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In a square $ABCD$ of diagonals $AC$ and $BD$ , we call $O$ at the center of the square. A square $PQRS$ is constructed with sides parallel to those of $ABCD$ with $P$ in segment $AO, Q$ in segment $BO, R$ in segment $CO, S$ in segment $DO$ . If area of $ABCD$ equals two times the area of $PQRS$ , and $M$ is the midpoint of the $AB$ side, calculate the measure of the angle $\angle AMP$ . | \frac{45^\circ}{2} | 0/8 |
Let \(n\ge3\) be a fixed integer, and let \(\alpha\) be a fixed positive real number. There are \(n\) numbers written around a circle such that there is exactly one \(1\) and the rest are \(0\)'s. An *operation* consists of picking a number \(a\) in the circle, subtracting some positive real \(x\le a\) from it, and adding \(\alpha x\) to each of its neighbors.
Find all pairs \((n,\alpha)\) such that all the numbers in the circle can be made equal after a finite number of operations.
*Proposed by Anthony Wang* | (n, \alpha) | 2/8 |
Let $n$ be a positive integer. Find the number of permutations $a_1$ , $a_2$ , $\dots a_n$ of the
sequence $1$ , $2$ , $\dots$ , $n$ satisfying $$ a_1 \le 2a_2\le 3a_3 \le \dots \le na_n $$ .
Proposed by United Kingdom | F_{n+1} | 2/8 |
Example 7 When is $x$ a rational number such that the algebraic expression $9 x^{2}$ $+23 x-2$ is exactly the product of two consecutive positive even numbers?
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | x = -17 | 2/8 |
A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| =|BC|= 2$ and$ |AB| = 4$. Let $E $ be the intersection point of lines $BC$ and $DA$. Suppose that $\angle AEB = 60^o$ and that $|CD| <|AB|$. Calculate the radius of the circle. | 2 | 2/8 |
Let $\Gamma$ be the circumcircle of a triangle $ABC$ and let $E$ and $F$ be the intersections of the bisectors of $\angle ABC$ and $\angle ACB$ with $\Gamma$. If $EF$ is tangent to the incircle $\gamma$ of $\triangle ABC$, then find the value of $\angle BAC$. | 60^\circ | 4/8 |
5. Two circles touch internally at point $A$. From the center of the larger circle $O$, a radius $O B$ of the larger circle is drawn, which is also tangent to the smaller circle at point $C$. Determine $\measuredangle B A C$. | 45^\circ | 3/8 |
10. Let two non-zero and non-collinear vectors $a, b$ in the same plane satisfy $b \perp (a-b)$. Then for any $x \in \mathbf{R}$, the range of $|a-b x|$ is $\qquad$ (expressed in terms of vectors $a, b$). | [|\mathbf{a} - \mathbf{b}|, +\infty) | 0/8 |
Fredek runs a private hotel. He claims that whenever $ n \ge 3$ guests visit the hotel, it is possible to select two guests that have equally many acquaintances among the other guests, and that also have a common acquaintance or a common unknown among the guests. For which values of $ n$ is Fredek right? (Acquaintance is a symmetric relation.) | n \geq 3 | 0/8 |
1. N1 (TWN) ${ }^{1 \mathrm{MO} 04}$ Find all pairs of positive integers $(x, p)$ such that $p$ is a prime, $x \leq 2 p$, and $x^{p-1}$ is a divisor of $(p-1)^{x}+1$. | (1, p) | 0/8 |
## [ equations in integers ] Decompositions and partitions $\quad]$ [ GCD and LCM. Mutual simplicity ]
Ostap Bender organized a giveaway of elephants to the population in the city of Fux. 28 union members and 37 non-members showed up for the giveaway, and Ostap distributed the elephants equally among all union members and equally among non-members.
It turned out that there was only one way to distribute the elephants (so that all elephants were distributed). What is the maximum number of elephants that Ostap Bender could have had? (It is assumed that each person who came received at least one elephant.) | 2072 | 0/8 |
53. As shown in the figure, in the rectangular paper piece $\mathrm{ABCD}$, $\mathrm{AB}=4, \mathrm{BC}=6$. The paper is folded so that point $\mathrm{B}$ coincides with point $\mathrm{D}$, and the fold line is $\mathrm{EF}$. The area of the part of the rectangle that does not overlap is $\qquad$ - | \dfrac{20}{3} | 1/8 |
Given that $\triangle ABC$ is an acute-angled triangle, and $L$ is any straight line in the coordinate plane containing the triangle. Let $u$, $v$, and $w$ be the perpendicular distances from points $A$, $B$, and $C$ to the line $L$, respectively. Prove that \( u^{2} \tan A + v^{2} \tan B + w^{2} \tan C \geq 2S \), where $S$ is the area of $\triangle ABC$. | u^{2} \tan A + v^{2} \tan B + w^{2} \tan C \geq 2S | 0/8 |
4. On the board, you can either write two ones, or erase two already written identical numbers $n$ and write the numbers $n+k$ and $n-k$, provided that $n-k \geqslant 0$. What is the minimum number of such operations required to obtain the number $2048?$ | 3071 | 0/8 |
[ Arithmetic progression ]
[ Arithmetic. Mental calculation, etc. ]
What is the sum of the digits of all numbers from one to a billion? | 40500000001 | 3/8 |
7. On the side $B C$ of triangle $A B C$, a point $K$ is marked.
It is known that $\angle B+\angle C=\angle A K B, \quad A K=4, \quad B K=9, \quad K C=3$.
Find the area of the circle inscribed in triangle $A B C$. | \dfrac{35}{13}\pi | 1/8 |
Problem 5. On the table, there are 10 stacks of playing cards (the number of cards in the stacks can be different, there should be no empty stacks). The total number of cards on the table is 2015. If a stack has an even number of cards, remove half of the cards. If the number of remaining cards in the stack is still even, remove half again, and so on, until the number of cards in the stack becomes odd. Do this for each stack. Explain:
a) What is the maximum possible number of cards remaining on the table?
b) What is the minimum possible number of cards remaining on the table?
In each case, provide an example of how the playing cards can be distributed among the stacks. | 2014 | 0/8 |
At a gathering, $n$ invitees arrived, including Mr. Balogh. Besides them, there is a journalist who wants to talk to Mr. Balogh. The journalist knows that no one knows Mr. Balogh, but Mr. Balogh knows everyone. The journalist can approach any of the invitees and ask if they know any other person.
a) Can the journalist definitely find Mr. Balogh with fewer than $n$ questions?
b) How many questions does the journalist need to find Mr. Balogh if they are lucky? | Yes | 0/8 |
A family of sets $\mathscr{F}$ is called "perfect" if for any three sets $X_{1}, X_{2}, X_{3} \in \mathscr{F}$,
$$
\left(X_{1} \backslash X_{2}\right) \cap X_{3} \text{ and } \left(X_{2} \backslash X_{1}\right) \cap X_{3}
$$
have at least one empty set. Prove that if $\mathscr{F}$ is a perfect subfamily of a given finite set $U$, then
$$
|\mathscr{F}| \leqslant |U|+1,
$$
where $|X|$ represents the number of elements in the finite set $X$. | |\mathscr{F}| \leqslant |U| + 1 | 0/8 |
In a triangle $ABC$ ($\angle{BCA} = 90^{\circ}$), let $D$ be the intersection of $AB$ with a circumference with diameter $BC$. Let $F$ be the intersection of $AC$ with a line tangent to the circumference. If $\angle{CAB} = 46^{\circ}$, find the measure of $\angle{CFD}$. | 92 | 3/8 |
3. (6 points) As shown in the figure, it is a geometric body composed of 26 identical small cubes, with its bottom layer consisting of $5 \times 4$ small cubes. If the entire outer surface (including the bottom) is painted red, then when this geometric body is disassembled, the number of small cubes that have 3 red faces is $\qquad$ pieces. | 14 | 0/8 |
2. As shown in the figure, $\angle A O B=\angle B O C=\angle C O D=\angle D O E=\angle E O F$, there are 15 acute angles in the figure, and the sum of these 15 acute angles is $525^{\circ}$, then $\angle A O D=$ $\qquad$ ${ }^{\circ}$. | 45 | 0/8 |
A string of 33 pearls has its largest and most valuable pearl in the middle. The other pearls are arranged so that their value decreases by $3000 \mathrm{Ft}$ per pearl towards one end and by $4500 \mathrm{Ft}$ per pearl towards the other end. How much is the middle pearl worth if the total value of the entire string of pearls is 25 times the value of the fourth pearl from the middle, on the more expensive side. | 90000 | 4/8 |
3. Let $N$ be a regular nonagon, $O$ the center of its circumscribed circle, $P Q$ and $Q R$ two consecutive sides of $N$, $A$ the midpoint of $P Q$, and $B$ the midpoint of the radius perpendicular to $Q R$. Find the angle between $A O$ and $A B$. | 30^\circ | 1/8 |
Thirty rays with the origin at the same point are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest possible value of $N$. | 210 | 1/8 |
In the interior of the square $ABCD$, the point $P$ is located such that $\angle DCP = \angle CAP = 25^{\circ}$. Find all possible values of $\angle PBA$. | 40 | 4/8 |
A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Take five good haikus. Scramble their lines randomly. What are the chances that you end up with five completely good haikus (with five, seven, five)?
Your answer will be \( \frac{m}{n} \) where \( m, n \) are numbers such that \( m, n \) are positive integers and \( \gcd(m, n) = 1 \).
Take this answer and add the numerator and denominator. | 3004 | 5/8 |
For a positive integer $k$, let $f_{1}(k)$ be the square of the sum of the digits of $k$. For example, $f_{1}(123) = (1+2+3)^{2} = 36$. Define $f_{n+1}(k) = f_{1}(f_{n}(k))$. Determine the value of $f_{2007}(2^{2006})$. Justify your claim. | 169 | 4/8 |
Georg is putting his $250$ stamps in a new album. On the first page, he places one stamp, and then on every subsequent page, he places either the same number of stamps or twice as many as on the preceding page. In this way, he ends up using precisely all $250$ stamps. How few pages are sufficient for him? | 8 | 0/8 |
The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A \cup B$. | 5 | 1/8 |
A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Take five good haikus. Scramble their lines randomly. What are the chances that you end up with five completely good haikus (with five, seven, five)?
Your answer will be $\frac{m}{n}$ where $m,n$ are numbers such that $m,n$ are positive integers and $\gcd(m,n) = 1$. Take this answer and add the numerator and denominator. | 3004 | 2/8 |
Ephram is growing $3$ different variants of radishes in a row of $13$ radishes total, but he forgot where he planted each radish variant and he can't tell what variant a radish is before he picks it. Ephram knows that he planted at least one of each radish variant, and all radishes of one variant will form a consecutive string, with all such possibilities having an equal chance of occurring. He wants to pick three radishes to bring to the farmers market, and wants them to all be of different variants. Given that he uses optimal strategy, the probability that he achieves this can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 17 | 0/8 |
There are $2010$ red cards and $2010$ white cards. All of these $4020$ cards are shuffled and dealt in pairs to each of the $2010$ players sitting around a round table. The game consists of several rounds, each of which players simultaneously hand over cards to each other according to the following rules:
- If a player holds at least one red card, they pass one red card to the player sitting to their left.
- Otherwise, they transfer one white card to the left.
The game ends after the round when each player has one red card and one white card. Determine the maximum number of rounds possible. | 2009 | 3/8 |
A pentagon has vertices labeled $A, B, C, D, E$ in that order counterclockwise, such that $AB$ and $ED$ are parallel, and \( \angle EAB = \angle ABD = \angle ACD = \angle CDA \). Furthermore, suppose that \( AB = 8 \), \( AC = 12 \), \( AE = 10 \). If the area of triangle $CDE$ can be expressed as \( \frac{a \sqrt{b}}{c} \), where $a$, $b$, $c$ are integers such that $b$ is square-free, and $a$, $c$ are relatively prime, find $a + b + c$. | 56 | 0/8 |
In a tournament with $n$ players, where $n < 10$, each player competes once against every other player. Players score 1 point for a win and 0 points for a loss, with no draws occurring. In a particular tournament, only one player finished with an odd number of points and was ranked fourth. Determine if this scenario is possible. If it is possible, calculate the number of wins the player achieved. | 3 | 5/8 |
Let $S = \{1, 2, \ldots, 100\}$. Consider a partition of $S$ into $S_1, S_2, \ldots, S_n$ for some $n$, i.e., $S_i$ are nonempty, pairwise disjoint sets, and $S = \bigcup_{i=1}^n S_i$. Let $a_i$ be the average of the elements of the set $S_i$. Define the score of this partition by:
\[
\frac{a_1 + a_2 + \ldots + a_n}{n}.
\]
Among all $n$ and partitions of $S$, determine the minimum possible score. | 10 | 4/8 |
Let $P_k(x) = (x-k)(x-(k+1))$. Kara picks four distinct polynomials from the set $\{P_1(x), P_2(x), P_3(x), \ldots , P_{12}(x)\}$ and discovers that when she computes the six sums of pairs of chosen polynomials, exactly two of the sums have two (not necessarily distinct) integer roots! How many possible combinations of four polynomials could Kara have picked? | 108 | 1/8 |
At a certain pizzeria, there are five different toppings available, and a pizza can be ordered with any (possibly empty) subset of them on it. Determine the number of ways to order an unordered pair of pizzas such that at most one topping appears on both pizzas and at least one topping appears on neither. | 271 | 2/8 |
10. Transporting utility poles from a construction site by the roadside along a straight road in the same direction to plant them 500 m away on the roadside, plant one at the 500 m mark, and then plant one every 50 m along the roadside. Knowing that the transport vehicle can carry a maximum of 3 poles at a time, to complete the task of transporting and planting 20 poles, and returning to the construction site, the minimum total distance the transport vehicle must travel is $\qquad$ m. | 14000 | 2/8 |
19. Given an arbitrary triangle. On each side of the triangle, 10 points are marked. Each vertex of the triangle is connected by segments to all the marked points on the opposite side. Into what maximum number of parts could the segments divide the triangle? | 331 | 3/8 |
Let \( ABCDE \) be a convex pentagon such that \( BC = DE \). Assume there is a point \( T \) inside \( ABCDE \) with \( TB = TD \), \( TC = TE \), and \( \angle TBA = \angle AET \). Let lines \( CD \) and \( CT \) intersect line \( AB \) at points \( P \) and \( Q \), respectively, and let lines \( CD \) and \( DT \) intersect line \( AE \) at points \( R \) and \( S \), respectively. Assume that points \( P, B, A, Q \) and \( R, E, A, S \), respectively, are collinear and occur on their lines in this order. Prove that the points \( P, S, Q, R \) are concyclic. | P, S, Q, R \text{ are concyclic} | 1/8 |
2. Find all integer solutions $x$ and $y$ that satisfy the equation $x^{2} + xy + y^{2} = 1$. | (1, 0) | 1/8 |
Problem 4. We say a 2023-tuple of nonnegative integers $\left(a_{1}, a_{2}, \ldots a_{2023}\right)$ is sweet if the following conditions hold:
- $a_{1}+a_{2}+\ldots+a_{2023}=2023$,
- $\frac{a_{1}}{2^{1}}+\frac{a_{2}}{2^{2}}+\ldots+\frac{a_{2023}}{2^{2023}} \leqslant 1$.
Determine the greatest positive integer $L$ such that
$$
a_{1}+2 a_{2}+\ldots+2023 a_{2023} \geqslant L
$$
holds for every sweet 2023-tuple $\left(a_{1}, a_{2}, \ldots, a_{2023}\right)$.
(Ivan Novak) | 22228 | 0/8 |
3. A certain number of boys and girls went camping during the summer break. They planned an ecological action that they would finish in 29 days if each child worked evenly - working the same part of the job in any given days. The boys worked a bit faster; in the same time, 2 boys do as much work as 3 girls. Fortunately, after three days of starting the work, a larger group of children joined them: boys - 8 times more than the initial number of girls, and girls - 18 times more than the initial number of boys. The newly arrived children worked with the same dynamics as the initially arrived ones. How many days in total were working days? | 5 | 5/8 |
Variant 134. a) $36+8 \pi$ cm or $23+3 \sqrt{41}+8 \pi$ cm; b) not always.
Evaluation criteria: 20 points - correct (not necessarily the same as above) solution and correct answer; $\mathbf{1 5}$ points - mostly correct solution and correct answer, but there are defects (for example: comparison of numbers in the last part of the solution is not performed or is performed incorrectly; it is not proven that the sum of the lengths of the arcs is equal to the length of the circumference); 10 points - the solution is mostly correct, but contains computational errors; 5 points - only one of the two cases is considered, and the solution and answer (for this case) are completely correct.
## Problem 4 (Variant 131).
When studying the operation of a new type of heat engine working cyclically, it was found that for part of the period it receives heat, with the absolute value of the heat supply power expressed by the law: $P_{1}(t)=P_{0} \frac{\sin \omega t}{100+\sin t^{2}}, 0<t<\frac{\pi}{\omega}$. The gas performs work, developing mechanical power $P_{2}(t)=3 P_{0} \frac{\sin (2 \omega t)}{100+\sin (2 t)^{2}}, 0<t<\frac{\pi}{2 \omega}$. The work done on the gas by external bodies is $2 / 3$ of the magnitude of the work done by the gas. Determine the efficiency of the engine. | \dfrac{1}{2} | 4/8 |
4. Each diagonal of the convex pentagon $A B C D E$ cuts off a triangle of unit area from it. Calculate the area of the pentagon $A B C D E$.
Lemma. Let in a convex quadrilateral $A B C D$ the diagonals $A C$ and $B D$ intersect at point $O$. If triangles $A B O$ and $C O D$ are equal in area, then $A B C D$ is a trapezoid.
Proof. Since $S_{A O B}=S_{C O D}$, then $S_{A B D}=S_{C A D}$, which means the heights from points $B$ and $C$ to line $A D$ in triangles $A B D$ and $C A D$ are equal. This implies that all points on line $B C$ are equidistant from line $A D$, hence $B C \| A D$. | \frac{5 + \sqrt{5}}{2} | 0/8 |
5. On the edge $S B$ of a regular quadrilateral pyramid $S A B C D$ with vertex $S$, a point $L$ is marked such that $B L: L S=2: 5$. The point $L$ is the vertex of a right circular cone, on the circumference of the base of which lie three vertices of the pyramid $S A B C D$.
a) Find the ratio $A S: C D$.
b) Suppose it is additionally known that the height of the pyramid $S A B C D$ is 7. Find the volume of the cone. | \dfrac{\sqrt{15}}{3} | 0/8 |
* 6. Among $n(\geqslant 3)$ lines, exactly $p(\geqslant 2)$ are parallel, and no three lines among the $n$ lines intersect at the same point. Then the number of regions into which these $n$ lines divide the plane is | \dfrac{n^2 + n - p^2 + p + 2}{2} | 1/8 |
The diagonals of a convex quadrilateral are 12 and 18 and intersect at point \( O \).
Find the sides of the quadrilateral formed by the points of intersection of the medians of triangles \( AOB, BOC, COD \), and \( AOD \). | 4 | 1/8 |
49th Putnam 1988 Problem B3 α n is the smallest element of the set { |a - b√3| : a, b non-negative integers with sum n}. Find sup α n . Solution | \dfrac{1 + \sqrt{3}}{2} | 2/8 |
$\left[\begin{array}{l}\text { Symmetry helps solve the problem } \\ \text { [The inscribed angle is half the central angle]}\end{array}\right]$
$A B$ is the diameter of the circle; $C, D, E$ are points on the same semicircle $A C D E B$. On the diameter $A B$, points $F$ and $G$ are taken such that $\angle C F A = \angle D F B$ and $\angle D G A = \angle E G B$. Find $\angle F D G$, if the arc $A C$ is $60^{\circ}$ and the arc $B E$ is $20^{\circ}$. | 50^\circ | 0/8 |
Task 2. Find the number of ways to color all natural numbers from 1 to 20 in blue and red such that both colors are used and the product of all red numbers is coprime with the product of all blue numbers. | 62 | 2/8 |
Given an isosceles triangle \(ABC\) with \(AB = BC\). In the circumcircle \(\Omega\) of triangle \(ABC\), a diameter \(CC'\) is drawn. A line passing through point \(C'\) parallel to \(BC\) intersects segments \(AB\) and \(AC\) at points \(M\) and \(P\) respectively. Prove that \(M\) is the midpoint of segment \(C'P\). | M \text{ is the midpoint of } C'P | 0/8 |
17. (5 points)
As shown in the figure, there is a circular path and two straight paths on the Green Green Grassland. The two straight paths both pass through the center of the circle, with a radius of 50 meters. Village Chief Slow Sheep starts from point A and walks along the path. He needs to walk through all the paths and return to point A. He needs to walk at least $\qquad$ meters. (Take $\pi$ as 3.14) | 671 | 0/8 |
[Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external segment. Auxiliary similar triangles [Mean proportional in a right triangle
From point $A$, two tangents (with points of tangency $M$ and $N$) and a secant intersecting the circle at points $B$ and $C$, and the chord $MN$ at point $P$, are drawn, with $AB: BC=2: 3$. Find $AP: PC$.
# | \dfrac{4}{3} | 2/8 |
4. Given an isosceles triangle $A B C$, where $\angle A=30^{\circ}, A B=A C$. Point $D$ is the midpoint of $B C$. On segment $A D$, point $P$ is chosen, and on side $A B$, point $Q$ is chosen such that $P B=P Q$. What is the measure of angle $P Q C ?$ (S. S. Korechkova) | 15^\circ | 3/8 |
15. There are 10 students standing in a row, and their birthdays are in different months. There are $n$ teachers who will select these students to join $n$ interest groups. Each student is selected by exactly one teacher, and the order of the students is maintained. Each teacher must select students whose birthdays are in months that are either strictly increasing or strictly decreasing (selecting one or two students is also considered strictly increasing or decreasing). Each teacher should select as many students as possible. For all possible orderings of the students, find the minimum value of $n$. | 4 | 0/8 |
If for all \( n \in \mathbf{Z}_{+} \), \( a^{n+2} + 3a^n + 1 \) is not a prime number, then the positive integer \( a \) is called "beautiful". Prove that in the set \( \{1, 2, \ldots, 2018\} \) there are at least 500 beautiful positive integers. | 500 | 0/8 |
Let \( P \) be the intersection point of the directrix \( l \) of an ellipse and its axis of symmetry, and \( F \) be the corresponding focus. \( AB \) is a chord passing through \( F \). Find the maximum value of \( \angle APB \) which equals \( 2 \arctan e \), where \( e \) is the eccentricity. | 2\arctane | 4/8 |
A space probe, moving in a straight line at a constant speed, flies past Mars and measures the distance to the planet exactly at 12:00 noon every day. It is known that on February 1st, the distance was 5 million km, on February 10th, it was 2 million km, and on February 13th, it was 3 million km. Determine when the probe will be at the minimum distance from Mars. In this problem, Mars can be considered as a point. | 9 | 0/8 |
Place the numbers $1,2,\cdots,n$ on a circle such that the absolute difference between any two adjacent numbers is either 3, 4, or 5. Find the smallest $n$ that satisfies these conditions. | 7 | 5/8 |
Positive integers \( a, b \), and \( c \) have the property that \( a^b \), \( b^c \), and \( c^a \) end in 4, 2, and 9, respectively. Compute the minimum possible value of \( a + b + c \). | 17 | 5/8 |
In the Magic Land, one of the magical laws of nature states: "A flying carpet will only fly if it has a rectangular shape." Prince Ivan had a flying carpet of dimensions $9 \times 12$. One day, an evil dragon cut off a small piece of this carpet with dimensions $1 \times 8$. Prince Ivan was very upset and wanted to cut off another piece $1 \times 4$ to make a rectangle of $8 \times 12$, but the wise Vasilisa suggested a different approach. She cut the carpet into three parts and used magic threads to sew them into a square flying carpet of dimensions $10 \times 10$. Can you figure out how Vasilisa the Wise transformed the damaged carpet? | 10\times10 | 3/8 |
There are 29 students in a class. It is known that for any pair of students, there is at least one other student (not from the considered pair) who is friends with exactly one person from that pair. What is the minimum number of pairs of friends that can exist in the class? | 21 | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Bob`, `Eric`, `Arnold`, `Alice`, `Peter`
- Each person has a favorite color: `blue`, `green`, `white`, `yellow`, `red`
- People use unique phone models: `huawei p50`, `samsung galaxy s21`, `oneplus 9`, `iphone 13`, `google pixel 6`
- Each person has an occupation: `artist`, `teacher`, `doctor`, `engineer`, `lawyer`
## Clues:
1. The person who is an engineer is somewhere to the right of the person who is a lawyer.
2. Bob is in the second house.
3. The person who uses a Samsung Galaxy S21 is the person who is a doctor.
4. The person who is a doctor is the person who loves blue.
5. The person whose favorite color is green is not in the fifth house.
6. The person who is a lawyer is the person who uses a OnePlus 9.
7. The person who loves blue is directly left of the person whose favorite color is red.
8. The person who is a lawyer is somewhere to the right of the person who uses a Samsung Galaxy S21.
9. There is one house between the person who uses a Google Pixel 6 and the person who uses a Huawei P50.
10. Arnold is the person who is an engineer.
11. Alice is the person who loves yellow.
12. The person who uses a Google Pixel 6 is Eric.
13. The person who uses a Google Pixel 6 is the person who is a teacher.
14. The person whose favorite color is red is somewhere to the right of the person who is a teacher.
What is the value of attribute Color for the person whose attribute Occupation is teacher? Please reason step by step, and put your final answer within \boxed{} | green | 0/8 |
There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Arnold`, `Peter`
- Everyone has something unique for lunch: `spaghetti`, `grilled cheese`, `pizza`
- People have unique favorite sports: `basketball`, `tennis`, `soccer`
- People have unique hair colors: `black`, `blonde`, `brown`
- People have unique favorite music genres: `classical`, `rock`, `pop`
- People have unique favorite book genres: `science fiction`, `mystery`, `romance`
## Clues:
1. The person who loves basketball is somewhere to the left of the person who loves science fiction books.
2. The person who has brown hair is Eric.
3. The person who has brown hair is the person who is a pizza lover.
4. The person who loves romance books is the person who is a pizza lover.
5. Peter is the person who has blonde hair.
6. The person who loves science fiction books is the person who loves soccer.
7. The person who loves eating grilled cheese is the person who loves basketball.
8. The person who loves pop music is the person who has blonde hair.
9. The person who has brown hair is somewhere to the left of the person who loves rock music.
10. The person who loves classical music is not in the first house.
What is the value of attribute FavoriteSport for the person whose attribute House is 2? Please reason step by step, and put your final answer within \boxed{} | tennis | 0/8 |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Arnold`, `Alice`, `Eric`
- Each person has an occupation: `engineer`, `artist`, `teacher`, `doctor`
## Clues:
1. The person who is an engineer is not in the third house.
2. The person who is a teacher is directly left of Peter.
3. The person who is an engineer is Arnold.
4. The person who is a doctor is somewhere to the right of Alice.
5. Arnold is somewhere to the left of Eric.
6. There is one house between Arnold and Alice.
What is the value of attribute Occupation for the person whose attribute House is 3? Please reason step by step, and put your final answer within \boxed{} | teacher | 2/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Bob`, `Peter`, `Eric`, `Arnold`, `Alice`
- The people keep unique animals: `dog`, `fish`, `bird`, `horse`, `cat`
- The people are of nationalities: `dane`, `swede`, `brit`, `german`, `norwegian`
- Each mother is accompanied by their child: `Meredith`, `Bella`, `Timothy`, `Samantha`, `Fred`
- Each person has a unique type of pet: `bird`, `fish`, `hamster`, `cat`, `dog`
## Clues:
1. The dog owner is not in the third house.
2. The person who keeps a pet bird is in the fourth house.
3. The German is Eric.
4. Bob is somewhere to the left of the person's child is named Fred.
5. The person with a pet hamster is Arnold.
6. The British person is the dog owner.
7. The person's child is named Meredith is not in the third house.
8. The fish enthusiast is the person's child is named Samantha.
9. The person who is the mother of Timothy is the German.
10. The person who has a cat is in the fifth house.
11. Peter is not in the fourth house.
12. The Swedish person and the person's child is named Samantha are next to each other.
13. The Swedish person is Peter.
14. The Dane is in the second house.
15. Eric and the cat lover are next to each other.
16. The German is in the first house.
17. The person with an aquarium of fish is somewhere to the right of the bird keeper.
What is the value of attribute House for the person whose attribute Children is Bella? Please reason step by step, and put your final answer within \boxed{} | 3 | 0/8 |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Bob`, `Arnold`, `Peter`, `Eric`, `Alice`
- People have unique favorite book genres: `mystery`, `science fiction`, `biography`, `fantasy`, `romance`
- Each mother is accompanied by their child: `Samantha`, `Timothy`, `Fred`, `Meredith`, `Bella`
## Clues:
1. The person's child is named Samantha is somewhere to the left of the person who loves biography books.
2. Arnold is the person who loves biography books.
3. The person who is the mother of Timothy is the person who loves science fiction books.
4. The person's child is named Bella is Peter.
5. There is one house between the person who loves biography books and Bob.
6. Alice is somewhere to the right of Eric.
7. The person who loves mystery books is in the fourth house.
8. The person who loves biography books is directly left of the person who is the mother of Timothy.
9. The person who loves fantasy books is directly left of the person's child is named Meredith.
What is the value of attribute House for the person whose attribute Children is Bella? Please reason step by step, and put your final answer within \boxed{} | 5 | 1/8 |
There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Arnold`, `Eric`, `Peter`
- Everyone has a unique favorite cigar: `pall mall`, `blue master`, `prince`
- The people keep unique animals: `horse`, `cat`, `bird`
- Each mother is accompanied by their child: `Bella`, `Fred`, `Meredith`
- People have unique favorite book genres: `science fiction`, `romance`, `mystery`
- People use unique phone models: `google pixel 6`, `iphone 13`, `samsung galaxy s21`
## Clues:
1. The person who loves mystery books is the person's child is named Fred.
2. The cat lover is Eric.
3. The person partial to Pall Mall is in the second house.
4. The person who keeps horses is the person's child is named Meredith.
5. The person's child is named Bella is the Prince smoker.
6. The person who uses an iPhone 13 is directly left of the person who uses a Samsung Galaxy S21.
7. The person's child is named Fred is directly left of Arnold.
8. Peter is somewhere to the left of Eric.
9. The person who loves science fiction books is the person who uses a Samsung Galaxy S21.
10. The person who loves science fiction books is in the third house.
11. The person who loves mystery books is not in the second house.
What is the value of attribute House for the person whose attribute Children is Fred? Please reason step by step, and put your final answer within \boxed{} | 1 | 4/8 |
There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Arnold`, `Peter`, `Eric`
- People have unique favorite book genres: `romance`, `mystery`, `science fiction`
- Each person has a unique birthday month: `sept`, `april`, `jan`
- The people are of nationalities: `brit`, `dane`, `swede`
## Clues:
1. Arnold is the person who loves mystery books.
2. Peter is directly left of the person who loves romance books.
3. Peter is somewhere to the left of the person who loves mystery books.
4. The person whose birthday is in April is somewhere to the right of the person whose birthday is in January.
5. The person whose birthday is in September is the Swedish person.
6. The Dane is in the third house.
7. The person who loves science fiction books is the person whose birthday is in September.
What is the value of attribute House for the person whose attribute Birthday is sept? Please reason step by step, and put your final answer within \boxed{} | 1 | 4/8 |
Let $ABCD$ be a convex cyclic quadrilateral. Suppose $P$ is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of $ABCD$ is the least. If $\{PA, PB, PC, PD\} = \{3, 4, 6, 8\}$, what is the maximum possible area of $ABCD$? | 54 | 1/8 |
Given in the figure:
- The line through $E$ and $F$ is parallel to the line passing through $A$ and $B$.
- The area of the triangle $\triangle FEC$ is $40$ cm$^2$.
- The sum of the areas of the triangles $\triangle ADF$ and $\triangle DBE$ is $150$ cm$^2$.
Determine whether the area of the triangle $\triangle DEF$ follows clearly from this information and, if necessary, calculate this area. | 60 | 3/8 |
The incircle of $\triangle ABC$ is tangent to sides $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ at $D$, $E$, and $F$, respectively. Point $G$ is the intersection of lines $AC$ and $DF$. The sides of $\triangle ABC$ have lengths $AB = 73$, $BC = 123$, and $AC = 120$. Find the length $EG$. | 119 | 1/8 |
We say that an integer $a$ is a quadratic, cubic, or quintic residue modulo $n$ if there exists an integer $x$ such that $x^2 \equiv a \pmod{n}$, $x^3 \equiv a \pmod{n}$, or $x^5 \equiv a \pmod{n}$, respectively. Further, an integer $a$ is a primitive residue modulo $n$ if it is exactly one of these three types of residues modulo $n$.
How many integers $1 \le a \le 2015$ are primitive residues modulo $2015$? | 694 | 0/8 |
There are $2022$ distinct integer points on the plane. Let $I$ be the number of pairs among these points that are exactly $1$ unit apart. Find the maximum possible value of $I$. | 3954 | 3/8 |
Three bells begin to ring simultaneously. The intervals between strikes for these bells are, respectively, \( \frac{4}{3} \) seconds, \( \frac{5}{3} \) seconds, and 2 seconds. Impacts that coincide in time are perceived as one. How many beats will be heard in 1 minute? (Include first and last.) | 85 | 4/8 |
Let $\Gamma_1$ be a circle with radius $\frac{5}{2}$. Points $A$, $B$, and $C$ are on $\Gamma_1$ such that $\overline{AB} = 3$ and $\overline{AC} = 5$. Let $\Gamma_2$ be a circle tangent to $AB$ and $BC$ at $Q$ and $R$, and also internally tangent to $\Gamma_1$ at $P$. $\Gamma_2$ intersects $AC$ at $X$ and $Y$. The area $[PXY]$ can be expressed as $\frac{a\sqrt{b}}{c}$. Find $a+b+c$. | 19 | 4/8 |
Consider a string of $n$ 1's. We wish to place some $+$ signs in between so that the sum is $1000$. For instance, if $n=190$, one may put $+$ signs so as to get $11$ ninety times and $1$ ten times, and get the sum $1000$. If $a$ is the number of positive integers $n$ for which it is possible to place $+$ signs so as to get the sum $1000$, then find the sum of digits of $a$. | 9 | 1/8 |
Say an odd positive integer $n > 1$ is \textit{twinning} if $p - 2 \mid n$ for every prime $p \mid n$. Find the number of twinning integers less than 250. | 12 | 2/8 |
A convex equilateral pentagon with side length $2$ has two right angles. The greatest possible area of the pentagon is $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m + n$. | 407 | 2/8 |
A circular disc is divided into $12$ equal sectors, each of which is colored using one of $6$ different colors. No two adjacent sectors can have the same color. Determine the number of distinct ways to color the disc under these conditions. | 20346485 | 2/8 |
Six different small books and three different large books are placed on a shelf. Three children can each choose either two small books or one large book. Determine the number of ways the three children can select their books. | 1176 | 4/8 |
Let $N$ be the set $\{1, 2, \dots, 2018\}$. For each subset $A$ of $N$ with exactly $1009$ elements, define
\[f(A) = \sum\limits_{i \in A} i \sum\limits_{j \in N, j \notin A} j.\]
If $\mathbb{E}[f(A)]$ is the expected value of $f(A)$ as $A$ ranges over all the possible subsets of $N$ with exactly $1009$ elements, find the remainder when the sum of the distinct prime factors of $\mathbb{E}[f(A)]$ is divided by $1000$. | 441 | 2/8 |
In square $ABCD$ with $AB = 10$, point $P, Q$ are chosen on side $CD$ and $AD$ respectively such that $BQ \perp AP,$ and $R$ lies on $CD$ such that $RQ \parallel PA.$ $BC$ and $AP$ intersect at $X,$ and $XQ$ intersects the circumcircle of $PQD$ at $Y$. Given that $\angle PYR = 105^{\circ},$ $AQ$ can be expressed in simplest radical form as $b\sqrt{c}-a$ where $a, b, c$ are positive integers. Find $a+b+c.$ | 23 | 1/8 |
Let $p$ be a prime. It is given that there exists a unique nonconstant function $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$ such that $\chi(1) = 1$ and $\chi(mn) = \chi(m)\chi(n)$ for all $m, n \not\equiv 0 \pmod p$ (here the product $mn$ is taken mod $p$). For how many positive primes $p$ less than $100$ is it true that \[\sum_{a=1}^{p-1}a^{\chi(a)}\equiv 0\pmod p?\] Here as usual $a^{-1}$ denotes multiplicative inverse.
[i]Proposed by David Altizio[/i] | 24 | 5/8 |
11 Given that $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $C: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, and points $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$ are on the ellipse $C$. If $x_{1}+x_{2}=\frac{1}{2}$, and $\overrightarrow{A F_{2}}=$ $\lambda \overrightarrow{F_{2} B}$, find the value of $\lambda$. | \dfrac{3 - \sqrt{5}}{2} | 0/8 |
Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$ , and the area common to triangle $GEM$ and square $AIME$ is $80$ square units. Find the length of the altitude to $EM$ in $\triangle GEM$ .
Please give the answer directly without any intermediate steps. | 25 | 4/8 |
How many quadratic residues are there modulo $pq$? And modulo $p^n$? You can use the fact that $\left(\mathbb{Z} / p^{n} \mathbb{Z}\right)^{*}$ is cyclic. | \frac{(p-1)(q-1)}{4} | 0/8 |
B3. One of the four gnomes, Anne, Bert, Chris, and Dirk, has stolen gold from the king. Each of the gnomes, who know each other inside and out, makes two statements. If a gnome is a liar, at least one of those two statements is a lie. If a gnome is not a liar, both statements are true.
Anne says: "Bert is a liar." and "Chris or Dirk did it."
Bert says: "Chris is a liar." and "Dirk or Anne did it."
Chris says: "Dirk is a liar." and "Anne or Bert did it."
Dirk says: "Anne is a liar." and "Bert or Chris did it."
How many of these eight statements are true? | 5 | 4/8 |
## Problema 2.
Sea $\mathrm{P}$ un punto del lado $\mathrm{BC}$ de un triángulo $\mathrm{ABC}$. La paralela por $\mathrm{P}$ a $\mathrm{AB}$ corta al lado $\mathrm{AC}$ en el punto $\mathrm{Q}$ y la paralela por $\mathrm{P}$ a $\mathrm{AC}$ corta al lado $\mathrm{AB}$ en el punto $\mathrm{R}$. La razón entre las áreas de los triángulos RBP y QPC es $\mathrm{k}^{2}$.
Determínese la razón entre las áreas de los triángulos ARQ y ABC. | \dfrac{k}{(k + 1)^2} | 5/8 |
31. Consider the identity $1+2+\cdots+n=\frac{1}{2} n(n+1)$. If we set $P_{1}(x)=\frac{1}{2} x(x+1)$, then it is the unique polynomial such that for all positive integer $n, P_{1}(n)=1+2+\cdots+n$. In general, for each positive integer $k$, there is a unique polynomial $P_{k}(x)$ such that
$$
P_{k}(n)=1^{k}+2^{k}+3^{k}+\cdots+n^{k} \quad \text { for each } n=1,2, \ldots .
$$
Find the value of $P_{2010}\left(-\frac{1}{2}\right)$. | 0 | 3/8 |
Problem 4. Inside a rectangular grid with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut out along the cell boundaries (the hole does not contain any boundary cells). If the figure is cut along all horizontal grid lines, 20 strips 1 cell wide will be obtained. How many strips will be obtained if, instead, it is cut along all vertical grid lines? (A $1 \times 1$ square is also a strip!)
[6 points]
(A. V. Shapovalov) | 21 | 1/8 |
13. Teacher Li and three students, Xiao Ma, Xiao Lu, and Xiao Zhou, set off from the school one after another and walk along the same road to the cinema. The three students have the same walking speed, and Teacher Li's walking speed is 1.5 times that of the students. Now, Teacher Li is 235 meters away from the school, Xiao Ma is 87 meters away from the school, Xiao Lu is 59 meters away from the school, and Xiao Zhou is 26 meters away from the school. When they walk another $\qquad$ meters, the distance Teacher Li is from the school will be exactly the sum of the distances the three students are from the school. | 42 | 4/8 |
11. Given positive integers $x, y, z$ satisfying $x y z=(22-x)(22-y)(22-z)$, and $x+y+z<44, x^{2}+y^{2}+z^{2}$, the maximum and minimum values are denoted as $M$ and $N$, respectively, then $M+N=$ $\qquad$. | 926 | 3/8 |
5. Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with its left focus at $F$, the line $y=k x-1$ intersects the ellipse at points $A$ and $B$. When the perimeter of $\triangle F A B$ is maximized, the area of $\triangle F A B$ is $\qquad$ | \dfrac{12\sqrt{2}}{7} | 1/8 |
4. Extreme set (from 6th grade, $\mathbf{1}$ point). From the digits 1 to 9, three single-digit and three two-digit numbers are formed, with no digits repeating. Find the smallest possible arithmetic mean of the resulting set of numbers. | 16.5 | 2/8 |
6.2. On the plate, there were 15 doughnuts. Karlson took three times more doughnuts than Little Man, and Little Man's dog Bibbo took three times fewer than Little Man. How many doughnuts are left on the plate? Explain your answer. | 2 | 4/8 |
4. For a regular tetrahedron $V-ABC$ with base edge length $a$ and side edge length $b$, let $M$ be a point on the height $VO$ such that $\frac{VM}{MO}=\frac{b}{a}$. A plane passing through $M$ is parallel to the side edge $VA$ and the base edge $BC$. The area of the section formed by this plane cutting the regular tetrahedron is | \dfrac{2 a b^2 (3a + b)}{9 (a + b)^2} | 0/8 |
10.284. A square and an equilateral triangle are described around a circle of radius $R$, with one side of the square lying on a side of the triangle. Calculate the area of the common part of the triangle and the square. | \dfrac{(18 - 4\sqrt{3})R^2}{3} | 0/8 |
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