problem stringlengths 10 7.44k | answer stringlengths 1 270 | difficulty stringclasses 8
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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... | \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 add... | (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 ... | 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... | 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 \( u^{2} \tan A + v^{2} \tan B + w^{2} \tan C \geq 2S \), whe... | 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 ev... | 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... | 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 fini... | |\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 fac... | 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 o... | 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... | 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 $m,n$ ar... | 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 consecutiv... | 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 play... | 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}}{... | 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... | 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}... | 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... | 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 com... | 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 \( DT \) ... | 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 $L$ suc... | 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... | 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 ... | \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 trap... | \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 ... | \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 = \a... | 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 path... | 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 $... | \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... | 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 pro... | 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... | 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 col... | 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: `spag... | 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: `engin... | 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... | 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 bo... | 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 ma... | 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: `roma... | 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 $\triangle DEF$ fol... | 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 resi... | 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 are... | 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... | 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... | 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 sim... | 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 \[... | 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... | \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... | 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... | \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}... | 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 ob... | 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, ... | 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 tet... | \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 |
Subsets and Splits
Filtered Answers A-D
Retrieves 100 rows where the answer is a single letter from A to D, providing basic filtering of the dataset.