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If integers $a$ , $b$ , $c$ , and $d$ satisfy $ bc + ad = ac + 2bd = 1 $ , find all possible values of $ \frac {a^2 + c^2}{b^2 + d^2} $ . | 2 | 0.666667 |
Find the smallest positive integer, $n$ , which can be expressed as the sum of distinct positive integers $a,b,c$ such that $a+b,a+c,b+c$ are perfect squares. | 55 | 0.166667 |
If the integers $1,2,\dots,n$ can be divided into two sets such that each of the two sets does not contain the arithmetic mean of its any two elements, find the largest possible value of $n$. | 8 | 0.083333 |
For positive integer $n$ we define $f(n)$ as sum of all of its positive integer divisors (including $1$ and $n$ ). Find all positive integers $c$ such that there exists strictly increasing infinite sequence of positive integers $n_1, n_2,n_3,...$ such that for all $i \in \mathbb{N}$ holds $f(n_i)-n_i=c$ | c = 1 | 0.583333 |
How many positive integers less than $2010$ are there such that the sum of factorials of its digits is equal to itself? | 3 | 0.166667 |
Let $n\ge 3$ . Suppose $a_1, a_2, ... , a_n$ are $n$ distinct in pairs real numbers.
In terms of $n$ , find the smallest possible number of different assumed values by the following $n$ numbers: $$ a_1 + a_2, a_2 + a_3,..., a_{n- 1} + a_n, a_n + a_1 $$ | 3 | 0.166667 |
Given an acute triangle $ABC$ with altituties AD and BE. O circumcinter of $ABC$ .If o lies on the segment DE then find the value of $sinAsinBcosC$ | \frac{1}{2} | 0.166667 |
Let $ABC$ be an acute-angled triangle and $P$ be a point in its interior. Let $P_A,P_B$ and $P_c$ be the images of $P$ under reflection in the sides $BC,CA$ , and $AB$ , respectively. If $P$ is the orthocentre of the triangle $P_AP_BP_C$ and if the largest angle of the triangle that can be formed by the line segments $ PA, PB$ . and $PC$ is $x^o$ , determine the value of $x$ . | 120^\circ | 0.916667 |
Find all positive integers $m$ for which $2001\cdot S (m) = m$ where $S(m)$ denotes the sum of the digits of $m$ . | 36018 | 0.333333 |
For some real number $c,$ the graphs of the equation $y=|x-20|+|x+18|$ and the line $y=x+c$ intersect at exactly one point. What is $c$ ? | 18 | 0.333333 |
Given that $ n$ is any whole number, determine the smallest positive integer that always divides $ n^2(n^2 \minus{} 1)$. | 12 | 0.75 |
Let $A$ , $B$ , $C$ , $D$ , $E$ , $F$ , $G$ , $H$ , $I$ be nine points in space such that $ABCDE$ , $ABFGH$ , and $GFCDI$ are each regular pentagons with side length $1$ . Determine the lengths of the sides of triangle $EHI$ . | 1 | 0.166667 |
Given the equation $2\cos(2x)\left(\cos(2x)-\cos\left(\frac{2014\pi^2}{x}\right)\right)=\cos(4x)-1$, find the sum of all positive real solutions $x$ to this equation. | 1080\pi | 0.166667 |
Find all quadruples $(p, q, r, n)$ of prime numbers $p, q, r$ and positive integer numbers $n$ , such that $$ p^2 = q^2 + r^n $$ (Walther Janous) | (p, q, r, n) = (3, 2, 5, 1) \text{ and } (p, q, r, n) = (5, 3, 2, 4) | 0.25 |
For real numbers $a$ and $b$ , define $$ f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}. $$ Find the smallest possible value of the expression $$ f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b). $$ | 4 \sqrt{2018} | 0.25 |
For how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\frac{8n}{9999-n}$ an integer? | 1 | 0.416667 |
Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ( $\angle C=90^\circ$ ). It is known that the circumcircles of triangles $AEM$ and $CDM$ are tangent. Find the angle $\angle BMC.$ | 90^\circ | 0.5 |
Find the largest $n$ such that the last nonzero digit of $n!$ is $1$ . | 1 | 0.416667 |
Let $a$ , $b$ , $c$ , $d$ , $e$ be real strictly positive real numbers such that $abcde = 1$ . Then is true the following inequality: $$ \frac{de}{a(b+1)}+\frac{ea}{b(c+1)}+\frac{ab}{c(d+1)}+\frac{bc}{d(e+1)}+\frac{cd}{e(a+1)}\geq \frac{5}{2} $$ | \frac{5}{2} | 0.916667 |
Find all integers $n\geq 1$ such that there exists a permutation $(a_1,a_2,...,a_n)$ of $(1,2,...,n)$ such that $a_1+a_2+...+a_k$ is divisible by $k$ for $k=1,2,...,n$ | n = 1 | 0.5 |
Every cell of an $8\times8$ chessboard contains either $1$ or $-1$. Given that there are at least four rows such that the sum of numbers inside the cells of those rows is positive, determine the maximum number of columns such that the sum of numbers inside the cells of those columns is less than $-3$. | 6 | 0.25 |
$a_0, a_1, \ldots, a_{100}$ and $b_1, b_2,\ldots, b_{100}$ are sequences of real numbers, for which the property holds: for all $n=0, 1, \ldots, 99$ , either $$ a_{n+1}=\frac{a_n}{2} \quad \text{and} \quad b_{n+1}=\frac{1}{2}-a_n, $$ or $$ a_{n+1}=2a_n^2 \quad \text{and} \quad b_{n+1}=a_n. $$ Given $a_{100}\leq a_0$ , what is the maximal value of $b_1+b_2+\cdots+b_{100}$ ? | 50 | 0.5 |
In a pile you have 100 stones. A partition of the pile in $ k$ piles is *good* if:
1) the small piles have different numbers of stones;
2) for any partition of one of the small piles in 2 smaller piles, among the $ k \plus{} 1$ piles you get 2 with the same number of stones (any pile has at least 1 stone).
Find the maximum and minimal values of $ k$ for which this is possible. | k_{\text{max}} = 13 | 0.416667 |
Let $a,b,c$ be distinct real numbers such that $a+b+c>0$ . Let $M$ be the set of $3\times 3$ matrices with the property that each line and each column contain all given numbers $a,b,c$ . Find $\{\max \{ \det A \mid A \in M \}$ and the number of matrices which realise the maximum value.
*Mircea Becheanu* | 6 | 0.25 |
Let $ABC$ be an equilateral triangle. Denote by $D$ the midpoint of $\overline{BC}$ , and denote the circle with diameter $\overline{AD}$ by $\Omega$ . If the region inside $\Omega$ and outside $\triangle ABC$ has area $800\pi-600\sqrt3$ , find the length of $AB$ .
*Proposed by Eugene Chen* | 80 | 0.166667 |
What is the largest positive integer $n < 1000$ for which there is a positive integer $m$ satisfying \[\text{lcm}(m,n) = 3m \times \gcd(m,n)?\] | 972 | 0.666667 |
Find the number of ordered $64$ -tuples $\{x_0,x_1,\dots,x_{63}\}$ such that $x_0,x_1,\dots,x_{63}$ are distinct elements of $\{1,2,\dots,2017\}$ and
\[x_0+x_1+2x_2+3x_3+\cdots+63x_{63}\]
is divisible by $2017.$ | \frac{2016!}{1953!} | 0.333333 |
Find all integers $ n \ge 3$ such that there are $ n$ points in space, with no three on a line and no four on a circle, such that all the circles pass through three points between them are congruent. | n = 4 | 0.25 |
$p(m)$ is the number of distinct prime divisors of a positive integer $m>1$ and $f(m)$ is the $\bigg \lfloor \frac{p(m)+1}{2}\bigg \rfloor$ th smallest prime divisor of $m$ . Find all positive integers $n$ satisfying the equation: $$ f(n^2+2) + f(n^2+5) = 2n-4 $$ | n = 5 | 0.25 |
The towns in one country are connected with bidirectional airlines, which are paid in at least one of the two directions. In a trip from town A to town B there are exactly 22 routes that are free. Find the least possible number of towns in the country. | 7 | 0.166667 |
$2021$ points are given on a circle. Each point is colored by one of the $1,2, \cdots ,k$ colors. For all points and colors $1\leq r \leq k$ , there exist an arc such that at least half of the points on it are colored with $r$ . Find the maximum possible value of $k$ . | k = 2 | 0.166667 |
Let $x, y$ be two positive integers, with $x> y$ , such that $2n = x + y$ , where n is a number two-digit integer. If $\sqrt{xy}$ is an integer with the digits of $n$ but in reverse order, determine the value of $x - y$ | 66 | 0.416667 |
Let $\triangle ABC$ be a triangle with $AB = 7$ , $AC = 8$ , and $BC = 3$ . Let $P_1$ and $P_2$ be two distinct points on line $AC$ ( $A, P_1, C, P_2$ appear in that order on the line) and $Q_1$ and $Q_2$ be two distinct points on line $AB$ ( $A, Q_1, B, Q_2$ appear in that order on the line) such that $BQ_1 = P_1Q_1 = P_1C$ and $BQ_2 = P_2Q_2 = P_2C$ . Find the distance between the circumcenters of $BP_1P_2$ and $CQ_1Q_2$ . | 3 | 0.333333 |
Let $T_1$ and $T_2$ be the points of tangency of the excircles of a triangle $ABC$ with its sides $BC$ and $AC$ respectively. It is known that the reflection of the incenter of $ABC$ across the midpoint of $AB$ lies on the circumcircle of triangle $CT_1T_2$ . Find $\angle BCA$ . | 90^\circ | 0.5 |
Given positive integers $a$, $b$, and $c$ such that $a<b<c$, and the system of equations $2x+y=2003$ and $y=|x-a|+|x-b|+|x-c|$ has exactly one solution, calculate the minimum value of $c$. | 1002 | 0.166667 |
Triangle $\vartriangle ABC$ has circumcenter $O$ and orthocenter $H$ . Let $D$ be the foot of the altitude from $A$ to $BC$ , and suppose $AD = 12$ . If $BD = \frac14 BC$ and $OH \parallel BC$ , compute $AB^2$ .
. | 160 | 0.083333 |
We say that a polygon $P$ is inscribed in another polygon $Q$ when all the vertices of $P$ belong to the perimeter of $Q$ . We also say in this case that $Q$ is circumscribed to $P$ . Given a triangle $T$ , let $\ell$ be the largest side of a square inscribed in $T$ and $L$ is the shortest side of a square circumscribed to $T$ . Find the smallest possible value of the ratio $L/\ell$ . | 2 | 0.25 |
Perimeter of triangle $ABC$ is $1$ . Circle $\omega$ touches side $BC$ , continuation of side $AB$ at $P$ and continuation of side $AC$ in $Q$ . Line through midpoints $AB$ and $AC$ intersects circumcircle of $APQ$ at $X$ and $Y$ .
Find length of $XY$ . | \frac{1}{2} | 0.833333 |
Let $\{a_n\}$ be a sequence of integers satisfying the following conditions.
- $a_1=2021^{2021}$
- $0 \le a_k < k$ for all integers $k \ge 2$
- $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$ .
Determine the $2021^{2022}$ th term of the sequence $\{a_n\}$ . | 0 | 0.5 |
An infinite sequence of digits $1$ and $2$ is determined by the following two properties:
i) The sequence is built by writing, in some order, blocks $12$ and blocks $112.$ ii) If each block $12$ is replaced by $1$ and each block $112$ by $2$ , the same sequence is again obtained.
In which position is the hundredth digit $1$ ? What is the thousandth digit of the sequence? | 2 | 0.166667 |
$25.$ Let $C$ be the answer to Problem $27.$ What is the $C$ -th smallest positive integer with exactly four positive factors? $26.$ Let $A$ be the answer to Problem $25.$ Determine the absolute value of the difference between the two positive integer roots of the quadratic equation $x^2-Ax+48=0$ $27.$ Let $B$ be the answer to Problem $26.$ Compute the smallest integer greater than $\frac{B}{\pi}$ | 8 | 0.25 |
Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$ , Oleg randomly selected a pair $(X, Y)$ . Compute the probability that $X = Y$ . | \frac{1}{20} | 0.333333 |
Find all positive integers $n$ for which there exist real numbers $x_1, x_2,. . . , x_n$ satisfying all of the following conditions:
(i) $-1 <x_i <1,$ for all $1\leq i \leq n.$ (ii) $ x_1 + x_2 + ... + x_n = 0.$ (iii) $\sqrt{1 - x_1^2} +\sqrt{1 - x^2_2} + ... +\sqrt{1 - x^2_n} = 1.$ | n = 1 | 0.083333 |
A $5\times5$ grid of squares is filled with integers. Call a rectangle *corner-odd* if its sides are grid lines and the sum of the integers in its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid?
Note: A rectangles must have four distinct corners to be considered *corner-odd*; i.e. no $1\times k$ rectangle can be *corner-odd* for any positive integer $k$ . | 60 | 0.083333 |
Let $k$ , $m$ and $n$ be integers such that $1<n \leq m-1 \leq k$ . Find maximum size of subset $S$ of set $\{1,2,...,k\}$ such that sum of any $n$ different elements from $S$ is not: $a)$ equal to $m$ , $b)$ exceeding $m$ | \left\lfloor \frac{m}{n} + \frac{n-1}{2} \right\rfloor | 0.083333 |
Suppose $a$ and $b$ are positive integers with a curious property: $(a^3 - 3ab +\tfrac{1}{2})^n + (b^3 +\tfrac{1}{2})^n$ is an integer for at least $3$ , but at most finitely many different choices of positive integers $n$ . What is the least possible value of $a+b$ ? | 6 | 0.166667 |
Find the maximum value of $$ \int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx $$ over all continuously differentiable functions $f:[0,1]\to\mathbb R$ with $f(0)=0$ and $$ \int^1_0|f'(x)|^2dx\le1. $$ | \frac{2}{3} | 0.5 |
For each vertex of triangle $ABC$ , the angle between the altitude and the bisectrix from this vertex was found. It occurred that these angle in vertices $A$ and $B$ were equal. Furthermore the angle in vertex $C$ is greater than two remaining angles. Find angle $C$ of the triangle. | \gamma = 120^\circ | 0.333333 |
Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2+qx+r=0.$ Which primes appear in seven or more elements of $S?$ | 2 | 0.5 |
Simone begins working on three tasks at 8:00 AM. The first two tasks each take 45 minutes, and after completing these, she takes a 15-minute break. The third task takes twice as long as each of the first two tasks. Calculate the time at which she finishes the third task. | 11:15 \text{ AM} | 0.833333 |
A rectangle with dimensions 10 units by 5 units is cut in half across its length, creating two congruent squares. Determine the side length of one of these squares. | 5 | 0.916667 |
Alex paid $250 for a set of four tires with a deal that included a fourth tire for $10. What was the regular price of one tire? | 80 | 0.916667 |
Luis's flight took off from Chicago at 3:15 PM and landed in Denver at 4:42 PM. Chicago is in the Central Time Zone, while Denver is in the Mountain Time Zone, which is 1 hour behind Chicago. If his flight took $h$ hours and $m$ minutes, with $0 < m < 60$, calculate the value of $h + m$. | 29 | 0.666667 |
Given that the score of Elena's video was $150$ and that $60\%$ of the votes were likes, calculate the total number of votes cast on Elena's video at that point. | 750 | 0.583333 |
Given that the car ran exclusively on its battery for the first $60$ miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of $0.03$ gallons per mile, and the car averaged $75$ miles per gallon on the whole trip, determine the total length of the trip in miles. | 108 | 0.833333 |
A right circular cone and a sphere have the same base radius, denoted as \( r \). The volume of the cone is one-third that of the sphere. Find the ratio of the altitude of the cone (\( h \)) to the radius of its base (\( r \)). | \frac{4}{3} | 0.833333 |
Given that the first vessel was $\frac{3}{4}$ full of oil, and after transferring all the oil to the second vessel, the second vessel was $\frac{5}{8}$ full, determine the ratio of the volume of the first vessel to the volume of the second vessel. | \frac{5}{6} | 0.916667 |
Tom's age $T$ years is equal to the sum of the ages of his four children. His age $M$ years ago was three times the sum of their ages then. Find the value of $T/M$. | \frac{11}{2} | 0.916667 |
The cost of the shirts was $\frac{5}{6}$ of the price at which he actually sold them, and a discount of $\frac{1}{4}$ was advertised. Find the ratio of the cost to the marked price. | \frac{5}{8} | 0.916667 |
Suppose $M$ minutes of internet usage can be purchased from a service provider for $P$ pennies. Determine the number of minutes of internet usage that can be purchased for $E$ euros, where 1 euro is equivalent to 100 pennies. | \frac{100EM}{P} | 0.416667 |
Carlos took $60\%$ of a whole pie. Maria took one fourth of the remainder. What portion of the whole pie was left? | 30\% | 0.916667 |
If one side of a triangle is $16$ inches and the opposite angle is $45^{\circ}$, calculate the diameter of the circumscribed circle. | 16\sqrt{2}\text{ inches} | 0.833333 |
Given Tom's age is $T$ years, which is also the sum of the ages of his four children, and his age $N$ years ago was three times the sum of their ages then, calculate $T/N$. | \frac{11}{2} | 0.833333 |
Given Tom's age is $T$ years, and the combined age of his partner and his four children is $T$ years. If $N$ years ago, Tom's age was thrice the sum of his partner's and children's ages then, find the value of $T/N$. | 7 | 0.75 |
Alice commutes to her office daily, taking normally 30 minutes in moderate traffic and 18 minutes in less traffic with an increased speed of 12 miles per hour. Determine the distance from Alice's home to her office in miles. | 9 | 0.416667 |
A $9 \times 9$ board consists of alternating light and dark squares. Calculate the difference between the number of dark squares and the number of light squares. | 1 | 0.5 |
If the digit 2 is placed after a three-digit number whose hundreds' digit is $a$, tens' digit is $b$, and units' digit is $c$, calculate the resulting four-digit number. | 1000a + 100b + 10c + 2 | 0.833333 |
Given that Ella has two containers, the first is $\tfrac{3}{4}$ full of juice and the second is completely empty, she transfers all the juice from the first container into the second container, and then the second container is $\tfrac{5}{8}$ full. What is the ratio of the volume of the first container to the volume of the second container? | \frac{5}{6} | 0.916667 |
Given that $\frac{2}{3}$ of the marbles are blue and the rest are red, what fraction of the marbles will be red after the number of red marbles is doubled and the number of blue marbles is increased by 50%? | \frac{2}{5} | 0.916667 |
The point \( P(a, b) \) in the \( xy \)-plane is first rotated counterclockwise by \( 180^\circ \) around the point \( (2, 3) \) and then reflected about the line \( y = x \). The image of \( P \) after these two transformations is at \( (1, -4) \). Find the value of \( b - a \). | -3 | 0.916667 |
Given that a triangle and a trapezoid are on parallel bases and share the same height, the base of the triangle is 24 inches, and the area of the triangle is 192 square inches, find the length of the median of the trapezoid assuming it has the same area as the triangle. | 12 | 0.916667 |
Find the number of three-digit whole numbers where the sum of the digits is $27$. | 1 | 0.916667 |
Theresa's parents have decided to buy her a new phone if she spends an average of 12 hours per week helping around the house for 7 weeks. For the first 6 weeks, she helps around the house for 14, 10, 13, 9, 12, and 11 hours. Calculate the number of hours she must work in the final week to earn the phone. | 15 | 0.416667 |
Given a 6x6 formation consisting of black and white square tiles with 12 black tiles and 24 white tiles spaced out uniformly, the pattern is surrounded by a border of white tiles. What is the new ratio of black tiles to white tiles in the extended pattern? | \frac{3}{13} | 0.583333 |
Consider a regular 25-gon. It has a certain number of lines of symmetry, $L$, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. Calculate $L + \frac{R}{2}$. | 32.2 | 0.75 |
Given that Luis wants to arrange his sticker collection in rows with exactly 4 stickers in each row, and he has 29 stickers initially, find the minimum number of additional stickers Luis must purchase so that the total number of stickers can be exactly split into 5 equal groups without any stickers left over. | 11 | 0.083333 |
A laptop is originally priced at $1200$ dollars and is put on sale for $30\%$ off. If a $12\%$ tax was added to the sale price, calculate the total selling price (in dollars) of the laptop. | 940.8 | 0.666667 |
Mr. Lee sold two speakers at $\textdollar{1.44}$ each. One speaker was sold at a 20% profit, whereas the other was sold at a 10% loss. Did Mr. Lee break even, or how much did he gain or lose in the overall transaction?
A) Lost 8 cents
B) Gained 8 cents
C) Broke even
D) Lost 16 cents
E) Gained 24 cents | B) Gained 8 cents | 0.083333 |
During a storm, Alex and Jamie are fishing 2 miles from the shore when their boat begins to leak. Water enters the boat at a rate of 15 gallons per minute due to the leak. Additionally, because of the heavy rain, an extra 5 gallons of water per minute accumulates in the boat. The boat will sink if it takes in more than 60 gallons of water. Alex rows towards the shore at a constant rate of 3 miles per hour while Jamie tries to bail out water. Determine the minimum rate at which Jamie must bail water, in gallons per minute, to ensure they reach the shore without the boat sinking. | 18.5 | 0.833333 |
The ratio of $a$ to $b$ is $5:4$, the ratio of $c$ to $d$ is $4:1$, and the ratio of $d$ to $b$ is $2:5$. Find the ratio of $a$ to $c$. | \frac{25}{32} | 0.916667 |
Two pitchers, one holding 800 mL and another 500 mL, are partially filled with apple juice. The first pitcher is 1/4 full and the second pitcher is 1/5 full. Water is added to fill each pitcher completely, then both pitchers are poured into a container. What fraction of the mixture in the container is apple juice? | \frac{3}{13} | 0.916667 |
Evaluate the expression: $3 - (-1) + 4 - 5 + (-6) - (-7) + 8 - 9$. | 3 | 0.75 |
In order to average 85 for five tests, determine the lowest score Shauna could earn on one of the other two tests, given that her scores on the first three tests are 82, 88, and 93. | 62 | 0.916667 |
Given $2^{1999} - 2^{1998} - 2^{1997} + 2^{1996} - 2^{1995} = m \cdot 2^{1995}$, calculate the value of $m$. | 5 | 0.833333 |
Calculate the sum of the numbers on the top faces that occurs with the same probability as when the sum is $15$ when $8$ standard dice are rolled. | 41 | 0.75 |
Given Mr. Green's garden measures $18$ steps by $25$ steps, and each of his steps is $2.5$ feet long, calculate the area of the garden in square feet, and then determine how many pounds of potatoes Mr. Green can expect from his garden given that he expects half a pound of potatoes per square foot. | 1406.25 | 0.916667 |
Calculate the sum of the numbers from 80 to 100, excluding 90. | 1800 | 0.916667 |
Given that five fish can be traded for three loaves of bread, one loaf of bread can be exchanged for six apples, and two apples can be traded for a bag of rice, determine the value of one fish in bags of rice. | \frac{9}{5} | 0.666667 |
The ratio of $p$ to $q$ is $5:4$, the ratio of $r$ to $s$ is $4:3$, and the ratio of $s$ to $q$ is $1:8$. Express the ratio of $p$ to $r$. | \frac{15}{2} | 0.916667 |
Given the expression $1-(-2)-3-(-4)-5-(-6)-7-(-8)$, evaluate the expression. | 6 | 0.916667 |
Jonathan has 1000 coins, consisting of pennies (1-cent coins) and dimes (10-cent coins), with at least one penny and one dime. Determine the difference in cents between the greatest possible and least amounts of money that Jonathan can have. | 8982 | 0.666667 |
If $\frac{2}{3}$ of the marbles are blue and the rest are red, what fraction of the marbles will be blue after the number of blue marbles is doubled and the number of red marbles stays the same? | \frac{4}{5} | 0.916667 |
Given that $|x^2 - 14x + 44| = 4$, calculate the sum of all real numbers $x$. | 28 | 0.916667 |
Given that Alice's car averages 30 miles per gallon of gasoline, and Bob's car averages 20 miles per gallon of gasoline, and Alice drives 120 miles and Bob drives 180 miles, calculate the combined rate of miles per gallon of gasoline for both cars. | \frac{300}{13} | 0.833333 |
Given the operation $a @ b = \frac{a \times b}{a+b}$, and $(a, b)$ are positive integers, calculate $35 @ 77$. | \frac{2695}{112} | 0.666667 |
Evaluate the expression $\frac{36 - 6 \times 3}{6 \div 3 \times 2}$. | 4.5 | 0.416667 |
Marie completes four tasks consecutively without taking any breaks. Each subsequent task takes 10 minutes longer than the previous one. The first task begins at 8:00 AM and takes 30 minutes. Determine the completion time of the fourth task. | 11:00 \text{ AM} | 0.833333 |
For real numbers \( x \) and \( y \), define \( x \spadesuit y = (x+y)(x-y) \). Evaluate the expression \( 2 \spadesuit (3 \spadesuit (1 \spadesuit 4)) \). | -46652 | 0.583333 |
Jenny bakes a 24-inch by 20-inch pan of banana bread. The banana bread is cut into pieces that measure 3 inches by 4 inches. How many pieces of banana bread does the pan contain? | 40 | 0.916667 |
Given the equation $\frac{1}{x^2} - \frac{1}{y^2} = \frac{1}{z}$, express the value of \( z \) in terms of \( x \) and \( y $. | \frac{x^2y^2}{y^2 - x^2} | 0.833333 |
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