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Let $p$ be a prime and $a$ and $n$ be natural numbers such that \( \frac{p^a - 1}{p-1} = 2^n \). Find the number of natural divisors of $na$.
|
4
|
4/8
|
We call a number greater than $25$ a \textit{semi-prime} if it is the sum of two different prime numbers. What is the greatest number of consecutive natural numbers that can be \textit{semi-prime}?
|
5
|
4/8
|
Using $600$ cards, where $200$ of them have the number $5$, $200$ have the number $2$, and the other $200$ have the number $1$, a student wants to create groups of cards such that the sum of the card numbers in each group is $9$. What is the maximum number of groups that the student can create?
|
150
|
3/8
|
Let $ABC$ be a scalene triangle with inradius $1$ and exradii $r_A$, $r_B$, and $r_C$ such that
\[ 20\left(r_B^2r_C^2+r_C^2r_A^2+r_A^2r_B^2\right)=19\left(r_Ar_Br_C\right)^2. \]
If
\[ \tan\frac{A}{2}+\tan\frac{B}{2}+\tan\frac{C}{2}=2.019, \]
then the area of $\triangle{ABC}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
|
201925
|
3/8
|
A set $S$ of positive integers is \textit{sum-complete} if there are positive integers $m$ and $n$ such that an integer $a$ is the sum of the elements of some nonempty subset of $S$ if and only if $m \le a \le n$.
Let $S$ be a sum-complete set such that $\{1, 3\} \subset S$ and $|S| = 8$. Find the greatest possible value of the sum of the elements of $S$.
|
223
|
3/8
|
Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly \(\frac{m+1}{m+n+2}\). Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times can be expressed as \(\frac{m}{n}\) for relatively prime positive integers $m$ and $n$. Compute $m+n$.
|
12
|
3/8
|
A natural number is called a *prime power* if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$.
Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$.
|
7
|
2/8
|
An $8 \times 8 \times 8$ cube is painted red on 3 faces and blue on 3 faces such that no corner is surrounded by three faces of the same color. The cube is then cut into 512 unit cubes. How many of these cubes contain both red and blue paint on at least one of their faces?
|
56
|
1/8
|
A triangle is composed of circular cells arranged in $5784$ rows: the first row has one cell, the second has two cells, and so on. The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called **diagonal** if the two cells in it aren't in the same row. What is the minimum possible amount of diagonal pairs in the division?
|
2892
|
3/8
|
Triangle $ABC$ has side lengths $AB=20$, $AC=14$, and $BC=22$. The median from $B$ intersects $AC$ at $M$, and the angle bisector from $C$ intersects $AB$ at $N$ and the median from $B$ at $P$. Let \( \frac{p}{q} = \frac{[AMPN]}{[ABC]} \) for positive integers \( p \) and \( q \) that are coprime. Note that \([ABC]\) denotes the area of triangle $ABC$. Find \( p+q \).
|
331
|
5/8
|
Compute the remainder when $99989796\ldots 121110090807 \ldots 01$ is divided by $010203 \ldots 091011 \ldots 9798$. Note that the first sequence starts at $99$, and the second sequence ends at $98$.
|
9801
|
0/8
|
Let $x_1, x_2, \ldots, x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p + x_q + x_r$ (with $1 \le p < q < r \le 5$) are equal to $0$, then $x_1 = x_2 = \cdots = x_5 = 0$.
|
7
|
3/8
|
Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt{2}}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\prod_{n=1}^{100}\psi(3^n)$.
|
5792
|
0/8
|
Every week, Ben goes to the supermarket and buys the following: $7$ apples at $\$2$ each, $4$ bottles of milk at $\$4$ each, $3$ loaves of bread at $\$3$ each, and $3$ bags of sugar at $\$6$ each. This week the store has a $25\%$ discount on all dairy products. Ben also has a coupon for $\$10$ off any order of $\$50$ or over. How much money does Ben spend on this shopping trip?
|
\$43
|
4/8
|
In trapezoid $EFGH$, sides $EF$ and $GH$ are equal. $GH$ is 12 units long, and the height of the trapezoid is 5 units. The top base $EF$ is 10 units long. Calculate the perimeter of trapezoid $EFGH$.
|
22 + 2\sqrt{26}
|
5/8
|
Let's define two operations $\oplus$ and $\otimes$ for any number $y$, where $y \oplus = 9 - y$ and $\oplus y = y - 9$. What is the value of $\oplus(18 \oplus)$?
|
-18
|
4/8
|
Triangles $PQR$ and $PRS$ are right triangles with $PQ = 14$ units, $PR = 24$ units, and $PS = 26$ units. What is the area, in square units, of quadrilateral $PQRS$?
|
288\text{ square units}
|
2/8
|
After a hurricane in Miami, the estimated damage was $\$$45 million in local (US) currency. Later, it was discovered that some assets were valued in Canadian dollars. The total value of these assets was $\$$15 million. At that time, 1 US dollar was worth 1.25 Canadian dollars. Determine the damage in Canadian dollars including the assets valued in Canadian dollars. Additionally, there is a recovery tax of 10% on the total damage calculated in Canadian dollars that needs to be considered.
|
78,375,000 \text{ Canadian dollars}
|
5/8
|
In heptagon $GEOMETRY$, $\angle G \cong \angle E \cong \angle T \cong \angle R$ and $\angle M$ is supplementary to $\angle Y$, $\angle J$ is supplementary to $\angle O$. Find the measure of $\angle T$.
|
135^\circ
|
3/8
|
The polynomial \[x^4 + bx^3 + cx^2 + 7dx + e = 0\] has integer coefficients, where $d$ is nonzero. Let $m$ be the exact number of integer roots of the polynomial, counting multiplicities. Determine all possible values for $m$.
|
0, 1, 2, 4
|
0/8
|
Let $g(x) = \frac{2 - x}{1 + 2x}$, $g_1(x) = g(g(x))$, $g_2(x) = g(g_1(x))$, and in general $g_n(x) = g(g_{n-1}(x))$. What is the value of $g_{1989}(5)$?
|
5
|
4/8
|
Berengere and her exchange student Emily are at a French café that accepts both euros and American dollars. They want to buy a cake which now costs 8 euros, but individually, they don't have enough money. Emily has a ten-dollar bill, and the exchange rate is 1 euro = 1.10 USD. How many euros does Berengere need to contribute so they can buy the cake?
|
0 \text{ euros}
|
3/8
|
The stem-and-leaf plot shows the duration (in minutes and seconds) of songs played during a concert by a band. There are 19 songs in total. In the stem-and-leaf plot, $3 \ 15$ represents 3 minutes, 15 seconds, which is the same as 195 seconds. Determine the median duration of these songs. Express your answer in seconds.
\begin{tabular}{c|ccccc}
0&45&50&55&&\\
1&10&12&15&18&20\\
2&05&10&15&25&30\\
3&00&05&10&15&20\\
4&10&15&&&\\
\end{tabular}
|
130
|
5/8
|
Four primes \(p_1\), \(p_2\), \(p_3\), and \(p_4\) form an increasing arithmetic sequence with \(p_1 > 3\) and common difference 4. What is the ones digit of \(p_1\)?
|
9
|
1/8
|
In a circle with radius $6$, two chords $PQ$ and $RS$ intersect at a point $T$. Chord $RS$ bisects chord $PQ$ such that $PT = TQ = 8$ and $PQ$ is the only chord from $P$ that $RS$ bisects. It is known that the sine of the central angle subtended by minor arc $PR$ is a rational number. If expressed as a fraction $\frac{p}{q}$ in lowest terms, what is the product $pq$?
|
pq=20
|
1/8
|
Define an integer $N$ to be $4$-nice or $6$-nice if there exists a positive integer $a$ such that $a^4$ or $a^6$ respectively, has exactly $N$ positive divisors. Determine the number of positive integers less than or equal to $500$ that are neither $4$-nice nor $6$-nice.
|
333
|
2/8
|
Tracy initially had a bag of candies. She first ate $\frac{1}{4}$ of them, then gave $\frac{1}{3}$ of the remaining candies to her friend Rachel. After that, Tracy and her mom each ate 12 candies from what was left. Lastly, her brother took between two to six candies, leaving Tracy with five candies. How many candies did Tracy have at the beginning?
|
68
|
3/8
|
In the shown diagram, segment $CD$ is parallel to segment $WX$. If $CX = 30$ units, $DP = 15$ units, and $PW = 45$ units, find the length of segment $PX$. Diagram:
[asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8));
pair W = (0,0), X = (24,0), C = (0,12), D = (9,12);
draw(C--D--W--X--cycle);
label("$C$",C,W); label("$D$",D,E); label("$W$",W,W); label("$X$",X,E);
pair P = intersectionpoint(C--X,D--W);
label("$P$",P,E);
[/asy]
|
22.5 \text{ units}
|
0/8
|
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x^2 - y) + 2f(x) y\]for all real numbers $x$ and $y.$ Additionally, suppose that $f(1) = 1$. Calculate the sum of all possible values of $f(3)$.
|
9
|
1/8
|
5.3. Among all the irreducible fractions, where the numerator and denominator are two-digit numbers, find the smallest fraction greater than $\frac{4}{5}$. In your answer, specify its numerator.
|
77
|
5/8
|
8. (10 points) Every day, Xiaoming has to pass through a flat section $AB$, an uphill section $BC$, and a downhill section $CD$ (as shown in the figure). It is known that $AB: BC: CD=1: 2: 1$, and Xiaoming's speed ratio on the flat section, uphill section, and downhill section is 3: 2: 4. What is the ratio of the time Xiaoming spends going to school to the time he spends coming back home? $\qquad$
|
19:16
|
1/8
|
23. When Chief Black Cat was chasing One-Ear, he found a $4 \times 4$ number puzzle left by One-Ear. In this puzzle, the sum of the four numbers in each row, each column, and each diagonal is the same. Among them, $a+b=2018, c+d=2019$, and the sum of the 16 numbers on the puzzle is the house number of the Cat-Eating Mouse. Chief Black Cat quickly calculated the house number of the Cat-Eating Mouse and caught One-Ear and the Cat-Eating Mouse. The house number of the Cat-Eating Mouse is $\qquad$
|
16148
|
5/8
|
13. A and B are two car retailers (hereinafter referred to as A and B) who ordered a batch of cars from a car manufacturer. Initially, the number of cars A ordered was 3 times the number of cars B ordered. Later, due to some reason, A transferred 6 cars from its order to B. When picking up the cars, the manufacturer provided 6 fewer cars than the total number ordered by A and B. In the end, the number of cars A purchased was twice the number of cars B purchased. Try to find: what is the maximum number of cars that A and B finally purchased in total? And what is the minimum number of cars they purchased in total?
|
18
|
2/8
|
5. In the Cartesian coordinate system, grid points are numbered as follows: $(0,0)$ is No. 1, $(1,0)$ is No. 2, $(1,1)$ is No. 3, $(0,1)$ is No. 4, $(0,2)$ is No. 5, $(1,2)$ is No. 6, $(2,2)$ is No. 7, $(2,1)$ is No. 8, $(2,0)$ is No. 9, $\cdots$ (as shown in Figure 7-1). Following the order of the arrows in the figure, the coordinates of the 2006th point are . $\qquad$
|
(44,19)
|
0/8
|
## Task 34/65
There are four consecutive natural numbers for which the following is true:
The sum of the cubes of the two smaller numbers is equal to the difference of the cubes of the two larger numbers.
These four numbers are to be found.
|
3,4,5,6
|
3/8
|
6. Given that $O$ is a point inside the equilateral $\triangle A B C$, the ratio of the angles $\angle A O B$, $\angle B O C$, and $\angle A O C$ is 6:5:4. Then, in the triangle formed by $O A$, $O B$, and $O C$, the ratio of the angles opposite these sides is
|
5: 3: 7
|
1/8
|
10. 12 people gather for dinner at the same restaurant every week, each time divided into three tables, with four people at each table. It is required that any two people sit at the same table at least once. Question: What is the minimum number of weeks such dinners need to be held?
|
5
|
0/8
|
On the extensions of the sides $CA$ and $CB$ of triangle $ABC$, segments $AA_1$ and $BB_1$ are laid out equal to segments $AL_3$ and $BL_3$, respectively, where $L_3$ is the foot of the angle bisector of angle $C$. Prove that the point $L_3$ is the incenter of triangle $A_1B_1C$.
|
L_3 \text{ is the incenter of } \triangle A_1B_1C
|
0/8
|
9.64 Two identical chessboards $(8 \times 8$ squares $)$ overlap each other. Now, one of them is rotated $45^{\circ}$ around the center. If the area of each square is 1, find the total area of the intersection of all the black squares of the two chessboards.
|
32(\sqrt{2} - 1)
|
0/8
|
Due to the rainstorms, our basement filled with water. To remove the water, three pumps were set up. The first pump alone could pump out the water in 3 hours, the second in 4 hours, and the third in 6 hours. After 30 minutes of working together, the second pump broke down. The remaining water was pumped out by the first and third pumps. How long did it take in total to remove the water?
|
\dfrac{7}{4}
|
4/8
|
5. Given that $a, b, c$ are three real numbers in an arithmetic sequence. Then the locus of the midpoint of the chord formed by the intersection of the line $b x+a y+c=0$ and the parabola $y^{2}=-\frac{1}{2} x$ is $\qquad$ .
|
4y^2 - 4y + x + 2 = 0
|
1/8
|
29. Find a function $f(x)$ such that for any real $x$, except 0 and 1, $f(1 / x) + f(1 - x) = x$.
|
-\dfrac{x}{2} + \dfrac{1}{2x} - \dfrac{1}{2(x - 1)}
|
0/8
|
6. (10 points) Square $ABCD$ and rectangle $BEFG$ are placed as shown in the figure, with $AG=CE=2$ cm. The area of square $ABCD$ is larger than the area of rectangle $BEFG$ by $\qquad$ square centimeters.
|
4
|
2/8
|
3B. Determine the smallest natural number, the product of whose digits is equal to 75600.
|
556789
|
5/8
|
Five. (20 points) Given the sequence of positive integers $\left\{T_{m, n}\right\}$ satisfies:
(1) $T_{m, m}=2^{m}$;
(2) $T_{m, n}=T_{n, m}$;
(3) $\left(T_{m, n}\right)^{m+n}=\left(T_{m, m+n}\right)^{n}$.
Here $m, n$ are any positive integers. Try to find the general term formula for the sequence $\left\{T_{m, n}\right\}$.
|
2^{\operatorname{lcm}(m,n)}
|
0/8
|
Let $\omega$ be an incircle of triangle $ABC$. Let $D$ be a point on segment $BC$, which is tangent to $\omega$. Let $X$ be an intersection of $AD$ and $\omega$ against $D$. If $AX : XD : BC = 1 : 3 : 10$, a radius of $\omega$ is $1$, find the length of segment $XD$. Note that $YZ$ expresses the length of segment $YZ$.
|
\dfrac{3\sqrt{10}}{5}
|
1/8
|
12. Given that $F$ is the right focus of the ellipse $C: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, and the ratio of the distance from any point $P$ on the ellipse $C$ to the point $F$ to the distance from point $P$ to the line $l: x=m$ is $\frac{1}{2}$.
(1) Find the equation of the line $l$.
(2) Let $A$ be the left vertex of the ellipse $C$. A line passing through the point $F$ intersects the ellipse $C$ at points $D$ and $E$. The lines $A D$ and $A E$ intersect the line $l$ at points $M$ and $N$, respectively. Does the circle with diameter $M N$ always pass through a fixed point? If so, find the coordinates of the fixed point; if not, explain the reason.
|
x = 4
|
0/8
|
Given two points \( A \) and \( B \) on a plane, there are \( m \) lines passing through point \( A \) and \( n \) lines passing through point \( B \). What is the maximum number of regions into which these \( m+n \) lines can divide the plane? (where \( m \neq 0 \) and \( n \neq 0 \)).
|
mn + 2m + 2n - 1
|
0/8
|
## Task A-1.3.
Determine how many six-digit natural numbers exist such that by removing the first two, or the last two digits, we obtain two four-digit numbers that give the same remainder when divided by 99.
|
8190
|
0/8
|
Let $l$ be a tangent line at the point $T\left(t,\ \frac{t^2}{2}\right)\ (t>0)$ on the curve $C: y=\frac{x^2}{2}.$ A circle $S$ touches the curve $C$ and the $x$ axis.
Denote by $(x(t),\ y(t))$ the center of the circle $S$. Find $\lim_{r\rightarrow +0}\int_r^1 x(t)y(t)\ dt.$
|
\dfrac{1 + \sqrt{2}}{30}
|
1/8
|
5. We have cards with numbers 5, 6, 7, ..., 55 (each card has one number). What is the maximum number of cards we can select so that the sum of the numbers on any two selected cards is not a palindrome? (A palindrome is a number that reads the same backward as forward.)
|
25
|
0/8
|
Example 1. Find $\frac{1}{1-\frac{1}{1-\cdots \frac{1}{1-\frac{355}{113}}}}$.
|
\dfrac{355}{113}
|
2/8
|
$\left[\begin{array}{l}\text { Angles between lines and planes } \\ {[\underline{\text { Linear dependence of vectors }}]}\end{array}\right]$
The lateral face of a regular quadrilateral pyramid forms an angle of $45^{\circ}$ with the plane of the base. Find the angle between the apothem of the pyramid and the plane of the adjacent face.
|
30^\circ
|
4/8
|
In an isosceles right triangle, one of the legs is 36 units. Starting from the right-angle vertex on one of the legs, we draw an infinite sequence of equilateral triangles, each touching the next, such that the third vertex of each inscribed triangle always lies on the hypotenuse, and the sides opposite these vertices fill the leg. Determine the sum of the areas of the equilateral triangles.
|
324
|
3/8
|
[ Intersecting lines, angle between them]
In rectangle $A B C D$, the sides are given as $A B=3, B C=4$. Point $K$ is at distances $\sqrt{10}$, 2, and 3 from points $A, B$, and $C$ respectively. Find the angle between the lines $C K$ and $B D$.
|
\arccos\left(\dfrac{3}{5}\right)
|
0/8
|
Problem 8. For what values of the parameter a does the equation $x^{3}+6 x^{2}+a x+8=0$ have exactly three solutions?
|
(-\infty, -15)
|
4/8
|
p1. Given the set $H = \{(x, y)|(x -y)^2 + x^2 - 15x + 50 = 0$ where x and y are natural numbers $\}$.
Find the number of subsets of $H$.
p2. A magician claims to be an expert at guessing minds with following show. One of the viewers was initially asked to hidden write a five-digit number, then subtract it with the sum of the digits that make up the number, then name four of the five digits that make up the resulting number (in order of any). Then the magician can guess the numbers hidden. For example, if the audience mentions four numbers result: $0, 1, 2, 3$, then the magician will know that the hidden number is $3$.
a. Give an example of your own from the above process.
b. Explain mathematically the general form of the process.
p3. In a fruit basket there are $20$ apples, $18$ oranges, $16$ mangoes, $10$ pineapples and $6$ papayas. If someone wants to take $10$ pieces from the basket. After that, how many possible compositions of fruit are drawn?
p4. Inside the Equator Park, a pyramid-shaped building will be made with base of an equilateral triangle made of translucent material with a side length of the base $8\sqrt3$ m long and $8$ m high. A globe will be placed in a pyramid the. Ignoring the thickness of the pyramidal material, determine the greatest possible length of the radius of the globe that can be made.
p5. What is the remainder of $2012^{2012} + 2014^{2012}$ divided by $2013^2$?
|
64
|
1/8
|
4. Given an integer $n>0$. There is a balance scale and $n$ weights with masses $2^{0}, 2^{1}, \cdots, 2^{n-1}$.
Now, through $n$ steps, all the weights are to be placed on the balance one by one, ensuring that during the process, the weight on the left side never exceeds the weight on the right side. In each step, one of the weights not yet placed on the balance is chosen and placed on either the left or the right side of the balance, until all weights are placed on the balance. Find the number of different ways to perform this process.
|
(2n-1)!!
|
2/8
|
A tanker of milk was delivered to the store. The seller has balance scales without weights (fляги can be placed on the pans of the scales), and three identical flasks, two of which are empty, and the third one contains 1 liter of milk. How can you pour exactly 85 liters of milk into one flask, making no more than eight weighings?
#
|
85
|
0/8
|
47. There are the same number of black balls and white balls in the box. Each time, 5 black balls and 8 white balls are taken out. After several times, only 12 black balls are left in the box. The box originally had $\qquad$ balls.
|
64
|
5/8
|
6. There are 100 equally divided points on a circle. The number of obtuse triangles formed by these points as vertices is $\qquad$ .
|
117600
|
1/8
|
Given a sector that is one-fourth of a circle with radius $R$. Find the length of the tangent drawn from the midpoint of the arc to its intersection with the extensions of the sector's radial boundaries.
|
2R
|
2/8
|
A geometric sequence has its first element as 6, the sum of the first $n$ elements is $\frac{45}{4}$, and the sum of the reciprocals of these elements is $\frac{5}{2}$. Which is this geometric sequence?
|
6, 3, \frac{3}{2}, \frac{3}{4}
|
4/8
|
II. (Full marks 25 points) As shown in the figure, $\odot O_{1}$ and $\odot O_{2}$ are externally tangent at $M$, and the angle between their two external common tangents is $60^{\circ}$. The line connecting the centers intersects $\odot O_{1}$ and $\odot O_{2}$ at $A$ and $B$ (different from $M$), respectively. A line through $B$ intersects $\odot O_{1}$ at points $C$ and $D$. Find the value of $\operatorname{ctg} \angle B A C \cdot \operatorname{ctg} \angle B A D$.
|
4
|
3/8
|
Let $\triangle ABC$ satisfy $AB = 17$, $AC = \frac{70}{3}$, and $BC = 19$. Let $I$ be the incenter of $\triangle ABC$ and $E$ be the excenter of $\triangle ABC$ opposite $A$. (Note: this means that the circle tangent to ray $AB$ beyond $B$, ray $AC$ beyond $C$, and side $BC$ is centered at $E$.) Suppose the circle with diameter $IE$ intersects $AB$ beyond $B$ at $D$. If $BD = \frac{a}{b}$ where $a$ and $b$ are coprime positive integers, find $a + b$.
|
22
|
1/8
|
You are playing a game called "Hovse." Initially, you have the number $0$ on a blackboard. At any moment, if the number $x$ is written on the board, you can perform one of the following operations:
- Replace $x$ with $3x + 1$
- Replace $x$ with $9x + 1$
- Replace $x$ with $27x + 3$
- Replace $x$ with $\left \lfloor \frac{x}{3} \right \rfloor$
You are not allowed to write a number greater than $2017$ on the board. How many positive numbers can you make with the game of "Hovse?"
|
127
|
5/8
|
Using each of the ten digits exactly once, construct two five-digit numbers such that their difference is minimized. Determine this minimal difference.
|
247
|
2/8
|
How many rearrangements of the letters of "HMMTHMMT" do not contain the substring "HMMT"? For instance, one such arrangement is $HMMHMTMT$.
|
361
|
2/8
|
The roots of the polynomial $f(x) = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1$ are all roots of unity. We say that a real number $r \in [0, 1)$ is nice if $e^{2i \pi r} = \cos 2\pi r + i \sin 2\pi r$ is a root of the polynomial $f$ and if $e^{2i \pi r}$ has a positive imaginary part. Let $S$ be the sum of the values of nice real numbers $r$. If $S = \frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.
|
31
|
3/8
|
A bitstring of length $\ell$ is a sequence of $\ell$ $0$'s or $1$'s in a row. How many bitstrings of length $2014$ have at least $2012$ consecutive $0$'s or $1$'s?
|
16
|
3/8
|
Let $A=(a_{ij})$ be the $n \times n$ matrix, where $a_{ij}$ is the remainder of the division of $i^j + j^i$ by $3$ for $i,j=1,2,\ldots,n$. Find the greatest $n$ for which $\det A \neq 0$.
|
5
|
1/8
|
Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is friends with four, Elías is friends with five, and so on, with each subsequent person being friends with one more person than the previous person. This pattern continues until reaching Yvonne, the twenty-fifth person, who is friends with everyone. How many people is Zoila, the twenty-sixth person, friends with?
Clarification: If $A$ is a friend of $B$, then $B$ is a friend of $A$.
|
13
|
2/8
|
Al and Bill play a game involving a fair six-sided die. The die is rolled until either a number less than $5$ is rolled on consecutive tosses, or a number greater than $4$ is rolled on consecutive tosses. Al wins if the last roll is a $5$ or $6$. Bill wins if the last roll is a $2$ or lower. Let $m$ and $n$ be relatively prime positive integers such that $\frac{m}{n}$ is the probability that Bill wins. Find the value of $m+n$.
|
29
|
3/8
|
Henry rolls a fair die. If the die shows the number $k$, Henry will then roll the die $k$ more times. The probability that Henry will never roll a 3 or a 6 either on his first roll or on one of the $k$ subsequent rolls is given by $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
|
904
|
3/8
|
Given a board with size $25 \times 25$. Some $1 \times 1$ squares are marked, so that for each $13 \times 13$ and $4 \times 4$ sub-boards, there are at least $\frac{1}{2}$ of the squares in the sub-board that are marked. Find the least possible number of marked squares in the entire board.
|
313
|
3/8
|
Jacob's analog clock has 12 equally spaced tick marks on the perimeter, but all the digits have been erased, so he doesn't know which tick mark corresponds to which hour. Jacob takes an arbitrary tick mark and measures clockwise to the hour hand and minute hand. He measures that the minute hand is 300 degrees clockwise of the tick mark, and that the hour hand is 70 degrees clockwise of the same tick mark. If it is currently morning, how many minutes past midnight is it?
|
500
|
3/8
|
Define $\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the number of integers $2 \le n \le 50$ such that $n$ divides $\phi^{!}(n)+1$.
|
30
|
0/8
|
A4. In the figure, $A Q P B$ and $A S R C$ are squares, and $A Q S$ is an equilateral triangle. If $Q S=4$ and $B C=x$, what is the value of $x$ ?
|
4\sqrt{3}
|
1/8
|
In a certain logarithm system, $\log 450$ is 1.1639 more than $\log 40$. What is the base of the system and what are the logarithms of the two numbers in this system?
|
8
|
0/8
|
Let \( S \) be the set of all nondegenerate triangles formed from the vertices of a regular octagon with side length 1. Find the ratio of the largest area of any triangle in \( S \) to the smallest area of any triangle in \( S \).
|
3+2\sqrt{2}
|
3/8
|
For \(0 \leq y \leq 2\), let \(D_y\) be the half-disk of diameter 2 with one vertex at \((0, y)\), the other vertex on the positive \(x\)-axis, and the curved boundary further from the origin than the straight boundary. Find the area of the union of \(D_y\) for all \(0 \leq y \leq 2\).
|
\pi
|
3/8
|
Centipedes and three-headed dragons. In a certain herd of centipedes and three-headed dragons, there are a total of 26 heads and 298 legs. Each centipede has one head. How many legs does a three-headed dragon have?
|
14
|
0/8
|
As shown in the figure, square \(ABCD\) and rectangle \(BEFG\) share a common vertex \(B\). The side length of square \(ABCD\) is \(6 \, \text{cm}\) and the length of rectangle \(BEFG\) is \(9 \, \text{cm}\). Find the width of the rectangle in centimeters.
|
4
|
0/8
|
The Fibonacci numbers are defined by \( F_{0}=0, F_{1}=1 \), and \( F_{n}=F_{n-1}+F_{n-2} \) for \( n \geq 2 \). There exist unique positive integers \( n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6} \) such that
\[
\sum_{i_{1}=0}^{100} \sum_{i_{2}=0}^{100} \sum_{i_{3}=0}^{100} \sum_{i_{4}=0}^{100} \sum_{i_{5}=0}^{100} F_{i_{1}+i_{2}+i_{3}+i_{4}+i_{5}}=F_{n_{1}}-5 F_{n_{2}}+10 F_{n_{3}}-10 F_{n_{4}}+5 F_{n_{5}}-F_{n_{6}}.
\]
Find \( n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6} \).
|
1545
|
0/8
|
Players \( A \) and \( B \) play a game on a blackboard that initially contains 2020 copies of the number 1. In every round, player \( A \) erases two numbers \( x \) and \( y \) from the blackboard, and then player \( B \) writes one of the numbers \( x+y \) and \( |x-y| \) on the blackboard. The game terminates as soon as, at the end of some round, one of the following holds:
1. One of the numbers on the blackboard is larger than the sum of all other numbers.
2. There are only zeros on the blackboard.
Player \( B \) must then give as many cookies to player \( A \) as there are numbers on the blackboard. Player \( A \) wants to get as many cookies as possible, whereas player \( B \) wants to give as few as possible. Determine the number of cookies that \( A \) receives if both players play optimally.
|
7
|
0/8
|
Let \( n \) be a positive integer. Determine the smallest number \( k \) of colors needed to color the edges of any directed simple graph with \( n \) vertices such that there is no monochromatic cycle.
|
2
|
3/8
|
As shown in Figure 3, the side length of equilateral triangle \( \triangle ABC \) is 5. Extend \( BA \) to point \( P \) such that \( |AP| = 9 \). Let \( D \) be a point on segment \( BC \) (including endpoints). The line \( AD \) intersects the circumcircle of \( \triangle BPC \) at points \( E \) and \( F \), where \( |EA| < |ED| \).
(1) Let \( |BD| = x \). Express \( |EA| - |DF| \) as a function \( f(x) \).
(2) Find the minimum value of \( f(x) \).
|
4\sqrt{5}
|
1/8
|
$\triangle XYZ$ is similar to $\triangle MNP$. The side $\overline{YZ}$ measures 10 cm, and the side $\overline{XZ}$ measures 7 cm. In $\triangle MNP$, the side $\overline{MN}$ measures 4.2 cm. Find the length of $\overline{NP}$, rounding your answer to the nearest tenth of a cm.
|
6.0 \text{ cm}
|
3/8
|
Consider Abby the alpaca is tethered to the corner of a $4\text{ m}$ by $6\text{ m}$ barn on a $5\text{ m}$ leash. How much area (in square meters) can Abby roam if she can only walk around the barn?
|
19\pi\text{ m}^2
|
0/8
|
In Mr. Petrov's 30-student history class, each student’s initials are uniquely represented by the same letter for both the first and last name. Besides the typical vowels A, E, I, O, U, the letter "X" is also considered a vowel in this peculiar setting. What is the probability of randomly selecting a student whose initials are vowels, given this modification?
|
\frac{1}{5}
|
4/8
|
Sandy thinks of a number. She triples it, adds 20, squares the result, and the final outcome is 2500. What was her original number?
|
10
|
2/8
|
Let $x$ and $y$ be positive numbers such that $x^2 y$ is constant. Suppose $y = 8$ when $x = 3$. Find the value of $x$ when $y$ is $72$ and $x^2$ has increased by a factor of $z = 4$.
|
1
|
4/8
|
Convert \(142_{10}\) to base 7. Represent any values normally represented by new symbols in decimal as \(A\) (if 10), \(B\) (if 11), and so on.
|
262_7
|
2/8
|
A rectangular mirror (the shaded region) is perfectly centered within a larger frame, which itself is within an even larger frame. The outermost perimeter of the largest frame measures 100 cm by 140 cm. Each side of this frame has a width of 15 cm. Inside this, there is another frame, whose sides are also 15 cm wide. What is the area of the mirror?
|
3200 \text{ cm}^2
|
0/8
|
The increasing sequence of positive integers $a_1, a_2, a_3, \dots$ follows the recursive formula
\[a_{n + 2} = a_{n + 1} + a_n\]
for all $n \geq 1$. If $a_7 = 210$, determine $a_8$.
|
340
|
3/8
|
Let $Q$ be a quartic polynomial such that $Q(0) = k$, $Q(1) = 3k$, and $Q(-1) = 5k$. What is $Q(2) + Q(-2)$?
|
26k
|
2/8
|
A cylindrical tower has a diameter of 25 feet and a height of 60 feet. A green stripe with a width of 2 feet wraps twice around the cylinder horizontally. Calculate the area of the stripe in square feet.
|
100\pi \text{ square feet}
|
1/8
|
The time right now is precisely 6:00 a.m. What time will it be in 2700 minutes?
|
3\!:\!00 \text{ a.m.}
|
4/8
|
Cutting isosceles right triangle $BFC$ out of square $ABCD$ and translating it leftwards creates the shape $ABFCDE$. The perimeter of square $ABCD$ is 40 inches. What is the perimeter, in inches, of $ABFCDE$?
[asy]
import graph;
filldraw((0,0)--(sqrt(2)*2,2)--(2+2sqrt(2),2)--(2sqrt(2),0)--(2+2sqrt(2),-2)--(sqrt(2)*2,-2)--cycle,grey,linewidth(1));
draw((sqrt(2)*2,2)--(sqrt(2)*2,-2),linewidth(1));
draw((2+2sqrt(2),-2)--(2+2sqrt(2),2)--(2sqrt(2),0)--(2+2sqrt(2),-2)--cycle,dashed);
label("A",(sqrt(2)*2,2),N);
label("B",(2+2sqrt(2),2),N);
label("C",(2+2sqrt(2),-2),S);
label("D",(sqrt(2)*2,-2),S);
label("E",(0,0),W);
label("F",(2sqrt(2),0),N);
[/asy]
|
20 + 20\sqrt{2} \text{ inches}
|
0/8
|
The Gnollish language has expanded to include a fourth word, "kreeg". In a sentence, "splargh" cannot come directly before "glumph" or "kreeg". How many valid 3-word sentences are there now in this updated Gnollish language?
|
48
|
4/8
|
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.