prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
Real numbers $a, b, c, d$ satisfy $a^2 + b^2 + c^2 + d^2 = 4$. Find the greatest possible value of $E(a, b, c, d) = a^4 + b^4 + c^4 + d^4 + 4(a + b + c + d)^2$. | 68 |
A tank can be filled with cold water in 14 minutes using a particular cold water tap. It can be emptied in 21 minutes by opening a hole at the base. If both the cold and hot water taps are opened simultaneously and the hole is also open, the tank fills up in 12.6 minutes. Determine the time required to fill the tank wi... | 18 |
Compute the following expression: $$ \left( \frac{4 - \log_{36} 4 - \log_6 18}{\log_4 3} \right) \cdot \left( \log_8 27 + \log_2 9 \right). $$ | 12 |
Find the sum of all three-digit natural numbers that contain at least one odd digit and at least one even digit. | 370775 |
On a board, the numbers $1, 2, 3, \ldots, 98, 99$ are written. One must mark $50$ of them such that the sum of any two marked numbers is never equal to $99$ or $100$. How many ways can one mark these numbers? | 1 |
A $39$-tuple of real numbers $(x_1,x_2,\ldots, x_{39})$ satisfies the following conditions: \[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\] Determine the ratio between the maximum of $\cos(x_1)$ and the maximum of $\sin(x_1)$ over all tuples $(x_1,x_2,\ldots, x_{39})$ satisfying the given conditions. ... | 37 |
Two infinite geometric series have the same sum. The first term of the first series is $1$, and the first term of the second series is $4$. The fifth terms of the two series are equal. The sum of each series can be written as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$. | 25 |
Let $ABC$ be a triangle with $AB=11$, $BC=12$, and $CA=13$. Let $M$ and $N$ be the midpoints of sides $CA$ and $AB$, respectively. The incircle touches sides $CA$ and $AB$ at points $X$ and $Y$, respectively. Suppose that $R$, $S$, and $T$ are the midpoints of line segments $MN$, $BX$, and $CY$, respectively. Then $\si... | 5033 |
We draw a radius of a circle. We draw a second radius $23^\circ$ clockwise from the first radius. We draw a third radius $23^\circ$ clockwise from the second. This continues until we have drawn $40$ radii, each $23^\circ$ clockwise from the one before it. What is the measure in degrees of the smallest angle between any... | 7 |
The graph of the degree $2021$ polynomial $P(x)$, which has real coefficients and leading coefficient $1$, meets the $x$-axis at the points $(1,0), (2,0), (3,0), \dots, (2020,0)$ and nowhere else. The mean of all possible values of $P(2021)$ can be written in the form $\frac{a!}{b}$, where $a$ and $b$ are positive inte... | 2023 |
A trapezium in the plane is a quadrilateral in which a pair of opposite sides are parallel. A trapezium is said to be non-degenerate if it has positive area. Find the number of mutually non-congruent, non-degenerate trapeziums whose sides are four distinct integers from the set $\{5,6,7,8,9,10\}$. | 31 |
Compute the greatest $4$-digit number $\underline{ABCD}$ such that $(A^3 + B^2)(C^3 + D^2) = 2015$. | 4132 |
In triangle $ABC$, let $P$ and $R$ be the feet of the perpendiculars from $A$ onto the external and internal bisectors of $\angle ABC$, respectively; and let $Q$ and $S$ be the feet of the perpendiculars from $A$ onto the internal and external bisectors of $\angle ACB$, respectively. If $PQ = 7$, $QR = 6$, and $RS = 8$... | 84 |
Let $ABC$ be a non-degenerate triangle with perimeter $4$ such that $a = bc \sin^2 A$. If $M$ is the maximum possible area of $\triangle ABC$ and $m$ is the minimum possible area of $\triangle ABC$, then $M^2 + m^2$ can be expressed in the form $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $... | 59 |
If $x = \cos 1^\circ \cos 2^\circ \cos 3^\circ \ldots \cos 89^\circ$ and $y = \cos 2^\circ \cos 6^\circ \cos 10^\circ \ldots \cos 86^\circ$, then what is the integer nearest to $\frac{2}{7} \log_2 \frac{y}{x}$? | 19 |
Compute \( \sum^{999}_{x=1}\gcd (x, 10x + 9) \). | 2331 |
Find the number of ordered pairs $(a, b)$ of positive integers that are solutions of the following equation: \[ a^2 + b^2 = ab(a+b). \] | 1 |
Tom has a scientific calculator. Unfortunately, all keys are broken except for one row: 1, 2, 3, + and -. Tom presses a sequence of $5$ random keystrokes; at each stroke, each key is equally likely to be pressed. The calculator then evaluates the entire expression, yielding a result of $E$. Find the expected value of $... | 1866 |
Let $N_{10}$ be the answer to problem 10. Compute the number of ordered pairs of integers $(m, n)$ that satisfy the equation: \[ m^2 + n^2 = mn + N_{10}. \] | 0 |
Al told Bob that he was thinking of $2011$ distinct positive integers. He also told Bob the sum of those $2011$ distinct positive integers. From this information, Bob was able to determine all $2011$ integers. How many possible sums could Al have told Bob? | 2 |
There are $13$ cities in a certain kingdom. Between some pairs of the cities, two-way direct bus, train, or plane connections are established. What is the least possible number of connections to be established so that, by choosing any two means of transportation, one can travel from any city to any other without using ... | 18 |
In the coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a constant $c$. Find $c$. | 1012 |
In $\triangle ABC$, $AB = 3$, $BC = 5$, and $CA = 6$. Points $D$ and $E$ are chosen such that $ACDE$ is a square which does not overlap with $\triangle ABC$. The length of $BD$ can be expressed in the form $\sqrt{m + n\sqrt{p}}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of a ... | 83 |
Find an integral solution of the equation: \[ \left \lfloor \frac{x}{1!} \right \rfloor + \left \lfloor \frac{x}{2!} \right \rfloor + \left \lfloor \frac{x}{3!} \right \rfloor + \dots + \left \lfloor \frac{x}{10!} \right \rfloor = 2019. \] Note: \( \lfloor u \rfloor \) stands for the greatest integer less than or equal... | 1176 |
Say that a 4-digit positive integer is \textit{mixed} if it has 4 distinct digits, its leftmost digit is neither the biggest nor the smallest of the 4 digits, and its rightmost digit is not the smallest of the 4 digits. For example, 2013 is mixed. How many 4-digit positive integers are mixed? | 1680 |
Let $A, B, C$ be points in that order along a line, such that $AB = 20$ and $BC = 18$. Let $\omega$ be a circle of nonzero radius centered at $B$, and let $\ell_1$ and $\ell_2$ be tangents to $\omega$ through $A$ and $C$, respectively. Let $K$ be the intersection of $\ell_1$ and $\ell_2$. Let $X$ lie on segment $\overl... | 35 |
Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$, with $i = 1, 2, \ldots, 2015$. | 2016 |
Suppose sequence $\{a_i\} = a_1, a_2, a_3, \ldots$ satisfies $a_{n+1} = \frac{1}{a_n+1}$ for all positive integers $n$. Define $b_k$ for positive integers $k \ge 2$ to be the minimum real number such that the product $a_1 \cdot a_2 \cdot \ldots \cdot a_k$ does not exceed $b_k$ for any positive integer choice of $a_1$. ... | 88 |
Let $ABC$ be a right triangle with $\angle ACB = 90^{\circ}$. $D$ is a point on $AB$ such that $CD \perp AB$. If the area of triangle $ABC$ is $84$, what is the smallest possible value of $AC^2 + (3 \cdot CD)^2 + BC^2$? | 1008 |
\textbf{Problem 5.} Miguel has two clocks, one clock advances $1$ minute per day and the other one goes $\frac{15}{10}$ minutes per day. If you put them at the same correct time, what is the least number of days that must pass for both to give the correct time simultaneously? | 1440 |
How many ways can the integers from $-7$ to $7$ inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease? | 128 |
Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from $1$ to $50$ uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability $\frac{3}{5}$. If he flips heads, he adds $1$ to his s... | 751 |
Find the number of $4$-digit numbers (in base $10$) that have non-zero digits and are divisible by $4$ but not by $8$. | 729 |
Evaluate the expression: \[2^{7}\times 3^{0}+2^{6}\times 3^{1}+2^{5}\times 3^{2}+\cdots+2^{0}\times 3^{7}.\] | 6305 |
Let the sum $\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}$ be written in its lowest terms as $\frac{p}{q}$. Find the value of $q - p$. | 83 |
You are using a lock with four dials, each set to a number between 0 and 9 (inclusive). You typically leave the lock with each dial one higher than the correct value. However, last night, someone changed all the values to 5. You recall that none of the digits in your code are prime numbers, 0, or 1, and the average val... | 10 |
Yukihira is counting the minimum number of lines $m$ that can be drawn on the plane so that they intersect in exactly $200$ distinct points. What is $m$? | 21 |
Find the maximal number of crosses, each consisting of 5 squares, that can be placed on an 8x8 grid without overlapping. | 8 |
Compute the number of ordered pairs \((m,n)\) of positive integers that satisfy the equation \(\text{lcm}(m,n)+\gcd(m,n)=m+n+30\). | 16 |
Let $a = 111\ldots111$ (forty ones) and $b = 111\ldots111$ (twelve ones). Determine the greatest common divisor of $a$ and $b$. | 1111 |
Find the number of integers $n$ with $1 \leq n \leq 100$ for which $n - \phi(n)$ is prime. Here, $\phi(n)$ denotes the number of positive integers less than $n$ which are relatively prime to $n$. | 13 |
If $a_1, a_2, \ldots, a_{2013} \in [-2, 2]$ and $a_1 + a_2 + \cdots + a_{2013} = 0$, find the maximum of $a_1^3 + a_2^3 + \cdots + a_{2013}^3$. | 4026 |
Consider a triangle $ABC$ where $AB = AC = 15$ and $BC = 18$. Points $D$ and $E$ are chosen on $CA$ and $CB$, respectively, such that $CD = 5$ and $CE = 3$. The point $F$ is chosen on the half-line $\overrightarrow{DE}$ so that $EF = 8$. If $M$ is the midpoint of $AB$ and $N$ is the intersection of $FM$ and $BC$, what ... | 9 |
Let $T = TNFTPP$, and let $S$ be the sum of the digits of $T$. In triangle $ABC$, points $D$, $E$, and $F$ are the feet of the angle bisectors of $\angle A$, $\angle B$, $\angle C$ respectively. Let point $P$ be the intersection of segments $AD$ and $BE$, and let $p$ denote the perimeter of $ABC$. If $AP = 3PD$, $BE = ... | 6 |
Let $p(n) = n^4 - 6n^2 - 160$. If $a_n$ is the least odd prime dividing $q(n) = |p(n-30) \cdot p(n+30)|$, find $\sum_{n=1}^{2017} a_n$. ($a_n = 3$ if $q(n) = 0$.) | 7933 |
Compute the number of non-negative integers $k < 2^{20}$ such that $\binom{5k}{k}$ is odd. | 20736 |
A positive integer $n$ is called \textit{mythical} if every divisor of $n$ is two less than a prime. Find the unique mythical number with the largest number of divisors. | 135 |
Find the integer closest to the value of the following expression: $$\left((3 + \sqrt{1})^{2023} - \left(\frac{1}{3 - \sqrt{1}}\right)^{2023} \right) \cdot \left((3 + \sqrt{2})^{2023} - \left(\frac{1}{3 - \sqrt{2}}\right)^{2023} \right) \cdot \ldots \cdot \left((3 + \sqrt{8})^{2023} - \left(\frac{1}{3 - \sqrt{8}}\right... | 0 |
Let $O$ and $A$ be two points in the plane with $OA = 30$, and let $\Gamma$ be a circle with center $O$ and radius $r$. Suppose that there exist two points $B$ and $C$ on $\Gamma$ with $\angle ABC = 90^{\circ}$ and $AB = BC$. Compute the minimum possible value of $\lfloor r \rfloor.$ | 12 |
The integers from $1$ to $25$, inclusive, are randomly placed into a $5 \times 5$ grid such that in each row, the numbers are increasing from left to right. If the columns from left to right are numbered $1, 2, 3, 4,$ and $5$, then the expected column number of the entry $23$ can be written as $\frac{a}{b}$ where $a$ a... | 17 |
Consider the polynomial $x^3 - 3x^2 + 10$. Let $a$, $b$, and $c$ be its roots. Compute $a^2b^2c^2 + a^2b^2 + b^2c^2 + c^2a^2 + a^2 + b^2 + c^2$. | 169 |
Let $f$ be a function defined on the set of positive integers, with values in the same set, which satisfies: - $f(n + f(n)) = 1$ for all $n \ge 1$. - $f(1998) = 2$ Find the lowest possible value of the sum $f(1) + f(2) + \ldots + f(1999)$, and find the formula of $f$ for which this minimum is satisfied. | 1997003 |
Let $\Omega_1$ be a circle with center $O$, and let $AB$ be the diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$, different from $O$. Suppose another circle $\Omega_2$ with center $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$, intersecting $\Omega_1$ ... | 20 |
$x$, $y$, and $z$ are real numbers such that $xyz = 10$. What is the maximum possible value of $x^3 y^3 z^3 - 3x^4 - 12y^2 - 12z^4$? | 760 |
The diagram shows an $8 \times 7$ rectangle with a 3-4-5 right triangle drawn in each corner. The lower two triangles have their sides of length 4 along the bottom edge of the rectangle, while the upper two triangles have their sides of length 3 along the top edge of the rectangle. A circle is tangent to the hypotenuse... | 47 |
Determine the last two digits of the number $Q = 2^{2017} + 2017^2$. | 61 |
How many real numbers $x$ are solutions to the equation $|x - 2| - 4 = \frac{1}{|x - 3|}$? | 2 |
There is a circle with a circumference of 1 meter and a point marked on it. Two cockroaches start running in the same direction from the marked point at different speeds. - Whenever the faster cockroach catches up with the slower one, the slower one instantly turns around and starts running in the opposite direction wi... | 0 |
There are $2010$ people sitting around a round table. First, we give one person $x$ a candy. Next, we give candies to the $1^{st}$ person, $1+2^{nd}$ person, $1+2+3^{rd}$ person, $\cdots$, and $1+2+\cdots+2009^{th}$ person clockwise from $x$. Find the number of people who get at least one candy. | 408 |
From the digits $1, 2, \ldots, 9$, we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows: $123456789$, $123456798$, $\ldots$, $987654321$. What is the $100000^{th}$ number? | 358926471 |
Fill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is $2005$. What is the number that goes into the leftmost box? ``` \begin{asy} size(300); label("999",(2.5,.5)); label("888",(7.5,.5)); draw((0,0)--(9,0)); draw((0,1)--(9,1)); for (int i=0; i<=9; ++i) { draw((i,0)--(i,1));... | 118 |
The numbers $\frac{1}{1}, \frac{1}{2}, \ldots, \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$ and $y$, erases them, and then writes down $x + y + xy$. He continues this process until only one number remains on the blackboard. What is this number? | 2010 |
Marko lives on the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\frac{2}{9}$, $1$ unit right with probability $\frac{2}{9}$, $1$ unit up and $1$ unit right with probability $\frac{4}{9}$, and he doesn’t move with probability $\frac{1}{9}$. After $2019$ seconds, Marko ends up on... | 1811716 |
A group of \( n \) friends participated in a math contest consisting of eight short-answer problems \( S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8 \) and four full-solution problems \( F_1, F_2, F_3, F_4 \). Each person in the group correctly solved exactly 11 out of the 12 problems. We create an \( 8 \times 4 \) table. Ins... | 10 |
Find the greatest value of $C$ for which, for any $x, y, z, u$ such that $0 \le x \le y \le z \le u$, the inequality holds: $$(x + y + z + u)^2 \ge C yz.$$ | 9 |
Let $a$, $b$, and $c$ be nonzero real numbers such that $a + \frac{1}{b} = 5$, $b + \frac{1}{c} = 12$, and $c + \frac{1}{a} = 13$. Find $abc + \frac{1}{abc}$. | 750 |
Marisela is putting on a juggling show! She starts with $1$ ball, tossing it once per second. Lawrence tosses her another ball every five seconds, and she always tosses each ball that she has once per second. Compute the total number of tosses Marisela has made one minute after she starts juggling. | 390 |
Ranu starts with one standard die on a table. At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step. After 2013 such steps, let $D$ be the number of dice on the table. What is the expected value (averag... | 10071 |
Consider the addition problem: \[ \begin{array}{ccccc} & C & A & S & H \\ + & & & M & E \\ \hline O & S & I & D & E \end{array} \] where each letter represents a base-ten digit, and \(C, M, O \neq 0\). Distinct letters are allowed to represent the same digit. How many ways are there to assign values to the letters so t... | 0 |
In the obtuse triangle $ABC$ with $AB = 7$, $BC = 20$, and $CA = 15$, let point $D$ be the foot of the altitude from $C$ to line $AB$. Evaluate $[ACD] + [BCD]$. (Note that $[XYZ]$ denotes the area of triangle $XYZ$.) | 150 |
Given a positive integer $n$. The number $2n$ has $28$ positive factors, while the number $3n$ has $30$ positive factors. Find the number of positive divisors of $6n$. | 35 |
There exist positive integers $N$ and $M$ such that the remainders of $N$ when divided by the four integers $6$, $36$, $216$, and $M$ form an increasing nonzero geometric sequence in that order. Find the smallest possible value of $M$. | 2001 |
The people in an infinitely long line are numbered $1,2,3,\dots$. Then, each person says either "Karl" or "Lark" independently and at random. Let $S$ be the set of all positive integers $i$ such that people $i$, $i+1$, and $i+2$ all say "Karl," and define $X = \sum_{i \in S} 2^{-i}$. Then the expected value of $X^2$ is... | 1028 |
Find the number of 4-digit numbers with distinct digits chosen from the set $\{0,1,2,3,4,5\}$ in which no two adjacent digits are even. | 150 |
Find the smallest natural number $x > 0$ such that all the following fractions are simplified: $$\frac{3x+9}{8}, \frac{3x+10}{9}, \frac{3x+11}{10}, \ldots, \frac{3x+49}{48}$$ This means that the numerators and denominators are relatively prime. | 20 |
Determine all positive integers that are equal to $300$ times the sum of their digits. | 2700 |
Let $\mathbb{Z}_{\ge 0}$ denote the set of nonnegative integers. Define a function $f:\mathbb{Z}_{\ge 0} \to \mathbb{Z}$ with $f\left(0\right)=1$ and \[ f\left(n\right)=512^{\left\lfloor n/10 \right\rfloor}f\left(\left\lfloor n/10 \right\rfloor\right) \] for all $n \ge 1$. Determine the number of nonnegative integers $... | 10010 |
Find the minimum positive value of $1 \times 2 \times 3 \times \ldots \times 2022$ where you can replace $\times$ with $+$ or $-$. | 1 |
Find the number of pairs \((a, b)\) of positive integers such that the greatest common divisor (GCD) of \(a\) and \(b\) is equal to \(1 \cdot 2 \cdot 3 \cdot \ldots \cdot 50\), and the least common multiple (LCM) of \(a\) and \(b\) is \(1^2 \cdot 2^2 \cdot 3^2 \cdot \ldots \cdot 50^2\). | 32768 |
Find the least positive integer $N$ which is both a multiple of 19 and whose digits add to 23. | 779 |
Determine the largest integer $n$ such that $2^n$ divides the decimal representation given by some permutation of the digits $2$, $0$, $1$, and $5$. For example, $2^1$ divides $2150$. Note: The number may start with $0$. | 10 |
Let $\Gamma$ be the maximum possible value of $a+3b+9c$ among all triples $(a,b,c)$ of positive real numbers such that \[ \log_{30}(a+b+c) = \log_{8}(3a) = \log_{27} (3b) = \log_{125} (3c) .\] If $\Gamma = \frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers, then find $p+q$. | 16 |
Mr. Vader gave out a multiple choice test, and every question had an answer that was one of A, B, or C. After the test, he curved the test so that everybody got $+50$ (so a person who got $x\%$ right would get a score of $x+50$). In the class, a score in the range $[90, \infty)$ gets an A, a score in the range $[80, 90... | 64 |
A sequence $a_0, a_1, a_2, \dots$ of real numbers satisfies $a_0 = 999$, $a_1 = -999$, and $a_n = a_{n-1}a_{n+1}$ for each positive integer $n$. Compute $\left\lvert a_1 + a_2 + \dots + a_{1000} \right\rvert$. | 1332 |
The \emph{Stooge sort} is a particularly inefficient recursive sorting algorithm defined as follows: given an array $A$ of size $n$, we swap the first and last elements if they are out of order. We then (if $n \ge 3$) Stooge sort the first $\lceil \frac{2n}{3} \rceil$ elements, then the last $\lceil \frac{2n}{3} \rceil... | 243 |
The diagram below shows a triangle divided into sections by three horizontal lines, which divide the altitude of the triangle into four equal parts, and three lines connecting the top vertex with points that divide the opposite side into four equal parts. If the shaded region has an area of $100$, find the area of the ... | 640 |
Let $M$ be a subset of the set of $2021$ integers $\{1, 2, 3, \ldots, 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$, we have $|a + b - c| > 10$. Determine the largest possible number of elements of $M$. | 1006 |
For how many $x \in \{1,2,3,\dots, 2024\}$ is it possible that Bekhzod summed $2024$ non-negative consecutive integers, Ozod summed $2024+x$ non-negative consecutive integers and they got the same result? | 1138 |
Each number from the set $\{1, 2, 3, 4, 5, 6, 7\}$ must be written in each circle of the diagram, so that the sum of any three aligned numbers is the same (e.g., $A+D+E = D+C+B$). What number cannot be placed on the circle $E$? | 4 |
Point $A$, $B$, $C$, and $D$ form a rectangle in that order. Point $X$ lies on $CD$, and segments $\overline{BX}$ and $\overline{AC}$ intersect at $P$. If the area of triangle $BCP$ is 3 and the area of triangle $PXC$ is 2, what is the area of the entire rectangle? | 15 |
Call a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x^n$ starts with $2012$. For example, $586$ is a leader since $586^3 = 201230056$. How many leaders are there in the set $\{1, 2, 3, \ldots, 2012\}$? | 2008 |
A subset $S$ of the set of integers $0$ to $99$ is said to have property $A$ if it is impossible to fill a crossword-puzzle with $2$ rows and $2$ columns using numbers in $S$ (where $0$ is written as $00$, $1$ as $01$, and so on). Determine the maximal number of elements in the set $S$ with the property $A$. | 25 |
A single section at a stadium can hold either $7$ adults or $11$ children. When $N$ sections are completely filled, an equal number of adults and children will be seated in them. What is the least possible value of $N$? | 18 |
What is the smallest perfect square larger than $1$ with a perfect square number of positive integer factors? | 36 |
Define $a_k = (k^2 + 1)k!$ and $b_k = a_1 + a_2 + a_3 + \cdots + a_k$. Let \[\frac{a_{100}}{b_{100}} = \frac{m}{n}\] where $m$ and $n$ are relatively prime natural numbers. Find $n - m$. | 99 |
When counting from $3$ to $201$, $53$ is the $51^{\text{st}}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{\text{th}}$ number counted. What is $n$? \[ \textbf{(A) } 146 \quad \textbf{(B) } 147 \quad \textbf{(C) } 148 \quad \textbf{(D) } 149 \quad \textbf{(E) } 150 \] | 149 |
Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$. How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$? Note that $\lfloor x\rfloor$ is known as the \textit{floor} function, which returns the greatest integ... | 1973 |
There are $2024$ people, consisting of knights and liars, and some of them are friends. Each person was asked how many friends they have, and the possible answers were $0, 1, \ldots, 2023$. Every knight answered truthfully, while every liar altered the real answer by exactly $1$. What is the minimal number of liars? | 1012 |
It is given that $2^{333}$ is a 101-digit number whose first digit is 1. How many of the numbers $2^k$, where $1 \leq k \leq 332$, have the first digit as 4? | 32 |
Find the period of the repetend of the fraction $\frac{39}{1428}$ by using binary numbers, i.e., its binary decimal representation. (Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2\cdots ... | 24 |
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