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The vertices of triangle \( \triangle ABC \) are \( A(0,0), B(0,420), C(560,0) \). A die has its six faces marked with \( A, A, B, B, C, C \). A point \(P_{1}=(k, m)\) is chosen inside \( \triangle ABC \), and subsequent points \( P_{2}, P_{3}, P_{4}, \cdots \) are generated according to the following rule: If \( P_{n} \) is already chosen, roll the die to get a label \( L, L \in \{A, B, C\} \). Then \( P_{n+1} \) is the midpoint of \( P_{n} \) and \( L \). Given \( P_{7}=(14,92) \), find the value of \( k+m \). | 344 | 0.75 |
Points \( A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6} \) divide a circle of radius 1 into six equal parts. From \( A_{1} \), a ray \( l_{1} \) is drawn in the direction of \( A_{2} \); from \( A_{2} \), a ray \( l_{2} \) is drawn in the direction of \( A_{3} \), and so on, from \( A_{6} \), a ray \( l_{6} \) is drawn in the direction of \( A_{1} \). From a point \( B_{1} \), taken on the ray \( l_{1} \), a perpendicular is dropped onto ray \( l_{6} \), from the base of this perpendicular another perpendicular is dropped onto \( l_{5} \), and so on. The base of the sixth perpendicular coincides with \( B_{1} \). Find the segment \( B_{1} A_{1} \). | 2 | 0.375 |
In a class of 45 students, all students participate in a tug-of-war. Among the remaining three events, each student participates in at least one event. It is known that 39 students participate in the shuttlecock kicking event and 28 students participate in the basketball shooting event. How many students participate in all three events? | 22 | 0.25 |
Students of Class 4(1) are lined up for a field trip. Xiaohong is the 15th person from the front of the line, and Xiaoming is the 20th person from the end of the line. There are 3 people between Xiaohong and Xiaoming. How many students are there in Class 4(1) in total? | 38 | 0.375 |
Consider the following three arithmetic sequences:
\[2, 11, 20, 29, \ldots\]
\[7, 15, 23, 31, \ldots\]
\[8, 19, 30, 41, \ldots\]
Determine the smallest (positive) common term of these three sequences! | 767 | 0.375 |
Find a six-digit number that starts with the digit 1 and such that if this digit is moved to the end, the resulting number is three times the original number. | 142857 | 0.875 |
The sheriff believes that if he catches a number of bandits on a given day that is a prime number, then he is lucky. On Monday and Tuesday, the sheriff was lucky. Starting from Wednesday, the number of bandits he caught was equal to the sum of the number caught the day before yesterday and twice the number caught the day before. What is the maximum number of consecutive days the sheriff could be lucky this week? Justify your answer. | 5 | 0.625 |
There are 7 clean sheets of paper on a table. Every minute, Vasya selects any 4 out of them and draws one star on each of the chosen sheets. Vasya wants each sheet to end up with a different number of stars (meaning no two sheets have the same number of stars). What is the minimum total number of stars Vasya will have to draw to achieve this? | 28 | 0.5 |
If the positive real numbers \( x \) and \( y \) satisfy the equation \( x^3 + y^3 + 3xy = 1 \), then find the minimum value of \( \left(x + \frac{1}{x}\right)^3 + \left(y + \frac{1}{y}\right)^3 \). | \frac{125}{4} | 0.25 |
You are trapped in ancient Japan, and a giant enemy crab is approaching! You must defeat it by cutting off its two claws and six legs and attacking its weak point for massive damage. You cannot cut off any of its claws until you cut off at least three of its legs, and you cannot attack its weak point until you have cut off all of its claws and legs. In how many ways can you defeat the giant enemy crab? (Note that the legs are distinguishable, as are the claws.) | 14400 | 0.625 |
Find the value of
$$
\frac{2^{2}}{2^{2}-1} \cdot \frac{3^{2}}{3^{2}-1} \cdot \frac{4^{2}}{4^{2}-1} \cdots \cdot \frac{2006^{2}}{2006^{2}-1} .
$$ | \frac{4012}{2007} | 0.75 |
We intersected the unit cube $ABCDEFGH$ with a plane such that it intersects the edges $AB$ and $AD$ at the internal points $P$ and $Q$, respectively, both at a distance $x$ from $A$, and it intersects the edge $BF$ at the point $R$. What is the distance $BR$ if $\angle QPR = 120^\circ$? | 1-x | 0.5 |
Given points \( A(1, 3), B(5, 4), C(3, 7), D(7, 1), E(10, 2), F(8, 6) \). If line \( l \) is parallel to line \( DF \), and the shortest distance from line \( l \) to any point inside (including the boundary) of triangle \( ABC \) is equal to the shortest distance from line \( l \) to any point inside (including the boundary) of triangle \( DEF \), find the equation of line \( l \). | 10x - 2y - 55 = 0 | 0.5 |
What is the least 6-digit natural number that is divisible by 198? | 100188 | 0.875 |
PQR Entertainment wishes to divide their popular idol group PRIME, which consists of seven members, into three sub-units - PRIME-P, PRIME-Q, and PRIME-R - with each of these sub-units consisting of either two or three members. In how many different ways can they do this, if each member must belong to exactly one sub-unit? | 630 | 0.625 |
Suppose that \( \mathbf{G} = 10^{100} \). (\( \mathbf{G} \) is known as a googol.) How many times does the digit 9 occur in the integer equal to \( \mathbf{G} - 1009^{2} \)? | 96 | 0.25 |
The bisector of angle $B A D$ of the right trapezoid $A B C D$ (with bases $A D$ and $B C$, and $\angle B A D=90^{\circ}$) intersects the lateral side $C D$ at point $E$. Find the ratio $C E: E D$ if $A D+B C=A B$. | 1:1 | 0.125 |
King Arthur has two equally wise advisors – Merlin and Percival. Each of them finds the correct answer to any question with a probability \( p \), or an incorrect answer with a probability \( q = 1-p \).
If both advisors give the same answer, the king listens to them. If they give opposite answers, the king makes a decision by flipping a coin.
One day, Arthur wondered whether he needed two advisors or if one would suffice. He called the advisors and said:
- I think the probability of making the right decisions won't decrease if I only keep one advisor and follow him. If this is correct, I should fire one of you. If not, I will leave everything as it is. Answer me, should I fire one of you?
- Whom exactly are you going to fire, King Arthur? – asked the advisors.
- If I decide to fire one of you, I will choose by drawing lots, flipping a coin.
The advisors left to think about the answer. The advisors, it is to be repeated, are equally wise, but not equally honest.
Percival is very honest and will try to give the correct answer, even if he risks being fired. But Merlin, honest in all other matters, in this situation decides to give an answer that maximizes the probability of not being fired. What is the probability that Merlin will be fired? | \frac{1}{4} | 0.5 |
Four friends came back from fishing. Each pair of them counted the sum of their catches. They obtained six numbers: $7, 9, 14, 14, 19, 21$. How many fish were caught in total? | 28 | 0.875 |
What is the smallest number of tetrahedrons into which a cube can be partitioned? | 5 | 0.75 |
Two positive numbers \(a\) and \(b\), with \(a > b\), are such that twice their sum is equal to three times their difference. What is the ratio \(a : b\)? | 5 : 1 | 0.625 |
On a beautiful Sunday spring morning, the father of a family went for a walk with his sons.
- Have you noticed, - he said to them, - that the age of the oldest of you is equal to the sum of the ages of your two other brothers?
- Yes. And we also noticed - they answered in unison - that the product of our ages and your age equals the sum of the cube of the number of your sons, multiplied by a thousand, and ten times the square of this number.
Can you determine from this conversation the age of the father at the time of the birth of the second son? | 34 | 0.25 |
In right triangle \( \triangle ABC \) with \( \angle C = 90^{\circ} \) and \( AB = 1 \), point \( E \) is the midpoint of side \( AB \), and \( CD \) is the altitude from \( C \) to \( AB \). Find the maximum value of \( (\overrightarrow{CA} \cdot \overrightarrow{CD}) \cdot (\overrightarrow{CA} \cdot \overrightarrow{CE}) \). | \frac{2}{27} | 0.75 |
From the consecutive natural numbers \(1, 2, 3, \ldots, 2014\), select \(n\) numbers such that any two of the selected numbers do not include one number being seven times another. Find the maximum value of \(n\) and provide a justification. | 1763 | 0.25 |
There are 200 computers in a computer center, some of which are connected by cables, with a total of 345 cables used. We define a "cluster" as a set of computers such that a signal can travel from any computer in the set to all other computers in the set, possibly through intermediate computers. Initially, all the computers formed a single cluster. One night, a malicious hacker cut several cables, resulting in the formation of 8 clusters. Find the maximum possible number of cables that were cut. | 153 | 0.875 |
Inside triangle \( ABC \), a point \( D \) is chosen such that \(\angle BAD = 60^\circ\) and \(\angle ABC = \angle BCD = 30^\circ\). It is known that \( AB = 15 \) and \( CD = 8 \). Find the length of segment \( AD \). If necessary, round the answer to 0.01 or write the answer as a common fraction. | 3.5 | 0.125 |
In rectangle \(ABCD\), a point \(E\) is marked on the extension of side \(CD\) beyond point \(D\). The bisector of angle \(ABC\) intersects side \(AD\) at point \(K\), and the bisector of angle \(ADE\) intersects the extension of side \(AB\) at point \(M\). Find \(BC\) if \(MK = 8\) and \(AB = 3\). | \sqrt{55} | 0.625 |
The diagonals of a convex quadrilateral \(ABCD\) intersect at point \(E\). It is known that the area of each of the triangles \(ABE\) and \(DCE\) is equal to 1, and the area of the entire quadrilateral does not exceed 4. Given that \(AD = 3\), find the side \(BC\). | 3 | 0.75 |
Given a geometric sequence $\{a_{n}\}$ that satisfies $\lim_{n \rightarrow +\infty} \sum_{i=1}^{n} a_{4} = 4$ and $\lim_{n \rightarrow +\infty} \sum_{i=1}^{n} a_{i}^{2} = 8$, determine the common ratio $q$. | \frac{1}{3} | 0.875 |
If \( x_1, x_2, x_3, x_4, \) and \( x_5 \) are positive integers that satisfy \( x_1 + x_2 + x_3 + x_4 + x_5 = x_1 x_2 x_3 x_4 x_5 \), that is the sum is the product, find the maximum value of \( x_5 \). | 5 | 0.5 |
There are \( n \) pieces of paper, each containing 3 different positive integers no greater than \( n \). Any two pieces of paper share exactly one common number. Find the sum of all the numbers written on these pieces of paper. | 84 | 0.375 |
In a tournament with 25 chess players, each having different skill levels and where the stronger player always wins:
What is the minimum number of matches required to determine the two strongest players? | 28 | 0.25 |
If the value of the expression $(\square + 121 \times 3.125) \div 121$ is approximately 3.38, what natural number should be placed in $\square$? | 31 | 0.875 |
Given an integer sequence $a_{1}, a_{2}, \cdots, a_{10}$ satisfying $a_{10}=3 a_{1}$, $a_{2}+a_{8}=2 a_{5}$, and
$$
a_{i+1} \in\left\{1+a_{i}, 2+a_{i}\right\}, i=1,2, \cdots, 9,
$$
find the number of such sequences. | 80 | 0.25 |
At 8:30 AM, a helicopter was over point $A$. After flying in a straight line for $s$ kilometers, the helicopter found itself over point $B$. After hovering over point $B$ for 5 minutes, the helicopter set off on the return course along the same route. The helicopter returned to point $A$ at 10:35 AM. The helicopter flew with the wind from $A$ to $B$ and against the wind on the return. The wind speed remained constant the entire time. Determine the wind speed, given that the helicopter's own speed was also constant and equal to $\mathrm{v}$ km/h in the absence of wind.
Under what conditions among the given quantities does the problem have a solution? | \sqrt{v(v - s)} | 0.25 |
Find the volume of the body $\Omega$ bounded by the surfaces
$$
z=\frac{9}{2} \sqrt{x^{2}+y^{2}}, \quad z=\frac{11}{2}-x^{2}-y^{2}
$$ | V = 2\pi | 0.5 |
Several hundred years ago, Columbus discovered the Americas. The four digits of the year he discovered the new continent are all different and their sum is 16. If you add 1 to the tens digit, it will be exactly 5 times the units digit. In which year did Columbus discover the Americas? | 1492 | 0.75 |
Given integers \( m \) and \( n \) such that \( m, n \in \{1, 2, \cdots, 1981\} \) and
\[
\left(n^{2} - m n - m^{2}\right)^{2} = 1,
\]
find the maximum value of \( m^{2} + n^{2} \). | 3524578 | 0.5 |
The points $A, B, C, D$ are consecutive vertices of a regular polygon, and the following equation holds:
$$
\frac{1}{A B}=\frac{1}{A C}+\frac{1}{A D}
$$
How many sides does the polygon have? | 7 | 0.625 |
Given a plane's first (second) trace line and the angle that this plane forms with the second (first) projection plane, find the plane's second (first) trace line, assuming that the intersection point of the given trace line with the projection axis lies outside the paper's boundary. How many solutions does this problem have? | 2 | 0.375 |
For a natural number $N$, if at least five out of the nine natural numbers $1-9$ can divide $N$, then $N$ is called a "five-divisible number". What is the smallest "five-divisible number" greater than 2000? | 2004 | 0.5 |
Let \( p(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \ldots + a_{0} \), where each \( a_{i} \) is either 1 or -1. Let \( r \) be a root of \( p \). If \( |r| > \frac{15}{8} \), what is the minimum possible value of \( n \)? | 4 | 0.5 |
Given a regular tetrahedron \(ABCD\) with edge length 2, there is a point \(P\) on edge \(AB\) such that \(AP < 1\). A cut is made through point \(P\) perpendicular to the edge \(AB\) and continues through the faces but stops at a certain point. When the cut stops, the length of the cut on face \(ABD\) is \(PM = 1\) and the length of the cut on face \(ABC\) is \(PN = \frac{2}{3}\). Find the length \(MN\). | MN = 1 | 0.625 |
A circle is tangent to the extensions of two sides \(AB\) and \(AD\) of a square \(ABCD\) with side length \(2\sqrt{3} \text{ cm}\). From point \(C\), two tangents are drawn to this circle. Find the radius of the circle given that the angle between the tangents is \(30^{\circ}\), and it is known that \(\sin 15^{\circ} = \frac{\sqrt{3}-1}{2\sqrt{2}}\). | 2 | 0.125 |
Point \( O \) is the center of the circle circumscribed around triangle \( ABC \) with sides \( AB = 8 \) and \( AC = 5 \). Find the length of side \( BC \) if the length of the vector \(\overrightarrow{OA} + 3 \overrightarrow{OB} - 4 \overrightarrow{OC}\) is 10. | 4 | 0.5 |
Martin decided to spend all his savings on sweets. He found out that he could buy three cream puffs and $3 \mathrm{dl}$ of Kofola, or $18 \mathrm{dkg}$ of yogurt raisins, or $12 \mathrm{dkg}$ of yogurt raisins and half a liter of Kofola. In the end, he bought one cream puff and $6 \mathrm{dl}$ of Kofola. How many grams of yogurt raisins does he have left over? | 60 | 0.5 |
Find the smallest positive integer $n$ that satisfies the following two properties:
1. $n$ has exactly 144 distinct positive divisors.
2. Among the positive divisors of $n$, there are ten consecutive integers. | 110880 | 0.25 |
Given positive integers \( x, y, z \) and real numbers \( a, b, c, d \) such that \( x \leqslant y \leqslant z, x^a = y^b = z^c = 70^d \), and \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d} \), determine the relationship between \( x + y \) and \( z \). Fill in the blank with “ \( > \)”, “ \( < \)” or “ \( = \)”. | = | 0.25 |
Two faces of a tetrahedron are equilateral triangles with side lengths of one unit, and the other two faces are isosceles right triangles. What is the volume of the tetrahedron? | \frac{\sqrt{2}}{12} | 0.875 |
Find the smallest real number $\lambda$ such that
\[
\left(a_{1}^n + \prod_{i=1}^{n} a_{i}, a_{2}^n + \prod_{i=1}^{n} a_{i}, \ldots, a_{n}^n + \prod_{i=1}^{n} a_{i}\right) \leq \lambda\left(a_{1}, a_{2}, \ldots, a_{n}\right)^n
\]
holds for any positive odd number $n$ and any $n$ positive integers $a_{1}, a_{2}, \ldots, a_{n}$. | 2 | 0.875 |
Given \( x, y \in [0,+\infty) \) and satisfying \( x^{3} + y^{3} + 3xy = 1 \).
Find the maximum value of \( x^{2} y \). | \frac{4}{27} | 0.25 |
Given that the first pocket contains 2 white balls, 3 red balls, and 5 yellow balls, and the second pocket contains 2 white balls, 4 red balls, and 4 yellow balls, if one ball is drawn from each pocket, what is the probability that the colors of the drawn balls are different? | \frac{16}{25} | 0.875 |
The diagonal of a rectangular parallelepiped is 13, and the diagonals of its side faces are $4 \sqrt{10}$ and $3 \sqrt{17}$.
Find its volume. | 144 | 0.875 |
The newly admitted freshmen at the university were divided into study groups so that each group had an equal number of students. Due to a reduction in the number of specializations, the number of groups decreased by 9, and all freshmen were redistributed into groups; the groups again became equal in number, with each having less than 30 students. It is known that there were 2376 freshmen in total. How many groups are there now?
*Answer: 99.* | 99 | 0.625 |
The sum of four consecutive primes is itself prime. What is the largest of the four primes?
A) 37
B) 29
C) 19
D) 13
E) 7 | 7 | 0.875 |
In an acute-angled triangle \( ABC \), points \( D, E \), and \( F \) are the feet of the perpendiculars from \( A, B \), and \( C \) onto \( BC, AC \), and \( AB \), respectively. Suppose \(\sin A = \frac{3}{5}\) and \( BC = 39 \). Find the length of \( AH \), where \( H \) is the intersection of \( AD \) with \( BE \). | 52 | 0.875 |
A coach decided to award 12 students who ran the distance in the best time. Each of them needs to be awarded with a "gold," "silver," or "bronze" medal. All three types of medals must be used (at least one of each), and a student who finished earlier cannot be awarded a less valuable medal than the one who finished later.
How many ways can the coach distribute the medals (assuming all runners have different times)? | 55 | 0.625 |
Given 10 positive integers, the sums of any 9 of them take exactly 9 different values: 86, 87, 88, 89, 90, 91, 93, 94, 95. After arranging these 10 positive integers in descending order, find the sum of the 3rd and the 7th numbers. | 22 | 0.5 |
Find all solutions to the numerical puzzle
$$
\mathrm{AX}+\mathrm{YX}=\mathrm{YPA}
$$
(different letters correspond to different digits, and identical letters correspond to identical digits). | 89 + 19 = 108 | 0.125 |
Let \( S(n) \) be the sum of the digits in the decimal representation of the number \( n \). Find \( S\left(S\left(S\left(S\left(2017^{2017}\right)\right)\right)\right) \). | 1 | 0.5 |
X is a set with n elements. Show that we cannot find more than \(2^{n-1}\) subsets of X such that every pair of subsets has non-empty intersection. | 2^{n-1} | 0.75 |
In the multiplication problem shown in the figure, \( A, B, C, D, E, F, G, H, I \) each represent a different single-digit number. What is the five-digit number represented by "FIGAA"?
```
ABC
× DC
-----
BEA
------
FIGAA
``` | 15744 | 0.375 |
Two circles of the same radius 9 intersect at points \(A\) and \(B\). A point \(C\) is chosen on the first circle, and a point \(D\) is chosen on the second circle. It turns out that point \(B\) lies on the segment \(CD\), and \(\angle CAD = 90^\circ\). On the perpendicular to \(CD\) passing through point \(B\), a point \(F\) is chosen such that \(BF = BD\) (points \(A\) and \(F\) are on opposite sides of the line \(CD\)). Find the length of the segment \(CF\). | 18 | 0.75 |
In the expression $(1+x+y)^{20}$, after expanding the brackets but not combining like terms, how many terms will there be? | 3^{20} | 0.5 |
The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{1}{3}$, and for any $n \in \mathbf{N}^{*}, a_{n+1}=a_{n}^{2}+a_{n}$. Determine the integer part of $\sum_{n=1}^{2016} \frac{1}{a_{n}+1}$. | 2 | 0.5 |
Divide the sides of a unit square \(ABCD\) into 5 equal parts. Let \(D'\) denote the second division point from \(A\) on side \(AB\), and similarly, let the second division points from \(B\) on side \(BC\), from \(C\) on side \(CD\), and from \(D\) on side \(DA\) be \(A'\), \(B'\), and \(C'\) respectively. The lines \(AA'\), \(BB'\), \(CC'\), and \(DD'\) form a quadrilateral.
What is the area of this quadrilateral? | \frac{9}{29} | 0.75 |
A group of \( n \) students doing an art project used red, blue, and yellow paint. Every student used at least one color, and some students used more than one color.
- The yellow paint was used by a total of 46 students.
- The red paint was used by a total of 69 students.
- The blue paint was used by a total of 104 students.
- Exactly 14 students used both yellow and blue and did not use red.
- Exactly 13 students used both yellow and red and did not use blue.
- Exactly 19 students used both blue and red and did not use yellow.
- Exactly 16 students used all three colors.
What is the value of \( n \)? | 141 | 0.75 |
Find the shortest distance from the line \(3x + 4y = 25\) to the circle \(x^2 + y^2 = 6x - 8y\). | \frac{7}{5} | 0.625 |
If the real numbers \( x \) and \( y \) satisfy \(\frac{x + y}{1 - xy} = \sqrt{5}\), then what is the value of \(\frac{|1 - xy|}{\sqrt{1 + x^2} \sqrt{1 + y^2}}?\) | \frac{\sqrt{6}}{6} | 0.5 |
In quadrilateral \(ABCD\), it is known that \(AB = BC\) and \(\angle ABC = \angle ADC = 90^{\circ}\). From vertex \(B\), a perpendicular \(BH\) is dropped to side \(AD\). Find the area of quadrilateral \(ABCD\) given that \(BH = h\). | h^2 | 0.875 |
Let the functions \( f(\alpha, x) \) and \( g(\alpha) \) be defined as
\[ f(\alpha, x)=\frac{\left(\frac{x}{2}\right)^{\alpha}}{x-1} \]
\[ g(\alpha)=\left.\frac{d^{4} f}{d x^{4}}\right|_{x=2} \]
Then \( g(\alpha) \) is a polynomial in \( \alpha \). Find the leading coefficient of \( g(\alpha) \). | \frac{1}{16} | 0.875 |
Find all finite non-empty sets $S$ consisting of positive integers that satisfy the condition: If $m, n \in S$, then $\frac{m+n}{(m,n)} \in S$ (where $m$ and $n$ do not need to be distinct). | S = \{2\} | 0.75 |
How many positive integers divide at least one of \( 10^{40} \) and \( 20^{30} \)? | 2301 | 0.5 |
A positive integer is called mystical if it has at least two digits and every pair of two consecutive digits, read from left to right, forms a perfect square. For example, 364 is a mystical integer because 36 and 64 are both perfect squares, but 325 is not mystical because 32 is not a perfect square. What is the largest mystical integer? | 81649 | 0.625 |
Let $a \geq 2$ be a real number, and let $x_1$ and $x_2$ be the roots of the equation $x^{2} - a x + 1 = 0$. Define $S_{n} = x_{1}^{n} + x_{2}^{n}$ for $n = 1, 2, \cdots$.
(1) Determine the monotonicity of the sequence $\left\{\frac{S_{n}}{S_{n+1}}\right\}$ and provide a proof.
(2) Find all real numbers $a$ such that for any positive integer $n$, certain conditions are satisfied. | a = 2 | 0.125 |
Find all strictly positive integers \( (a, b, p) \) with \( p \) being a prime such that \( 2^{a} + p^{b} = 19^{a} \). | (1, 1, 17) | 0.5 |
The lateral edges of a triangular pyramid are pairwise perpendicular and equal to \(a\), \(b\), and \(c\). Find the volume of the pyramid. | \frac{1}{6} abc | 0.125 |
Given natural numbers \( m \) and \( n \). There are two piles on the table: the first pile contains \( n \) stones, and the second pile contains \( m \) stones. Petya and Vasya play the following game. Petya starts. In one move, a player can break one of the available piles on the table into several smaller piles. The player who cannot make a move loses. For which values of \( m \) and \( n \) can Petya ensure victory regardless of Vasya's play? | m \neq n | 0.375 |
A car departed from point A to point B, and with some delay, a second car followed. When the first car had traveled half the distance, the second car had traveled $26 \frac{1}{4}$ km. When the second car had traveled half the distance, the first car had traveled $31 \frac{1}{5}$ km. After overtaking the first car, the second car reached point B, immediately turned back, and after driving 2 km, met the first car. Find the distance between points A and B. Give your answer as a number without units. | 58 | 0.25 |
The number \(27,000,001\) has exactly four prime factors. Find their sum. | 652 | 0.625 |
Let \( \triangle ABC \) have side lengths \(a, b, c\) with \( b > \max \{a, c\} \). Given three nonzero real numbers \( x_0, y_0, z_0 \) that satisfy the line \( ax + by + c = 0 \) passing through the point \( \left(\frac{z_0}{x_0}, \frac{2 y_0}{x_0}\right) \), and the point \( \left(\frac{z_0}{y_0}, \frac{x_0}{y_0}\right) \) lying on the ellipse \( x^2 + \frac{y^2}{4} = 1 \), find the maximum value of \( \tan B \cdot \cot C \). | \frac{5}{3} | 0.5 |
The ant starts at point A and travels $1+\frac{1}{10}$ cm north, then $2+\frac{2}{10}$ cm west, then $3+\frac{3}{10}$ cm south, then $4+\frac{4}{10}$ cm east, then $5+\frac{5}{10}$ cm north, then $6+\frac{6}{10}$ cm west, and so on. After 1000 steps, the ant reaches point B. Find the straight-line distance between points A and B in centimeters. Record the square of the found distance as your answer. | 605000 | 0.25 |
Find the largest natural number that cannot be represented as the sum of two composite numbers. (Recall that a natural number is called composite if it is divisible by some natural number other than itself and one.) | 11 | 0.625 |
How many four-digit numbers contain the digit 9 followed immediately by the digit 5? | 279 | 0.25 |
A two-digit integer between 10 and 99, inclusive, is chosen at random. Each possible integer is equally likely to be chosen. What is the probability that its tens digit is a multiple of its units (ones) digit? | \frac{23}{90} | 0.625 |
Two players, A and B, play a game: A coin is tossed repeatedly. The game ends when either the total number of heads (or tails) reaches 5. If the total number of heads reaches 5 by the end of the game, player A wins; otherwise, player B wins. What is the probability that the game will conclude in fewer than 9 tosses? | \frac{93}{128} | 0.625 |
In a triangle, the points \( a \) (a > 0) denote the incenter, and also the points of tangency of the inscribed circle with the sides respectively. Using no more than three lines with a straightedge, construct a segment of length \( a - c \). | a - c | 0.25 |
Let \( n \) be an integer greater than 2, and \( a_n \) be the largest \( n \)-digit number that is neither the sum nor the difference of two perfect squares.
(1) Find \( a_n \) (expressed as a function of \( n \)).
(2) Find the smallest value of \( n \) such that the sum of the squares of the digits of \( a_n \) is a perfect square. | n = 66 | 0.75 |
Ladybugs gathered in a meadow. If a ladybug has six spots on its back, it always tells the truth, and if it has four spots, it always lies. There were no other types of ladybugs in the meadow. The first ladybug said, "Each of us has the same number of spots on our back." The second said, "Altogether, there are 30 spots on our backs." The third contradicted, "No, altogether there are 26 spots on our backs." "Among these three, exactly one is telling the truth," stated each of the remaining ladybugs. How many ladybugs gathered in the meadow? | 5 | 0.25 |
How many solutions does the cryptarithm \(\frac{B+O+C+b+M+O+\breve{U}}{K+J+A+C+C}=\frac{22}{29}\) have, where different letters represent different digits, the same letters represent the same digits, and it is known that the digit 0 is not used? | 0 | 0.625 |
Find the cosine of the angle between the slant height and the diagonal of the base of a regular four-sided pyramid where the lateral edge is equal to the side of the base. | \frac{\sqrt{6}}{6} | 0.375 |
In a tennis tournament, 512 schoolchildren participate. For a win, 1 point is awarded, for a loss, 0 points. Before each round, pairs are drawn from participants with an equal number of points (those who do not have a pair are awarded a point without playing). The tournament ends as soon as a sole leader is determined. How many schoolchildren will finish the tournament with 6 points? | 84 | 0.25 |
Given sets \(A, B, C\) (not necessarily distinct) satisfying \(A \cup B \cup C = \{1, 2, 3, \dots, 11\}\), find the number of ordered triples \((A, B, C)\) satisfying this condition. | 7^{11} | 0.125 |
Given that one root of the equation \( x^2 - 4x + b = 0 \) is the opposite of a root of the equation \( x^2 + 4x - b = 0 \), find the positive root of the equation \( x^2 + bx - 4 = 0 \). | 2 | 0.875 |
A segment connecting the centers of two intersecting circles is divided by their common chord into segments equal to 5 and 2. Find the length of the common chord, given that the radii of the circles are in the ratio \(4: 3\). | 2\sqrt{23} | 0.875 |
Vasya built a pyramid of balls as follows: there is 1 ball at the top of the pyramid, 4 balls in the second layer from the top, and so on. The balls lie along the border and inside the pyramid. Find the total number of balls lying in the third and fifth layers from the top. | 34 | 0.875 |
The distance between docks \( A \) and \( B \) is covered by a motor ship downstream in 5 hours and upstream in 6 hours. How many hours will it take for a raft to float downstream over this distance? | 60 \text{ hours} | 0.875 |
Let \( n \) be the answer to this problem. An urn contains white and black balls. There are \( n \) white balls and at least two balls of each color in the urn. Two balls are randomly drawn from the urn without replacement. Find the probability, in percent, that the first ball drawn is white and the second is black. | 19 | 0.375 |
Three people \(A\), \(B\), and \(C\) play a game of passing a basketball from one to another. Find the number of ways of passing the ball starting with \(A\) and reaching \(A\) again on the 11th pass. For example, one possible sequence of passing is:
\[ A \rightarrow B \rightarrow A \rightarrow B \rightarrow C \rightarrow A \rightarrow B \rightarrow C \rightarrow B \rightarrow C \rightarrow B \rightarrow A . \] | 682 | 0.125 |
The set $\{[x] + [2x] + [3x] \mid x \in \mathbb{R}\} \mid \{x \mid 1 \leq x \leq 100, x \in \mathbb{Z}\}$ has how many elements, where $[x]$ denotes the greatest integer less than or equal to $x$. | 67 | 0.25 |
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