 100$; set $B$ is: $1^{2}, 2^{2}, \cdots, 100^{2}$. For a number $x$ in set $A$, if there is a number $y$ in set $B$ such that $x+y$ is also a number in set $B$, then $x$ is called an "associated number". Therefore, the number of such associated numbers in set $A$ is $\qquad$.
|
73
|
3/8
|
20. Let $\lfloor x\rfloor$ be the greatest integer not exceeding $x$. For instance, $\lfloor 3.4\rfloor=3,\lfloor 2\rfloor=2$, and $\lfloor-2.7\rfloor=-3$. Determine the value of the constant $\lambda>0$ so that $2\lfloor\lambda n\rfloor=1-n+\lfloor\lambda\lfloor\lambda n\rfloor\rfloor$ for all positive integers $n$.
|
1 + \sqrt{2}
|
3/8
|
1. The pond has a rectangular shape. On the first frosty day, the part of the pond within 10 meters of the nearest shore froze. On the second day, the part within 20 meters froze, on the third day, the part within 30 meters, and so on. On the first day, the area of open water decreased by 20.2%, and on the second day, it decreased by 18.6% of the original area. On which day will the pond be completely frozen?
|
7
|
5/8
|
6. There are infinitely many cards, each with a real number written on it. For each real number $x$, there is exactly one card with the number $x$ written on it. Two players each select a set of 100 cards, denoted as $A$ and $B$, such that the sets are disjoint. Formulate a rule to determine which of the two players wins, satisfying the following conditions:
(1) The winner depends only on the relative order of these 200 cards: if these 200 cards are placed face down in increasing order, and the audience is informed which card belongs to which player, but not what number is written on each card, the audience can still determine who will win;
(2) If the elements of the two sets are written in increasing order as
$$
A=\left\{a_{1}, a_{2}, \cdots, a_{100}\right\}, B=\left\{b_{1}, b_{2}, \cdots, b_{100}\right\},
$$
where, for all $i \in\{1,2, \cdots, 100\}$, $a_{i}>b_{i}$, then $A$ defeats $B$;
(3) If three players each select a set of 100 cards, denoted as $A$, $B$, and $C$, and $A$ defeats $B$, and $B$ defeats $C$, then $A$ defeats $C$.
Question: How many such rules are there?
[Note] Two different rules mean that there exist two sets $A$ and $B$ such that in one rule, $A$ defeats $B$, and in the other rule, $B$ defeats $A$.
|
100
|
4/8
|
In a roulette wheel, any number from 0 to 2007 can come up with equal probability. The roulette wheel is spun repeatedly. Let $P_{k}$ denote the probability that at some point the sum of the numbers that have come up in all the spins equals $k$. Which number is greater: $P_{2007}$ or $P_{2008} ?$
|
P_{2007}
|
3/8
|
1. For a regular triangular prism with base edge length and side edge length both 1, a section is made through one side of the base and the midpoint of the line connecting the centers of the top and bottom bases. What is the area of the section? $\qquad$ .
|
\dfrac{4\sqrt{3}}{9}
|
4/8
|
An excursion group of 6 tourists is sightseeing. At each attraction, three people take photos while the others photograph them. After visiting the minimal number of attractions, how many attractions will it take for each tourist to have photos of all the other participants?
|
4
|
2/8
|
Kanel-Belov A.Y.
The periods of two sequences are 7 and 13. What is the maximum length of the initial segment that can coincide?
|
18
|
0/8
|
9. In a convex quadrilateral $A B C D, \angle B A C=\angle C A D, \angle A B C=\angle A C D$, the extensions of $A D$ and $B C$ meet at $E$, and the extensions of $A B$ and $D C$ meet at $F$. Determine the value of
$$
\frac{A B \cdot D E}{B C \cdot C E} .
$$
|
1
|
4/8
|
5. A semicircle of radius 1 is drawn inside a semicircle of radius 2, as shown in the diagram, where $O A=O B=2$.
A circle is drawn so that it touches each of the semicircles and their common diameter, as shown.
What is the radius of the circle?
|
\dfrac{8}{9}
|
3/8
|
Exercise 7. Let $n$ be a strictly positive integer. Domitille has a rectangular grid divided into unit squares. Inside each unit square is written a strictly positive integer. She can perform the following operations as many times as she wishes:
- Choose a row and multiply each number in the row by $n$.
- Choose a column and subtract $n$ from each integer in the column.
Determine all values of $n$ for which the following property is satisfied:
Regardless of the dimensions of the rectangle and the integers written in the cells, Domitille can end up with a rectangle containing only 0s after a finite number of operations.
|
2
|
1/8
|
4. How many numbers at most can be chosen from the set $M=\{1,2, \ldots, 2018\}$ such that the difference of any two chosen numbers is not equal to a prime number?
The regional round of category B takes place
## on Tuesday, April 10, 2018
so that it starts no later than 10:00 AM and the contestants have 4 hours of pure time to solve the problems. Allowed aids are writing and drawing supplies and school MF tables. Calculators, laptops, and any other electronic aids are not allowed. Each problem can earn the contestant 6 points; the logical correctness and completeness of the written solution are also evaluated. The point threshold (higher than 7 points) for determining successful solvers will be centrally set after evaluating the statistics of results from all regions. This information will be communicated to the students before the competition begins.
|
1009
|
0/8
|
In triangle $ABC$ , $AC=13$ , $BC=14$ , and $AB=15$ . Points $M$ and $D$ lie on $AC$ with $AM=MC$ and $\angle ABD = \angle DBC$ . Points $N$ and $E$ lie on $AB$ with $AN=NB$ and $\angle ACE = \angle ECB$ . Let $P$ be the point, other than $A$ , of intersection of the circumcircles of $\triangle AMN$ and $\triangle ADE$ . Ray $AP$ meets $BC$ at $Q$ . The ratio $\frac{BQ}{CQ}$ can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m-n$ .
Please give the answer directly without any intermediate steps.
|
218
|
0/8
|
1. Find all five-digit numbers $\overline{a b c d e}$ that are divisible by 9 and for which $\overline{a c e}-\overline{b d a}=760$.
|
81828
|
2/8
|
5. A subset $M$ of the set $S=\{1928,1929, \cdots, 1949\}$ is called "red" if and only if the sum of any two distinct elements in $M$ is not divisible by 4. Let $x$ and $y$ represent the number of red four-element subsets and red five-element subsets of $S$, respectively. Try to compare the sizes of $x$ and $y$, and explain the reason.
---
The translation preserves the original text's formatting and structure.
|
y > x
|
3/8
|
We define the *Fibonacci sequence* $\{F_n\}_{n\ge0}$ by $F_0=0$ , $F_1=1$ , and for $n\ge2$ , $F_n=F_{n-1}+F_{n-2}$ ; we define the *Stirling number of the second kind* $S(n,k)$ as the number of ways to partition a set of $n\ge1$ distinguishable elements into $k\ge1$ indistinguishable nonempty subsets.
For every positive integer $n$ , let $t_n = \sum_{k=1}^{n} S(n,k) F_k$ . Let $p\ge7$ be a prime. Prove that \[ t_{n+p^{2p}-1} \equiv t_n \pmod{p} \] for all $n\ge1$ .
*Proposed by Victor Wang*
|
t_{n+p^{2p}-1} \equiv t_n \pmod{p}
|
2/8
|
Given \( n (\geqslant 4) \) points in a plane, where no three points are collinear, connected by \( l \left( \geqslant \left[\frac{n^{2}}{4} \right] + 1\right) \) line segments. If two triangles share a common edge, they are called an (unordered) triangle pair. Let \( T \) be the number of triangle pairs formed by these \( l \) line segments. Prove that \( T \geqslant \frac{l\left(4 l - n^{2}\right)\left(4 l - n^{2} - n\right)}{2 n^{2}} \).
|
T \geqslant \frac{l\left(4 l - n^{2}\right)\left(4 l - n^{2} - n\right)}{2 n^{2}}
|
0/8
|
6. Let $a_{1}, a_{2}, \cdots, a_{2014}$ be a permutation of the positive integers $1,2, \cdots$, 2014. Denote
$$
S_{k}=a_{1}+a_{2}+\cdots+a_{k}(k=1,2, \cdots, 2014) \text {. }
$$
Then the maximum number of odd numbers in $S_{1}, S_{2}, \cdots, S_{2014}$ is $\qquad$
|
1511
|
1/8
|
In regular octagon $ABCDEFGH$, $M$ and $N$ are midpoints of $\overline{BC}$ and $\overline{FG}$ respectively. Compute $[ABMO]/[EDCMO]$. ($[ABCD]$ denotes the area of polygon $ABCD$.) [asy]
pair A,B,C,D,E,F,G,H;
F=(0,0); E=(2,0); D=(2+sqrt(2),sqrt(2)); C=(2+sqrt(2),2+sqrt(2));
B=(2,2+2sqrt(2)); A=(0,2+2*sqrt(2)); H=(-sqrt(2),2+sqrt(2)); G=(-sqrt(2),sqrt(2));
draw(A--B--C--D--E--F--G--H--cycle);
draw(A--E);
pair M=(B+C)/2; pair N=(F+G)/2;
draw(M--N);
label("$A$",A,N); label("$B$",B,NE); label("$C$",C,E); label("$D$",D,E);
label("$E$",E,S); label("$F$",F,S); label("$G$",G,W); label("$H$",H,W);
label("$M$",M,NE); label("$N$",N,SW);
label("$O$",(1,2.4),E);
[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
|
8
|
4/8
|
A few (at least $5$) integers are placed on a circle such that each integer is divisible by the sum of its neighboring integers. If the sum of all the integers is positive, what is the minimal value of this sum?
|
2
|
1/8
|
Let $S = \{1, 2, \dots, 9\}$. Compute the number of functions $f : S \rightarrow S$ such that, for all $s \in S$, $f(f(f(s))) = s$ and $f(s) - s$ is not divisible by $3$.
|
288
|
5/8
|
An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M \in \overline{AB}$, $Q \in \overline{AC}$, and $N, P \in \overline{BC}$.
Suppose that $ABC$ is an equilateral triangle with side length $2$, and that $AMNPQ$ has a line of symmetry perpendicular to $BC$. The area of $AMNPQ$ is given by $n - p \sqrt{q}$, where $n, p, q$ are positive integers and $q$ is not divisible by the square of a prime. Compute $100n + 10p + q$.
|
5073
|
5/8
|
A group of 6 students decided to form "study groups" and "service activity groups" based on the following principle:
- Each group must have exactly 3 members.
- For any pair of students, there are the same number of study groups and service activity groups that both students are members of.
Given that there is at least one group and no three students belong to both the same study group and service activity group, find the minimum number of groups.
|
8
|
0/8
|
A rectangle-shaped puzzle is assembled with $2000$ pieces that are all equal rectangles and similar to the large rectangle. The sides of the small rectangles are parallel to those of the large one. The shortest side of each piece measures $1$. Determine the minimum possible value of the area of the large rectangle.
|
2000
|
1/8
|
Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$. How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$? Note that $\lfloor x \rfloor$ is known as the $\textit{floor}$ function, which returns the greatest integer less than or equal to $x$.
|
1973
|
0/8
|
In a triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpendicular to each other. If $AD = 7$ and $BE = 9$, find the integer nearest to the area of triangle $ABC$.
|
47
|
5/8
|
Let $N = \prod_{k=1}^{1000} (4^k - 1)$. Determine the largest positive integer $n$ such that $5^n$ divides evenly into $N$.
|
624
|
4/8
|
Felix chooses a positive integer as the starting number and writes it on the board. He then repeats the following step:
- If the number $n$ on the board is even, he replaces it with $\frac{1}{2}n$.
- If the number $n$ on the board is odd, he replaces it with $n^2 + 3$.
Determine for how many choices of starting numbers below $2023$ Felix will never write a number of more than four digits on the board.
|
21
|
4/8
|
Integers $a$, $b$, $c$ are selected independently and at random from the set $\{ 1, 2, \cdots, 10 \}$, with replacement. If $p$ is the probability that $a^{b-1}b^{c-1}c^{a-1}$ is a power of two, compute $1000p$.
|
136
|
1/8
|
Two congruent equilateral triangles $\triangle ABC$ and $\triangle DEF$ lie on the same side of line $BC$ so that $B$, $C$, $E$, and $F$ are collinear as shown. A line intersects $\overline{AB}$, $\overline{AC}$, $\overline{DE}$, and $\overline{EF}$ at $W$, $X$, $Y$, and $Z$, respectively, such that $\frac{AW}{BW} = \frac{2}{9}$, $\frac{AX}{CX} = \frac{5}{6}$, and $\frac{DY}{EY} = \frac{9}{2}$. The ratio $\frac{EZ}{FZ}$ can then be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
|
33
|
4/8
|
An arithmetic sequence of exactly $10$ positive integers has the property that any two elements are relatively prime. Compute the smallest possible sum of the $10$ numbers.
|
1360
|
1/8
|
Let $N$ be the number of positive divisors of $2010^{2010}$ that end in the digit 2. What is the remainder when $N$ is divided by 2010?
|
503
|
2/8
|
Let $a > 0$. If the inequality $22 < ax < 222$ holds for precisely $10$ positive integers $x$, find how many positive integers satisfy the inequality $222 < ax < 2022$?
|
90
|
2/8
|
For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes, respectively. A positive integer $n$ is called fair if $s(n) = c(n) > 1$. Find the number of fair integers less than $100$.
|
7
|
4/8
|
The vertices of a rhombus are located on the sides of a parallelogram, and the sides of the rhombus are parallel to the diagonals of the parallelogram. Find the ratio of the areas of the rhombus and the parallelogram if the ratio of the diagonals of the parallelogram is \( k \).
|
\frac{2k}{(1+k)^2}
|
0/8
|
Point \( K \) lies on edge \( AB \) of pyramid \( ABCD \). Construct the cross-section of the pyramid with a plane passing through point \( K \) parallel to lines \( BC \) and \( AD \).
|
K L M N
|
2/8
|
Five children and a monkey want to share a bag of walnuts. One of the children—unbeknownst to the others—divided the walnuts in the bag into five equal parts. He found one extra walnut, gave it to the monkey, and left with one-fifth of the walnuts. The second child then divided the remaining walnuts into five equal parts, also found one extra walnut, gave it to the monkey, and took one-fifth of the portion. The remaining three children did the same. Finally, the five children gathered the remaining walnuts, divided them into five equal parts, found one extra walnut, and gave it to the monkey.
What is the minimum number of walnuts that were originally in the bag?
|
15621
|
2/8
|
The height of a triangle divides an angle of the triangle in the ratio \(2: 1\), and the base into segments whose ratio (of the larger to the smaller) is equal to \(k\). Find the sine of the smaller angle at the base and the permissible values of \(k\).
|
\frac{1}{k-1}
|
4/8
|
Given the angles of triangle ABC and the radius R of the circumscribed circle passing through the vertices, connect the feet of the altitudes of the triangle. Calculate the sides, angles, and the circumradius of the triangle formed by these feet (the orthic triangle).
|
\frac{R}{2}
|
3/8
|
There are 11 kilograms of cereal. How can one measure out 1 kilogram of cereal using a balance scale and a single 3-kilogram weight in just two weighings?
|
1 \text{ kg}
|
3/8
|
Through every three vertices of a cube, located at the ends of each trio of edges meeting at one vertex, a plane is drawn. Find the volume of the solid bounded by these planes if the edge length of the cube is $a$.
|
\frac{a^3}{6}
|
2/8
|
Let \( n \) be a positive integer. Let \( V_{n} \) be the set of all sequences of 0's and 1's of length \( n \). Define \( G_{n} \) to be the graph having vertex set \( V_{n} \), such that two sequences are adjacent in \( G_{n} \) if and only if they differ in either 1 or 2 places. For instance, if \( n=3 \), the sequences \((1,0,0)\), \((1,1,0)\), and \((1,1,1)\) are mutually adjacent, but \((1,0,0)\) is not adjacent to \((0,1,1)\).
Show that, if \( n+1 \) is not a power of 2, then the chromatic number of \( G_{n} \) is at least \( n+2 \).
|
n+2
|
4/8
|
There are 100 light bulbs arranged in a row on a Christmas tree. The bulbs are then toggled according to the following algorithm: initially, all the bulbs are turned on; after one second, every second bulb is turned off; after another second, every third bulb is toggled (turned off if it was on, turned on if it was off); after another second, every fourth bulb is toggled, and so on. After 100 seconds, the process stops. Find the probability that a randomly selected bulb is on after this process (the bulbs do not burn out or break).
|
0.1
|
0/8
|
Consider the numbers \( k \) and \( n \) such that they are relatively prime, with \( n \geq 5 \) and \( k < \frac{n}{2} \). A regular \((n ; k)\)-star is defined as a closed broken line obtained by replacing every \( k \) consecutive sides of a regular \( n \)-gon with a diagonal connecting those endpoints. For example, a \((5 ; 2)\)-star has 5 self-intersection points as illustrated by the bold points in the figure. How many self-intersections does a \((2018 ; 25)\)-star have?
|
48432
|
3/8
|
The length of the road is 300 km. Car $A$ starts at one end of the road at noon and moves at a constant speed of 50 km/h. At the same time, car $B$ starts at the other end of the road with a constant speed of 100 km/h, and a fly starts with a speed of 150 km/h. Upon meeting car $A$, the fly turns around and heads towards car $B$.
1) When does the fly meet car $B$?
2) If upon meeting car $B$, the fly turns around, heads towards car $A$, meets it, turns around again, and continues flying back and forth between $A$ and $B$ until they collide, when do the cars crush the fly?
|
2 \ \text{hours}
|
2/8
|
The microcalculator MK-97 can perform only three operations on numbers stored in memory:
1) Check if two chosen numbers are equal,
2) Add the chosen numbers,
3) For chosen numbers \( a \) and \( b \), find the roots of the equation \( x^2 + ax + b = 0 \), and if there are no roots, display a message indicating so.
All results of actions are stored in memory. Initially, one number \( x \) is stored in the memory. Using the MK-97, how can you determine if this number is equal to one?
|
x = 1
|
4/8
|
Let \( a \otimes b = a \times \sqrt{b + \sqrt{b + 1 + \sqrt{b + 2 + \sqrt{b + 3 + ...}}}} \). If \( 3 \otimes h = 15 \), find the value of \( h \).
|
20
|
1/8
|
Three congruent isosceles triangles \( EFO \), \( OFG \), and \( OGH \) have \( EF=EO=OG=GH=15 \) and \( FG=EO=OH=18 \). These triangles are arranged to form trapezoid \( EFGH \), as shown. Point \( Q \) is on side \( FG \) such that \( OQ \) is perpendicular to \( FG \).
Point \( Z \) is the midpoint of \( EF \) and point \( W \) is the midpoint of \( GH \). When \( Z \) and \( W \) are joined, the trapezoid is divided into two smaller trapezoids. Find the ratio of the area of trapezoid \( EFZW \) to the area of trapezoid \( ZWGH \) in simplified form, and calculate \( r+s \) where the ratio is \( r:s \).
|
2
|
1/8
|
The numbers $\sqrt{3u-2}$, $\sqrt{3u+2}$, and $2\sqrt{u}$ are given as the side lengths of a triangle. What is the measure of the largest angle in this triangle?
|
90
|
2/8
|
Consider the function $y = g(x)$ where $g(x) = \frac{x^2}{Dx^2 + Ex + F}$, and $D, E, F$ are integers. The graph has vertical asymptotes at $x = -3$ and $x = 2$. It is also given that for all $x > 5$, $g(x) > 0.5$. If additionally, $g(0) = \frac{1}{4}$, find $D + E + F$.
|
-4
|
4/8
|
Marguerite drove 100 miles in 2.4 hours. If Sam drove for 4 hours at the same average rate as Marguerite but rested for 1 hour during this period, how many miles did he drive in total?
|
125 \text{ miles}
|
4/8
|
In a square layout, the vertices of the square are the centers of four circles. Each side of the square is 10 cm, and the radius of each circle is 3 cm. Calculate the area in square centimeters of the shaded region formed by the exclusion of the parts of the circles that overlap with the square.
|
100 - 9\pi \text{ cm}^2
|
0/8
|
Each triangle is a 45-45-90 triangle, and the hypotenuse of one triangle is the leg of an adjacent triangle. The hypotenuse of the largest triangle is 10 centimeters. What is the length of the leg of the smallest triangle? Express your answer as a common fraction.
|
\frac{5}{2}
|
4/8
|
A bowling ball is redesigned with a spherical surface and a diameter of 24 cm. For custom fitting, three holes are drilled for a new bowler, each 6 cm deep. The diameters of the holes are 2 cm, 2.5 cm, and 4 cm. Assuming the holes are right circular cylinders, calculate the volume of the fitted bowling ball in cubic centimeters, expressed in terms of \(\pi\).
|
2264.625\pi \text{ cubic cm}
|
0/8
|
Let
\[ g(x) = x^3 + 5x^2 + 15x + 35. \]
The graphs of \( y = g(x) \) and \( y = g^{-1}(x) \) intersect at exactly one point \((c,d)\). Enter the ordered pair \((c,d)\).
|
(-5, -5)
|
1/8
|
In $\Delta ABC$, $AC = BC$, $m\angle DCB = 50^{\circ}$, and $CD \parallel AB$. Point $E$ is on extension of $CD$ such that $DE \parallel BC$. Determine $m\angle ECD$.
|
50^\circ
|
2/8
|
Jamie paid for a $2.40 meal using exactly 50 coins consisting of pennies, nickels, and dimes. How many dimes did Jamie use if no change was received?
|
10
|
1/8
|
Let a square have a side length of \(10^{0.2}\) meters. Compute the area of the square and then multiply it by \(10^{0.1}\), \(10^{-0.3}\), and \(10^{0.4}\).
|
10^{0.6}
|
1/8
|
A plane's passengers consist of 50% women and 50% men. Twenty percent of the women and fifteen percent of the men are in first class. What is the total number of passengers in first class if the plane is carrying 300 passengers?
|
53
|
0/8
|
What is the least number of colors needed to shade the tessellation shown, such that no two tiles sharing a side or overlapping vertically are the same color? The tessellation consists of a base layer of rectangles uniformly placed and a top layer consisting of circles, where each circle overlaps four rectangles (one on each side, without side sharing).
|
3
|
4/8
|
As shown in the figure, in triangle $\mathrm{ABC}$, points $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$ are on segments $\mathrm{AZ}, \mathrm{BX}, \mathrm{CY}$ respectively, and $Y Z=2 Z C, Z X=3 X A, X Y=4 Y B$. The area of triangle $X Y Z$ is 24. Find the area of triangle $\mathrm{ABC}$.
|
59
|
2/8
|
Inside a circular pancake with a radius of 10, a coin with a radius of 1 has been baked. What is the minimum number of straight cuts needed to surely hit the coin?
|
10
|
0/8
|
A nine-digit number is odd. The sum of its digits is 10. The product of the digits of the number is non-zero. The number is divisible by seven. When rounded to three significant figures, how many millions is the number equal to?
|
112
|
5/8
|
A point is randomly thrown onto the segment [6, 11], and let \( k \) be the resulting value. Find the probability that the roots of the equation \(\left(k^{2}-2k-15\right)x^{2}+(3k-7)x+2=0\) satisfy the condition \( x_{1} \leq 2x_{2} \).
|
\frac{1}{3}
|
4/8
|
4. Let the function $y=f(x)$ have a domain of $\mathbf{R}$, and for $x>1$, and for any real numbers $x, y \in \mathbf{R}$, $f(x+y)=f(x) f(y)$ holds. The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=f(0)$, and $f\left(a_{n+1}\right)=\frac{1}{f\left(-2-a_{n}\right)}(n \in \mathbf{N})$. If the inequality $\left(1+\frac{1}{a_{1}}\right)\left(1+\frac{1}{a_{2}}\right) \cdots\left(1+\frac{1}{a_{n}}\right) \geqslant k \cdot$ $\sqrt{2 n+1}$ holds for all $n \in \mathbf{N}$, find the minimum value of $k$. (2003 Hunan Province Mathematics Competition Problem)
|
\dfrac{2\sqrt{3}}{3}
|
4/8
|
There are four circles. In the first one, a chord $A B$ is drawn, and the distance from the midpoint of the smaller of the two arcs formed to $A B$ is 1. The second, third, and fourth circles are located inside the larger segment and are tangent to the chord $A B$. The second and fourth circles are tangent to the inside of the first and externally to the third. The sum of the radii of the last three circles is equal to the radius of the first circle. Find the radius of the third circle, given that the line passing through the centers of the first and third circles is not parallel to the line passing through the centers of the other two circles.
|
\dfrac{1}{2}
|
2/8
|
8.1. Of all numbers with the sum of digits equal to 25, find the one whose product of digits is maximal. If there are several such numbers, write the smallest of them in the answer.
|
33333334
|
4/8
|
Example 1 Let the base edge length of the regular tetrahedron $V-A B C$ be 4, and the side edge length be 8. Construct a section $A E D$ through $A$ that intersects the side edges $V B, V C$. Find the minimum perimeter of the section $\triangle A E D$.
|
11
|
2/8
|
A chess knight has injured his leg and is limping. He alternates between a normal move and a short move where he moves to any diagonally neighbouring cell.
The limping knight moves on a $5 \times 6$ cell chessboard starting with a normal move. What is the largest number of moves he can make if he is starting from a cell of his own choice and is not allowed to visit any cell (including the initial cell) more than once?
|
24
|
1/8
|
In a circle with center $O$, the diameter $A B$ is extended to $C$ such that $B C = \frac{1}{2} \cdot A B = \frac{r}{2}$. A tangent line $s$ is drawn at point $B$, and from an arbitrary point $M$ on this tangent $s$, another tangent $M D$ to the circle is drawn. Show that $\angle B M C = \frac{1}{3} \angle D M C$.
|
\angle BMC = \frac{1}{3} \angle DMC
|
1/8
|
Four identical pieces, in the shape of right triangles, were arranged in two different ways, as shown in the given figures. The squares $A B C D$ and $E F G H$ have sides respectively equal to $3 \mathrm{~cm}$ and $9 \mathrm{~cm}$. Determine the measure of the side of the square $I J K L$.
|
3\sqrt{5}
|
0/8
|
1. Daria Dmitrievna is preparing a test on number theory. She promised to give each student as many problems as the number of addends they create in the numerical example
$$
a_{1}+a_{2}+\ldots+a_{n}=2021
$$
where all numbers $a_{i}$ are natural numbers, greater than 10, and are palindromes (do not change if their digits are written in reverse order). If a student does not find any such example, they will receive 2021 problems on the test. What is the minimum number of problems a student can receive? (20 points)
|
3
|
5/8
|
## Task B-2.4.
Pinocchio tells the truth on Mondays and Tuesdays, always lies on Saturdays, and on the other days of the week, he either tells the truth or lies. In response to the question "What is your favorite subject at school?" over six consecutive days of the week, he gave the following answers in order: "History", "Mathematics", "Geography", "Physics", "Chemistry", "Physics". What subject does Pinocchio like the most? Explain your answer.
|
History
|
2/8
|
## Task B-1.7.
The length of the base of an isosceles triangle is $6 \mathrm{~cm}$, and the cosine of the angle at the base is $\frac{5}{13}$. Determine the radius of the circle that touches both legs of the triangle and the circle circumscribed around it.
|
\dfrac{169}{72}
|
2/8
|
8th Chinese 1993 Problem A2 α > 0 is real and n is a positive integer. What is the maximum possible value of α a + α b + α c + ... , where a, b, c, ... is any sequence of positive integers with sum n?
|
\max(n \alpha, \alpha^n)
|
1/8
|
3. In a right-angled triangle $ABC$ with hypotenuse $AB$, let $I$ and $U$ be the center of the inscribed circle and the point of tangency of this circle with the leg $BC$, respectively. Determine the ratio $|AC|:|BC|$, if the angles $\angle CAU$ and $\angle CBI$ are equal.
(Jaroslav Zhouf)
|
\dfrac{3}{4}
|
4/8
|
[ Motion problems]
The road from home to school takes Petya 20 minutes. One day, on his way to school, he remembered that he had forgotten his pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell, but if he returns home for the pen, he will, walking at the same speed, be 7 minutes late for the start of the lesson. What part of the way had he walked before he remembered the pen?
|
\dfrac{1}{4}
|
3/8
|
13. A and B are partners in a business, and together they made a profit of $a^{2}$ yuan (where $a$ is a two-digit natural number). When dividing the money, A takes 100 yuan first, then B takes 100 yuan, followed by A taking another 100 yuan, B taking another 100 yuan, and so on, until the last amount taken is less than 100 yuan. To ensure that both end up with the same total amount, the one who took more gave 35.5 yuan to the one who took less. The total profit they made has $\qquad$ possible values.
|
4
|
2/8
|
Dudeney, Amusements in Mathematics Problem 25 The common cash consists of brass coins of varying thickness, with a round, square, or triangular hold in the center. These are strung on wires like buttons. Supposing that eleven coins with round holes are worth fifteen ching-changs, that eleven with square holes are worth sixteen ching-changs, and that eleven with triangular holes are worth seventeen ching-changs, how can a Chinaman give me change for half a crown, using no coins other than the three mentioned? A ching-chang is worth exactly twopence and four-fifteenths of a ching-chang.
|
7
|
1/8
|
6. It is given that the sequence $\left(a_{n}\right)_{n=1}^{\infty}$, with $a_{1}=a_{2}=2$, is given by the recurrence relation
$$
\frac{2 a_{n-1} a_{n}}{a_{n-1} a_{n+1}-a_{n}^{2}}=n^{3}-n
$$
for all $n=2,3,4, \ldots$. Find the integer that is closest to the value of $\sum_{k=2}^{2011} \frac{a_{k+1}}{a_{k}}$.
|
3015
|
3/8
|
Subsets and Splits
Filtered Answers A-D
Retrieves 100 rows where the answer is a single letter from A to D, providing basic filtering of the dataset.