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At a sandwich shop, there are five types of bread, seven kinds of meat, and six kinds of cheese. A sandwich is made up of one type of bread, one type of meat, and one type of cheese. Turkey, salami, Swiss cheese, and rye bread are available at the shop. If a customer does not order a sandwich with a turkey/Swiss cheese combination nor a sandwich with rye bread/salami, how many different sandwiches can the customer order? | 199 | 4/8 |
Rosa received a perfume bottle in the shape of a cylinder with a base radius of $7 \mathrm{~cm}$ and a height of $10 \mathrm{~cm}$. After two weeks of using the perfume, $0.45$ liters remained in the bottle. What fraction represents the volume of perfume that Rosa has already used? | \frac{49\pi-45}{49\pi} | 1/8 |
Several circles are depicted in the picture, connected by line segments. Tanya selects a natural number \( n \) and arranges different natural numbers in the circles so that the following property holds for all these numbers:
If the numbers \( a \) and \( b \) are not connected by a line segment, then the sum \( a^{2}+b^{2} \) must be coprime with \( n \). If they are connected, the numbers \( a^{2}+b^{2} \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists? | 65 | 0/8 |
The lateral faces of a regular triangular prism are squares. Find the angle between the diagonal of a lateral face and a non-intersecting side of the base of the prism. | \arccos\frac{\sqrt{2}}{4} | 2/8 |
Let \(\omega_{1}, \omega_{2}, \ldots, \omega_{100}\) be the roots of \(\frac{x^{101}-1}{x-1}\) (in some order). Consider the set
\[
S=\left\{\omega_{1}^{1}, \omega_{2}^{2}, \omega_{3}^{3}, \ldots, \omega_{100}^{100}\right\}.
\]
Let \(M\) be the maximum possible number of unique values in \(S\), and let \(N\) be the minimum possible number of unique values in \(S\). Find \(M-N\). | 98 | 0/8 |
Every week, based on listeners' SMS votes, the ten most popular songs are selected. It is known that: 1) the same set of songs in the same order is never selected two weeks in a row; 2) a song that has once dropped in the ranking does not climb up again. What is the maximum number of weeks that the same 10 songs can remain in the ranking? | 46 | 0/8 |
In the game of Galactic Dominion, players compete to amass cards, each of which is worth a certain number of points. You are playing a version of this game with only two kinds of cards: planet cards and hegemon cards. Each planet card is worth 2010 points, and each hegemon card is worth four points per planet card held. Starting with no planet cards and no hegemon cards, on each turn (starting at turn one), you take either a planet card or a hegemon card, whichever is worth more points given your current hand. Define a sequence \(\{a_{n}\}\) for all positive integers \(n\) by setting \(a_{n}\) to be 0 if on turn \(n\) you take a planet card and 1 if you take a hegemon card. What is the smallest value of \(N\) such that the sequence \(a_{N}, a_{N+1}, \ldots\) is necessarily periodic (meaning that there is a positive integer \(k\) such that \(a_{n+k}=a_{n}\) for all \(n \geq N\))? | 503 | 1/8 |
Let \( P(n) = \left(n-1^3\right)\left(n-2^3\right) \ldots\left(n-40^3\right) \) for positive integers \( n \). Suppose that \( d \) is the largest positive integer that divides \( P(n) \) for every integer \( n > 2023 \). If \( d \) is a product of \( m \) (not necessarily distinct) prime numbers, compute \( m \). | 48 | 0/8 |
I5.1 In Figure 2, find \( a \).
I5.2 If \( b = \log_{2} \left( \frac{a}{5} \right) \), find \( b \).
I5.3 A piece of string, \( 20 \text{ m} \) long, is divided into 3 parts in the ratio of \( b-2 : b : b+2 \). If \( N \text{ m} \) is the length of the longest portion, find \( N \).
I5.4 Each interior angle of an \( N \)-sided regular polygon is \( x^{\circ} \). Find \( x \). | 144 | 0/8 |
A square and four smaller squares are contained within it, each smaller square circumscribing a circle with a radius of 3 cm. What is the area of the larger square and the total area occupied by the four smaller squares? | 144 \text{ cm}^2 | 0/8 |
Consider the polynomial $g(x) = x^{2009} + 19x^{2008} + 1$ with distinct roots $s_1, s_2, ..., s_{2009}$. A polynomial $Q$ of degree $2009$ has the property that
\[ Q\left(s_j + \dfrac{1}{s_j}\right) = 0 \] for $j = 1, \ldots, 2009$. Determine the value of $\frac{Q(1)}{Q(-1)}$. | 1 | 1/8 |
In a basketball league, the number of games won by five teams is displayed in a graph, but the team names are missing. The clues below provide information about the teams:
1. The Sharks won more games than the Falcons.
2. The Knights won more games than the Wolves, but fewer than the Dragons.
3. The Wolves won more than 22 games.
Given the bar graph data: 24 wins, 27 wins, 33 wins, 36 wins, 38 wins. How many games did the Knights win? | 33 | 4/8 |
In the Zorgian language, there are 4 words: "zor", "glib", "mek", and "troz". In a sentence, "zor" cannot come directly before "glib", and "mek" cannot come directly before "troz"; all other sentences are grammatically correct (including sentences with repeated words). How many valid 3-word sentences are there in Zorgian? | 48 | 4/8 |
Consider the polynomial equation \[ x^4 + ax^3 + bx^2 + cx + d = 0, \] where \( a, b, c, \) and \( d \) are rational numbers. This polynomial has roots \( 2-\sqrt{5} \) and \( 1 \). Find the value of \( c \) if the polynomial also has a root that is three times an integer. | -8 | 4/8 |
In the right-angled triangle $LMN$, suppose $\sin N = \frac{3}{5}$. If the length of $LM$ is 15, calculate the length of $LN$. | 25 | 4/8 |
The score on a certain 150-point test varies directly with the square of the time a student spends preparing for the test. If a student receives 90 points on a test for which she spent 2 hours preparing, what score would she receive on the next test if she spent 3 hours preparing? | 202.5 | 2/8 |
In triangle $PQR$, $PQ = PR$, and $\overline{PS}$ is an altitude. Point $T$ is on the extension of $\overline{QR}$ such that $PT = 12$. The values of $\tan \angle RPT$, $\tan \angle SPT$, and $\tan \angle QPT$ form a geometric progression, and the values of $\cot \angle SPT$, $\cot \angle RPT$, $\cot \angle SPQ$ form an arithmetic progression. Determine the area of triangle $PQR$. | 24 | 0/8 |
Humanity discovers 15 habitable planets, of which 7 are Earth-like and 8 are Mars-like. Earth-like planets require 3 units of colonization effort, whereas Mars-like planets need only 1 unit. If a total of 18 units of colonization effort is available, how many different combinations of planets can be colonized, given that all planets are unique? | 2163 | 2/8 |
Suppose in a right triangle $PQR$, with $\angle PQR = 90^\circ$, we have $\cos Q = \frac{3}{5}$. If $PR = 5$, find the length of $PQ$. | 3 | 3/8 |
14. As shown in the figure, in acute $\triangle A B C$, $A C=B C=10$, and $D$ is a point on side $A B$. The incircle of $\triangle A C D$ and the excircle of $\triangle B C D$ that is tangent to side $B D$ both have a radius of 2. Find the length of $A B$.
untranslated text remains the same as the source text. | 4\sqrt{5} | 1/8 |
Through the altitude of an equilateral triangle with side length \(a\), a plane is drawn perpendicular to the plane of the triangle, and within this plane a line \(l\) is taken parallel to the altitude of the triangle. Find the volume of the solid obtained by rotating the triangle around the line \(l\). | \dfrac{\sqrt{3}}{24} \pi a^3 | 0/8 |
Let $ABC$ be an equilateral triangle. Find all positive integers $n$, for which the function $f$, defined on all points $M$ from the circle $S$ circumscribed to triangle $ABC$, defined by the formula $f:S \rightarrow R, f(M)=MA^n+MB^n+MC^n$, is a constant function. | 2 | 0/8 |
$ABCD$ - a convex quadrilateral. It is known that $\angle CAD = \angle DBA = 40^{\circ}, \angle CAB = 60^{\circ}, \angle CBD = 20^{\circ}$. Find the angle $\angle CDB$. | 30^{\circ} | 3/8 |
23. The birth date of Albert Einstein is 14 March 1879. If we denote Monday by 1, Tuesday by 2, Wednesday by 3, Thursday by 4, Friday by 5, Saturday by 6 and Sunday by 7, which day of the week was Albert Einstein born? Give your answer as an integer from 1 to 7 . | 5 | 4/8 |
Let $\frac{3}{2} \leqslant x \leqslant 5$. Prove the inequality $2 \sqrt{x+1}+\sqrt{2x-3}+\sqrt{15-3x}<2 \sqrt{19}$. | 2 \sqrt{x+1}+\sqrt{2x-3}+\sqrt{15-3x} < 2 \sqrt{19} | 1/8 |
101. The number $N=2^{4 n+1}-4^{n}-1$ is divisible by 9 for an integer $n \geqslant 0$. The number $8 N+9$ is a perfect square.
10 | (4^{n+1} - 1)^2 | 1/8 |
## Task 2 - 191242
Let $M$ be the set of all square areas $Q$ that lie in a given plane $\epsilon$, have a given point $Z$ of the plane $\epsilon$ as their center, and have a given segment length $a$ as their side length.
For arbitrary square areas $Q, Q^{\prime}$ from this set $M$, let $u\left(Q \cup Q^{\prime}\right)$ denote the perimeter of the polygonal area that results from the intersection of the square areas $Q$ and $Q^{\prime}$.
Investigate whether there are square areas $Q, Q^{\prime}$ in $M$ with the smallest possible $u\left(Q \cap Q^{\prime}\right)$.
If this is the case, determine (in dependence on $a$) this smallest possible value of $u\left(Q \cap Q^{\prime}\right)$. | 8a(\sqrt{2} - 1) | 4/8 |
9.6. What is the smallest natural number $a$ for which there exist integers $b$ and $c$ such that the quadratic trinomial $a x^{2}+b x+c$ has two distinct positive roots, each not exceeding $\frac{1}{1000} ?$
(A. Khryabrov) | 1001000 | 3/8 |
11.4. In a row, $n$ integers are written such that the sum of any seven consecutive numbers is positive, and the sum of any eleven consecutive numbers is negative. For what largest $n$ is this possible? | 16 | 1/8 |
13 The line $l$ is the left directrix of the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, $F_{1}$ is the left focus, and the moving point $P$ is on $l$. The third vertex of the equilateral triangle with side $P F_{1}$ is $M$, and the points $P$, $F_{1}$, and $M$ are arranged counterclockwise. Find the equation of the trajectory of point $M$. | x - \sqrt{3} y + 7 = 0 | 2/8 |
3. Find the smallest natural number $k$ such that every $k$-element set of pairwise coprime three-digit numbers contains at least one prime number. | 12 | 1/8 |
1. If we consider a pair of skew lines as one pair, then among the 12 lines formed by the edges of a regular hexagonal pyramid, the number of pairs of skew lines is $\qquad$ pairs. | 24 | 4/8 |
Given a square with side length \(a\), on each side of the square, a trapezoid is constructed outside such that the top bases and the lateral sides of these trapezoids form a regular dodecagon. Compute its area. | \dfrac{3}{2} a^2 | 0/8 |
A natural number $n$ was alternately divided by $29$, $41$ and $59$. The result was three nonzero remainders, the sum of which equals $n$. Find all such $n$ | 79 | 0/8 |
Task B-3.4. A sphere is circumscribed around a right circular cone. A plane containing the center of this sphere and parallel to the base of the cone divides the cone into two parts of equal volumes. Determine the cosine of the angle at the vertex of the axial section of the cone. | \sqrt[3]{2} - 1 | 3/8 |
2.4. Find all values of $x$ for which the smallest of the numbers $8-x^{2}$ and $\operatorname{ctg} x$ is not less than -1. In the answer, record the total length of the found intervals on the number line, rounding it to hundredths if necessary. | 4.57 | 5/8 |
Four. (50 points) $n$ points $P_{1}, P_{2}, \cdots, P_{n}$ are sequentially on a straight line, each point is colored one of white, red, green, blue, or purple. If for any two adjacent points $P_{i}, P_{i+1}$ $(i=1,2, \cdots, n-1)$, either these two points are the same color, or at least one of them is white, it is called a good coloring. Find the total number of good colorings. | \dfrac{3^{n+1} - (-1)^n}{2} | 1/8 |
Bakayev E.V.
Cheburashka has a set of 36 stones with masses of 1 g, 2 g, ..., 36 g, and Shapoklyak has super glue, one drop of which can glue two stones together (thus, three stones can be glued with two drops, and so on). Shapoklyak wants to glue the stones in such a way that Cheburashka cannot select one or several stones from the resulting set with a total mass of 37 g. What is the minimum number of drops of glue she needs to achieve her goal? | 9 | 2/8 |
Rhombus $ABCD$ is similar to rhombus $BFDE$. The area of rhombus $ABCD$ is 24, and $\angle BAD=60^\circ$. What is the area of rhombus $BFDE$?
[asy]
pair A,B,C,D,I,F;
A=(0,0);
B=(10,0);
C=(15,8.7);
D=(5,8.7);
I=(5,2.88);
F=(10,5.82);
draw(A--B--C--D--cycle,linewidth(0.7));
draw(D--I--B--F--cycle,linewidth(0.7));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
label("$E$",I,W);
label("$F$",F,E);
[/asy] | 8 | 4/8 |
A square has sides of length 2. Set $\cal S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $\cal S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$. | 86 | 0/8 |
The same eight people sit in a certain church pew every week, but not always in the same order. Every week, each person hugs the people immediately to his or her left and right. How many weeks does it take (at a minimum) for every pair of people to hug at least once? | 4 | 4/8 |
In the graph below, each grid line counts as one unit. The line shown below passes through the point $(1001,n)$ (not shown on graph). Find $n$.
[asy]size(250,0);
add(shift(-10,-10)*grid(20,20));
draw((-10,0)--(10,0),linewidth(2));
draw((0,-10)--(0,10),linewidth(2));
label("x",(10,0),E);
label("y",(0,10),N);
draw((-10,-2.71) -- (10,8.71),blue,Arrows);[/asy] | 575 | 4/8 |
Find the last two digits of $\binom{200}{100}$. Express the answer as an integer between $0$ and $99$. (e.g., if the last two digits are $05$, just write $5$). | 20 | 0/8 |
Suppose that $a$, $b$, $c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of
\[20(a^2+b^2+c^2+d^2)-\sum_{\text{sym}}a^3b,\]
where the sum is over all $12$ symmetric terms. | 112 | 2/8 |
Let $p$ and $q$ be positive integers such that $p$ is a prime, $p$ divides $q-1$, and $p+q$ divides $p^2+2020q^2$. Find the sum of the possible values of $p$. | 141 | 2/8 |
There is a pack of 27 distinct cards, and each card has three values on it. The first value is a shape from $\{\Delta,\square,\odot\}$; the second value is a letter from $\{A,B,C\}$; and the third value is a number from $\{1,2,3\}$.
In how many ways can we choose an unordered set of 3 cards from the pack, so that no two of the chosen cards have two matching values?
For example, we can choose $\{\Delta A1, \Delta B2, \odot C3\}$, but we cannot choose $\{\Delta A1, \square B2, \Delta C1\}$. | 1278 | 0/8 |
On the blackboard, there are $25$ points arranged in a grid as shown in the figure below. Gastón needs to choose $4$ points that will serve as the vertices of a square. How many different ways can he make this choice?
$$
\begin{matrix}
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet \\
\bullet & \bullet & \bullet & \bullet & \bullet
\end{matrix}
$$ | 50 | 3/8 |
Let $x$ be the minimal root of the equation $x^2 - 4x + 2 = 0$. Find the first two digits after the decimal point of the number $\{x + x^2 + \cdots + x^{20}\}$, where $\{a\}$ denotes the fractional part of $a$. | 41 | 4/8 |
A polynomial $P(x)$ is a \emph{base-$n$ polynomial} if it is of the form $a_d x^d + a_{d-1} x^{d-1} + \cdots + a_1 x + a_0$, where each $a_i$ is an integer between $0$ and $n-1$ inclusive and $a_d > 0$. Find the largest positive integer $n$ such that for any real number $c$, there exists at most one base-$n$ polynomial $P(x)$ for which $P(\sqrt{2} + \sqrt{3}) = c$. | 9 | 0/8 |
Two ducks, Wat and Q, are taking a math test with $1022$ other ducklings. The test consists of $30$ questions, and the $n$th question is worth $n$ points. The ducks work independently on the test. Wat answers the $n$th problem correctly with a probability of $\frac{1}{n^2}$, while Q answers the $n$th problem correctly with a probability of $\frac{1}{n+1}$. Unfortunately, the remaining ducklings each answer all $30$ questions incorrectly.
Just before turning in their test, the ducks and ducklings decide to share answers. For any question on which Wat and Q have the same answer, the ducklings change their answers to agree with them. After this process, what is the expected value of the sum of all $1024$ scores? | 1020 | 3/8 |
On Thursday, January 1, 2015, Anna buys one book and one shelf. For the next two years, she buys one book every day and one shelf on alternate Thursdays, so she next buys a shelf on January 15th. On how many days in the period from Thursday, January 1, 2015, until (and including) Saturday, December 31, 2016, is it possible for Anna to arrange all her books on all her shelves such that there is an equal number of books on each shelf? | 89 | 1/8 |
For every integer $a > 1$, an infinite list of integers is constructed $L(a)$, as follows:
1. $a$ is the first number in the list $L(a)$.
2. Given a number $b$ in $L(a)$, the next number in the list is $b + c$, where $c$ is the largest integer that divides $b$ and is smaller than $b$.
Find all the integers $a > 1$ such that $2002$ is in the list $L(a)$. | 2002 | 3/8 |
Mr. Jones teaches algebra. He has a whiteboard with a pre-drawn coordinate grid that runs from $-10$ to $10$ in both the $x$ and $y$ coordinates. Consequently, when he illustrates the graph of a quadratic, he likes to use a quadratic with the following properties:
1. The quadratic sends integers to integers.
2. The quadratic has distinct integer roots both between $-10$ and $10$, inclusive.
3. The vertex of the quadratic has integer $x$ and $y$ coordinates both between $-10$ and $10$, inclusive.
How many quadratics are there with these properties? | 478 | 4/8 |
Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and
\[
a_{n} = 2 a_{n-1}^2 - 1
\]
for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality
\[
a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}.
\]
What is the value of $100c$, rounded to the nearest integer? | 167 | 5/8 |
For an integer $k$, let $T_k$ denote the number of $k$-tuples of integers $(x_1, x_2, \ldots, x_k)$ with $0 \le x_i < 73$ for each $i$, such that $73 \mid x_1^2 + x_2^2 + \ldots + x_k^2 - 1$. Compute the remainder when $T_1 + T_2 + \ldots + T_{2017}$ is divided by $2017$. | 2 | 1/8 |
There are 6 positive integers \( a, b, c, d, e, f \) arranged in order, forming a sequence, where \( a=1 \). If a positive integer is greater than 1, then the number that is one less than it must appear to its left. For example, if \( d > 1 \), then one of \( a, b, \) or \( c \) must have the value \( d-1 \). For instance, the sequence \( 1,1,2,1,3,2 \) satisfies the requirement; \( 1,2,3,1,4,1 \) satisfies the requirement; \( 1,2,2,4,3,2 \) does not satisfy the requirement. How many different permutations satisfy the requirement? | 203 | 3/8 |
Doug and Ryan are competing in the 2005 Wiffle Ball Home Run Derby. In each round, each player takes a series of swings. Each swing results in either a home run or an out, and an out ends the series. When Doug swings, the probability that he will hit a home run is \( \frac{1}{3} \). When Ryan swings, the probability that he will hit a home run is \( \frac{1}{2} \). In one round, what is the probability that Doug will hit more home runs than Ryan? | \frac{1}{5} | 5/8 |
Let \(a \neq b\) be positive real numbers and \(m, n\) be positive integers. An \(m+n\)-gon \(P\) has the property that \(m\) sides have length \(a\) and \(n\) sides have length \(b\). Further suppose that \(P\) can be inscribed in a circle of radius \(a+b\). Compute the number of ordered pairs \((m, n)\), with \(m, n \leq 100\), for which such a polygon \(P\) exists for some distinct values of \(a\) and \(b\). | 940 | 1/8 |
Some \(1 \times 2\) dominoes, each covering two adjacent unit squares, are placed on a board of size \(n \times n\) so that no two of them touch (not even at a corner). Given that the total area covered by the dominoes is 2008, find the least possible value of \(n\). | 77 | 0/8 |
One hundred points labeled 1 to 100 are arranged in a \(10 \times 10\) grid such that adjacent points are one unit apart. The labels are increasing left to right, top to bottom (so the first row has labels 1 to 10, the second row has labels 11 to 20, and so on).
Convex polygon \(\mathcal{P}\) has the property that every point with a label divisible by 7 is either on the boundary or in the interior of \(\mathcal{P}\). Compute the smallest possible area of \(\mathcal{P}\). | 63 | 2/8 |
If \( a \) is the sum of all the positive factors of 2002, find the value of \( a \).
It is given that \( x>0, y>0 \) and \( \sqrt{x}(\sqrt{x}+\sqrt{y})=3 \sqrt{y}(\sqrt{x}+5 \sqrt{y}) \). If \( b=\frac{2 x+\sqrt{x y}+3 y}{x+\sqrt{x y}-y} \), find the value of \( b \).
Given that the equation \(||x-2|-1|=c\) has only 3 integral solutions, find the value of \( c \).
If \( d \) is the positive real root of the equation \(\frac{1}{2}\left\{\frac{1}{2}\left[\frac{1}{2}\left(\frac{1}{2} x^{2}+2\right)+2\right]+2\right\}=2\), find the value of \( d \). | 2 | 4/8 |
Let $n$ be a strictly positive integer. Denote by $\mathfrak{S}_{n}$ the set of bijections from the set $\{1,2, \ldots, n\}$ to itself. Let $\sigma \in \mathfrak{S}_{n}$; we say that $(i, j)$ (for $1 \leq i, j \leq n$) is an inversion of $\sigma$ if $i\sigma(j)$. We also say that $i$ (for $1 \leq i \leq n$) is a record of $\sigma$ if for all $j$ with $1 \leq j \leq i-1$ we have $\sigma(j)<\sigma(i)$ (thus, 1 is always a record). Finally, denote $\operatorname{inv}(\sigma)$ the number of inversions of $\sigma$ and $\operatorname{rec}(\sigma)$ its number of records.
Calculate the following sum:
$$
\sum_{\sigma \in \mathfrak{S}_{n}} 2^{\operatorname{inv}(\sigma)+\operatorname{rec}(\sigma)}
$$ | 2^{n(n+1)/2} | 0/8 |
Consider two intersecting circles of radius $r$, whose centers are a distance $O_{1} O_{2}=1$ apart. The common chord is bisected at point $O$. A ray starting from $O$ and making an angle $\alpha$ with the common chord intersects the given circles at points $A$ and $B$. What is the length of $A B$? | \sin \alpha | 2/8 |
Pat and Mat dug a well. On the first day, Pat dug a hole $40 \mathrm{~cm}$ deep. The second day, Mat continued and dug to three times the depth. On the third day, Pat dug as much as Mat did the previous day and hit water. At that moment, the ground was $50 \mathrm{~cm}$ above the top of his head.
Determine how tall Pat was.
(M. Dillingerová) | 150 | 0/8 |
Let $a>1$ be a positive integer. The sequence of natural numbers $\{a_n\}_{n\geq 1}$ is defined such that $a_1 = a$ and for all $n\geq 1$, $a_{n+1}$ is the largest prime factor of $a_n^2 - 1$. Determine the smallest possible value of $a$ such that the numbers $a_1$, $a_2$,$\ldots$, $a_7$ are all distinct. | 46 | 2/8 |
5. In a row, all natural numbers from 1 to 100 inclusive are written in ascending order. Under each number in this row, the product of its digits is written. The same procedure is applied to the resulting row, and so on. How many odd numbers will be in the fifth row? | 19 | 0/8 |
3. If on the interval $[1,2]$, the function $f(x)=x^{2}+p x+q$ and $g(x)=x+\frac{1}{x^{2}}$ take the same minimum value at the same point, then the maximum value of $f(x)$ on this interval is $(\quad)$. | 4 + \sqrt[3]{4} - \dfrac{5}{2}\sqrt[3]{2} | 0/8 |
Consider the natural numbers from 0 to $2^n - 1$. These numbers are divided into two categories: $A$ contains all numbers whose binary representation has an odd number of 1s, and $B$ contains all the other numbers. Find the value of $\sum_{x \in A} x^n - \sum_{y \in B} y^n$. | (-1)^{n-1} \cdot n! \cdot 2^{\frac{n(n-1)}{2}} | 0/8 |
Problem 10.1. Find all real numbers $x$ such that $\tan \left(\frac{\pi}{12}-x\right)$, $\tan \frac{\pi}{12}$ and $\tan \left(\frac{\pi}{12}+x\right)$ form (in some order) a geometric progression. | x = k\pi | 1/8 |
On the side $AB$ of the convex quadrilateral $ABCD$, a point $M$ is chosen such that $\angle AMD = \angle ADB$ and $\angle ACM = \angle ABC$. The tripled square of the ratio of the distance from point $A$ to the line $CD$ to the distance from point $C$ to the line $AD$ is $2$, and $CD = 20$. Find the radius of the inscribed circle of triangle $ACD$. | 4\sqrt{10} - 2\sqrt{15} | 1/8 |
1. Given that $a$, $b$, $c$, and $d$ are prime numbers, and $a b c d$ is the sum of 77 consecutive positive integers. Then the minimum value of $a+b+c+d$ is $\qquad$ | 32 | 5/8 |
26. A sphere passes through the vertices of one face of a cube and touches the sides of the opposite face of the cube. Find the ratio of the volumes of the sphere and the cube. | \dfrac{41\sqrt{41}}{384} \pi | 2/8 |
44. As shown in the figure, on the graph of the inverse proportion function $y=\frac{2}{x}(x>0)$, there are points $P_{1}, P_{2}, P_{3}, P_{4}$, whose x-coordinates are $1,2,3,4$ respectively. Perpendicular lines to the x-axis and y-axis are drawn through these points, and the areas of the shaded parts from left to right in the figure are $S_{1}, S_{2}, S_{3}$, respectively. Then $S_{1}+S_{2}+S_{3}=$ $\qquad$ . | \dfrac{3}{2} | 0/8 |
Lynne is tiling her long and narrow rectangular front hall. The hall is 2 tiles wide and 13 tiles long. She is going to use exactly 11 black tiles and exactly 15 white tiles. Determine the number of distinct ways of tiling the hall so that no two black tiles are adjacent (that is, share an edge).
## PART B
For each question in Part B, your solution must be well organized and contain words of explanation or justification when appropriate. Marks are awarded for completeness, clarity, and style of presentation. A correct solution, poorly presented, will not earn full marks. | 486 | 0/8 |
For a triangle with angles $\alpha, \beta, \gamma$,
$$
\sin \alpha+\sin \beta=(\cos \alpha+\cos \beta) \sin \gamma
$$
is satisfied. What is the measure of angle $\gamma$? | 90^\circ | 0/8 |
In an ancient cipher known as "Scytale," a strip of papyrus was wound around a cylindrical rod, coil to coil, without gaps or overlaps. Then, with the rod in a horizontal position, the text of the message was written line by line on the papyrus. After that, the strip of papyrus with the written text was sent to the recipient, who had an identical rod, allowing them to read the message. We received a message encrypted using the "Scytale" cipher. However, its author, concerned with ensuring even lines, drew horizontal lines while writing, which remained on the strip in the form of dashes between the letters. The angle of inclination of these dashes to the edge of the strip is $\alpha$, the width of the strip is $d$, and the width of each line is $h$. Indicate how, using the given data, to read the text. | \frac{d}{h} | 2/8 |
1. Two trains, each containing 15 identical cars, were moving towards each other at constant speeds. Exactly 28 seconds after the first cars of the trains met, passenger Sasha, sitting in the third car, passed passenger Valera from the oncoming train, and another 32 seconds later, the last cars of these trains had completely passed each other. In which car, by count, was Valera traveling?
Answer: 12. | 12 | 3/8 |
A rope of length 2018 is subjected to the following operation: it is first divided into two ropes of positive integer lengths, and the lengths of the two ropes are recorded. One of these ropes is then selected and the procedure is repeated until there are 2018 ropes each of length 1. If the lengths of the two ropes obtained in a particular operation are not equal, then that operation is called "good".
(1) Find the maximum and minimum number of good operations;
(2) Prove that in all processes in which the number of good operations reaches its minimum, the number of different lengths of the ropes recorded is the same. | 2016 | 0/8 |
In the quadrilateral $A B C D$, $A B=C D, A B C \angle=90^{\circ}$ and $B C D \angle=100^{\circ}$. The intersection point of the perpendicular bisectors of sides $A D$ and $B C$ is $M$. Determine the angle $B M C$.
---
The translation is provided as requested, maintaining the original formatting and structure. | 10^\circ | 5/8 |
Two lines intersect at an angle $\gamma$. A grasshopper jumps from one line to the other; the length of each jump is 1 meter, and the grasshopper does not jump back unless it is possible. Prove that the sequence of jumps is periodic if and only if $\gamma / \pi$ is a rational number. | \gamma / \pi \text{ is a rational number} | 1/8 |
6. The base of the pyramid $TABC$ is the triangle $ABC$, all sides of which are equal to $\sqrt{3}$, and the height of the pyramid coincides with the lateral edge $TA$. Find the area of the section of the pyramid by a plane passing through the center of the sphere circumscribed around the pyramid, forming an angle of $60^{\circ}$ with the base plane, intersecting the edge $AB$ at point $M$ such that $MB=2AM$, and intersecting the edge $BC$. It is known that the distance from point $A$ to the section plane is 0.25. | \dfrac{11\sqrt{3}}{30} | 0/8 |
Given a quadratic function \( f(x) = ax^2 + bx + 2 \) (where \( a \neq 0 \)), and for any real number \( x \), it holds that \( |f(x)| \geq 2 \). Find the coordinates of the focus of this parabolic curve. | \left(0, 2 + \dfrac{1}{4a}\right) | 4/8 |
Let $K$ and $N>K$ be fixed positive integers. Let $n$ be a positive integer and let $a_{1}, a_{2}, \ldots, a_{n}$ be distinct integers. Suppose that whenever $m_{1}, m_{2}, \ldots, m_{n}$ are integers, not all equal to 0, such that $\left|m_{i}\right| \leqslant K$ for each $i$, then the sum
$$
\sum_{i=1}^{n} m_{i} a_{i}
$$
is not divisible by $N$. What is the largest possible value of $n$?
## Proposed by North Macedonia | \left\lfloor \log_{K+1} N \right\rfloor | 1/8 |
10. In a square lawn with side length $a$, there are sprinkler devices installed at each of the four corners. The sprinkler devices can rotate to spray water at a $90^{\circ}$ angle, and each sprinkler can rotate from one side of its corner to the other side, with an effective range of $a$. What is the proportion of the lawn that can be covered simultaneously by all four sprinklers to the entire lawn? $\qquad$ . | 1 + \dfrac{\pi}{3} - \sqrt{3} | 0/8 |
Problem 2. (2 points)
The sum of the sines of five angles from the interval $\left[0 ; \frac{\pi}{2}\right]$ is 3. What are the greatest and least integer values that the sum of their cosines can take? | 4 | 1/8 |
## Task A-3.3.
Given is a triangle $A B C$. Let point $D$ be the foot of the altitude from vertex $A$, and point $E$ be the intersection of the angle bisector of $\varangle C B A$ with the opposite side. If $\varangle B E A=45^{\circ}$, determine $\varangle E D C$. | 45^\circ | 4/8 |
## 31. Child Psychology
The brightest hours of a little child are spent drawing. He depicts either Indians or Eskimos and draws a dwelling for each of them. Thus, next to the Eskimos, he usually draws an igloo, and next to the Indians - wigwams. But sometimes the child makes mistakes, and not infrequently, for example, an igloo can be found next to the Indians. One female psychologist took these drawings very seriously; she discovered that there were twice as many Indians as Eskimos, that there were as many Eskimos with wigwams as there were Indians with igloos, and that, speaking of Eskimos, one wigwam corresponds to three igloos. Then she decided to determine what part of the inhabitants of the wigwams are Indians (and derive from this a very complex IQ $^{1}$ formula for the child). But she has not yet succeeded in doing so. Help her. | \dfrac{7}{8} | 0/8 |
Example 3 In $\triangle A B C$, $A B=A C, \angle A=$ $20^{\circ}, D$ is a point on $A C$, $\angle D B C=60^{\circ}, E$ is a point on $A B$, $\angle E C B=50^{\circ}$. Find the degree measure of $\angle B D E$. | 30 | 5/8 |
Question As shown in Figure 1, two rectangles $R$ and $S$ are placed inside an acute triangle $T$. Let $A(X)$ denote the area of polygon $X$. Find the maximum value of $\frac{A(R)+A(S)}{A(T)}$ or prove that the maximum value does not exist. | \dfrac{2}{3} | 5/8 |
4. Place 27 balls numbered $1 \sim 27$ into three bowls, Jia, Yi, and Bing, such that the average values of the ball numbers in bowls Jia, Yi, and Bing are $15$, $3$, and $18$, respectively, and each bowl must contain no fewer than 4 balls. Then the maximum value of the smallest ball number in bowl Jia is $\qquad$ | 10 | 2/8 |
## Task A-2.5.
Let $n$ be a natural number. If a regular $n$-gon is divided into $n-2$ triangles by drawing $n-3$ diagonals that do not have common interior points, we say that we have obtained a triangulation. A triangulation of an $n$-gon, some of whose vertices are red, is good if each of those $n-2$ triangles has at least two red vertices.
Determine the smallest natural number $k$, depending on $n$, such that we can color $k$ vertices of a regular $n$-gon red so that there exists at least one good triangulation. | \left\lceil \frac{n}{2} \right\rceil | 5/8 |
3.95. The base of the pyramid is an isosceles triangle with an angle $\alpha$ between the lateral sides. The pyramid is placed in a certain cylinder such that its base is inscribed in the base of this cylinder, and the vertex coincides with the midpoint of one of the cylinder's generators. The volume of the cylinder is $V$. Find the volume of the pyramid. | \dfrac{V \sin \alpha \cos^2 \dfrac{\alpha}{2}}{3\pi} | 0/8 |
4. Quadrilateral $ABCD$ is inscribed in a circle. The tangents to this circle, drawn at points $A$ and $C$, intersect on the line $BD$. Find the side $AD$, if $AB=2$ and $BC: CD=4: 5$.
Answer: $\frac{5}{2}$. | \dfrac{5}{2} | 5/8 |
$\underline{115625}$ topics: [ Tangent circles [Mean proportionals in a right triangle]
Two circles touch each other externally at point $C$. A line is tangent to the first circle at point $A$ and to the second circle at point $B$. The line $A C$ intersects the second circle at point $D$, different from $C$. Find $B C$, if $A C=9$, $C D=4$.
# | 6 | 0/8 |
7.2 On the Island of Truth and Lies, there are knights who always tell the truth, and liars who always lie. One day, 15 residents of the island lined up by height (from tallest to shortest, the tallest being the first) for a game. Each had to say one of the following phrases: "There is a liar below me" or "There is a knight above me." In the end, those standing from the fourth to the eighth position said the first phrase, while the rest said the second. How many knights were among these 15 people, given that all residents have different heights? | 11 | 1/8 |
Let $A$ be a set of natural numbers, for which for $\forall n\in \mathbb{N}$ exactly one of the numbers $n$ , $2n$ , and $3n$ is an element of $A$ . If $2\in A$ , show whether $13824\in A$ . | 13824 \notin A | 1/8 |
Let n be the natural number ( $n\ge 2$ ). All natural numbers from $1$ up to $n$ ,inclusive, are written on the board in some order: $a_1$ , $a_2$ , $...$ , $a_n$ . Determine all natural numbers $n$ ( $n\ge 2$ ), for which the product $$ P = (1 + a_1) \cdot (2 + a_2) \cdot ... \cdot (n + a_n) $$ is an even number, whatever the arrangement of the numbers written on the board. | n | 5/8 |
Let $n$ be a positive integer. There are $n$ purple and $n$ white cows queuing in a line in some order. Tim wishes to sort the cows by colour, such that all purple cows are at the front of the line. At each step, he is only allowed to swap two adjacent groups of equally many consecutive cows. What is the minimal number of steps Tim needs to be able to fulfill his wish, regardless of the initial alignment of the cows? | n | 4/8 |
A set of $k$ integers is said to be a *complete residue system modulo* $k$ if no two of its elements are congruent modulo $k$ . Find all positive integers $m$ so that there are infinitely many positive integers $n$ wherein $\{ 1^n,2^n, \dots , m^n \}$ is a complete residue system modulo $m$ . | m | 0/8 |
Let $N \geqslant 3$ be an integer. In the country of Sibyl, there are $N^2$ towns arranged as the vertices of an $N \times N$ grid, with each pair of towns corresponding to an adjacent pair of vertices on the grid connected by a road. Several automated drones are given the instruction to traverse a rectangular path starting and ending at the same town, following the roads of the country. It turned out that each road was traversed at least once by some drone. Determine the minimum number of drones that must be operating.
*Proposed by Sutanay Bhattacharya and Anant Mudgal* | N | 0/8 |
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