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For every integer $n \ge 2$ let $B_n$ denote the set of all binary $n$ -nuples of zeroes and ones, and split $B_n$ into equivalence classes by letting two $n$ -nuples be equivalent if one is obtained from the another by a cyclic permutation.(for example 110, 011 and 101 are equivalent). Determine the integers $n \ge 2$ for which $B_n$ splits into an odd number of equivalence classes. | n = 2 | 0.166667 |
The five numbers $17$ , $98$ , $39$ , $54$ , and $n$ have a mean equal to $n$ . Find $n$ . | 52 | 0.583333 |
For a function satisfying $ f'(x) > 0$ for $ a\leq x\leq b$ , let $ F(x) \equal{} \int_a^b |f(t) \minus{} f(x)|\ dt$ . For what value of $ x$ is $ F(x)$ is minimized? | x = \frac{a + b}{2} | 0.75 |
If $ a\equal{}2b\plus{}c$ , $ b\equal{}2c\plus{}d$ , $ 2c\equal{}d\plus{}a\minus{}1$ , $ d\equal{}a\minus{}c$ , what is $ b$ ? | b = \frac{2}{9} | 0.916667 |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $$ \displaystyle{f(f(x)) = \frac{x^2 - x}{2}\cdot f(x) + 2-x,} $$ for all $x \in \mathbb{R}.$ Find all possible values of $f(2).$ | 2 | 0.5 |
Given $1 \le n \le 455$ and $n^3 \equiv 1 \pmod {455}$, calculate the number of solutions. | 9 | 0.416667 |
If we write $ |x^2 \minus{} 4| < N$ for all $ x$ such that $ |x \minus{} 2| < 0.01$, find the smallest value we can use for $ N$. | 0.0401 | 0.916667 |
Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $ . Here $[x]$ denotes the greatest integer less than or equal to $x.$ | 2997 | 0.666667 |
Given that $0.025$ mol of the isotope ${ }_{24}^{54}\text{Cr}$ contains $N$ neutrons, calculate the total number of neutrons in this quantity. | 4.5 \times 10^{23} | 0.083333 |
Determine all composite positive integers $n$ with the following property: If $1 = d_1 < d_2 < \cdots < d_k = n$ are all the positive divisors of $n$ , then $$ (d_2 - d_1) : (d_3 - d_2) : \cdots : (d_k - d_{k-1}) = 1:2: \cdots :(k-1) $$ (Walther Janous) | n = 4 | 0.833333 |
A 10-digit arrangement $ 0,1,2,3,4,5,6,7,8,9$ is called *beautiful* if (i) when read left to right, $ 0,1,2,3,4$ form an increasing sequence, and $ 5,6,7,8,9$ form a decreasing sequence, and (ii) $ 0$ is not the leftmost digit. For example, $ 9807123654$ is a beautiful arrangement. Determine the number of beautiful arrangements. | 126 | 0.5 |
Let $h_1$ and $h_2$ be the altitudes of a triangle drawn to the sides with length 5 and $2\sqrt 6$, respectively. If $5 + h_1 \leq 2\sqrt 6 + h_2$, determine the length of the third side of the triangle. | 7 | 0.916667 |
Comparing the numbers $10^{-49}$ and $2 \cdot 10^{-50}$, by how much does the first number exceed the second? | 8 \cdot 10^{-50} | 0.916667 |
We construct three circles: $O$ with diameter $AB$ and area $12+2x$ , $P$ with diameter $AC$ and area $24+x$ , and $Q$ with diameter $BC$ and area $108-x$ . Given that $C$ is on circle $O$ , compute $x$ . | 60 | 0.75 |
If the polynomial $ P(x)$ satisfies $ 2P(x) \equal{} P(x \plus{} 3) \plus{} P(x \minus{} 3)$ for every real number $x$, determine the maximum possible degree of $P(x)$. | 1 | 0.333333 |
Let $a,b,c\in \mathbb{R}$ with $a^2+b^2+c^2=1$ and $\lambda\in \mathbb{R}_{>0}\setminus\{1\}$ . Then for each solution $(x,y,z)$ of the system of equations:
\[
\begin{cases}
x-\lambda y=a,
y-\lambda z=b,
z-\lambda x=c.
\end{cases}
\]
we have $\displaystyle x^2+y^2+z^2\leqslant \frac1{(\lambda-1)^2}$ .
*Radu Gologan* | \frac{1}{(1 - \lambda)^2} | 0.083333 |
Given a cuboctahedron with 6 square faces and 8 equilateral triangle faces, find the value of 100 times the square of the ratio of the volume of an octahedron to a cuboctahedron with the same side length. | 4 | 0.75 |
$\textbf{Problem 5.}$ Miguel has two clocks, one clock advances $1$ minute per day and the other one goes $15/10$ 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 | 0.5 |
The measures (in degrees) of the interior angles of a convex hexagon form an arithmetic sequence of positive integers. Let $m^{\circ}$ be the measure of the largest interior angle of the hexagon. Find the largest possible value of $m^{\circ}$. | 175 | 0.666667 |
A subset of the real numbers has the property that for any two distinct elements of it such as x and y, we have $ (x+y-1)^2 = xy+1 $. Find the maximum number of elements in this set. | 3 | 0.583333 |
Determine the smallest positive integer $a$ for which there exist a prime number $p$ and a positive integer $b \ge 2$ such that $$ \frac{a^p -a}{p}=b^2. $$ | 9 | 0.25 |
For each real number $p > 1$ , find the minimum possible value of the sum $x+y$ , where the numbers $x$ and $y$ satisfy the equation $(x+\sqrt{1+x^2})(y+\sqrt{1+y^2}) = p$ . | \frac{p - 1}{\sqrt{p}} | 0.083333 |
The number 1 is a solution of the equation $(x + a)(x + b)(x + c)(x + d) = 16$ ,
where $a, b, c, d$ are positive real numbers. Find the largest value of $abcd$ . | 1 | 0.916667 |
Positive integers $a$ and $b$ satisfy $a^3 + 32b + 2c = 2018$ and $b^3 + 32a + 2c = 1115$ . Find $a^2 + b^2 + c^2$ . | 226 | 0.25 |
Let $T$ be the triangle in the $xy$ -plane with vertices $(0, 0)$ , $(3, 0)$ , and $\left(0, \frac32\right)$ . Let $E$ be the ellipse inscribed in $T$ which meets each side of $T$ at its midpoint. Find the distance from the center of $E$ to $(0, 0)$ . | \frac{\sqrt{5}}{2} | 0.833333 |
Let $\omega$ be the unit circle centered at the origin of $R^2$ . Determine the largest possible value for the radius of the circle inscribed to the triangle $OAP$ where $ P$ lies the circle and $A$ is the projection of $P$ on the axis $OX$ . | \frac{\sqrt{2} - 1}{2} | 0.666667 |
Given that $a$ , $b$ , and $c$ are the roots of the equation $x^3-3x^2+mx+24=0$ , and that $-a$ and $-b$ are the roots of the equation $x^2+nx-6=0$ , determine the value of $n$ . | -1 | 0.75 |
Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$ . | f(x) = \frac{1}{x} | 0.583333 |
Find all positive integers $n$ such that there exists a prime number $p$ , such that $p^n-(p-1)^n$ is a power of $3$ .
Note. A power of $3$ is a number of the form $3^a$ where $a$ is a positive integer. | n = 2 | 0.916667 |
On a table near the sea, there are $N$ glass boxes where $N<2021$ , each containing exactly $2021$ balls. Sowdha and Rafi play a game by taking turns on the boxes where Sowdha takes the first turn. In each turn, a player selects a non-empty box and throws out some of the balls from it into the sea. If a player wants, he can throw out all of the balls in the selected box. The player who throws out the last ball wins. Let $S$ be the sum of all values of $N$ for which Sowdha has a winning strategy and let $R$ be the sum of all values of $N$ for which Rafi has a winning strategy. What is the value of $\frac{R-S}{10}$ ? | 101 | 0.666667 |
$f : \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies $m+f(m+f(n+f(m))) = n + f(m)$ for every integers $m,n$. Given that $f(6) = 6$, determine $f(2012)$. | -2000 | 0.083333 |
Given the average of the numbers $1,2,3,...,98,99$, and $x$ is $100x$. Find the value of $x$. | \frac{50}{101} | 0.833333 |
Suppose a non-identically zero function $f$ satisfies $f\left(x\right)f\left(y\right)=f\left(\sqrt{x^2+y^2}\right)$ for all $x$ and $y$ . Compute $$ f\left(1\right)-f\left(0\right)-f\left(-1\right). $$ | -1 | 0.833333 |
Find all monic polynomials $P(x)=x^{2023}+a_{2022}x^{2022}+\ldots+a_1x+a_0$ with real coefficients such that $a_{2022}=0$ , $P(1)=1$ and all roots of $P$ are real and less than $1$ . | P(x) = x^{2023} | 0.75 |
All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values. | 100 | 0.75 |
Let points $A,B,$ and $C$ lie on a line such that $AB=1, BC=1,$ and $AC=2.$ Let $C_1$ be the circle centered at $A$ passing through $B,$ and let $C_2$ be the circle centered at $A$ passing through $C.$ Find the area of the region outside $C_1,$ but inside $C_2.$ | 3\pi | 0.916667 |
Note that $12^2=144$ ends in two $4$ s and $38^2=1444$ end in three $4$ s. Determine the length of the longest string of equal nonzero digits in which the square of an integer can end. | 3 | 0.75 |
Patrick started walking at a constant rate along a straight road from school to the park. One hour after Patrick left, Tanya started running along the same road from school to the park. One hour after Tanya left, Jose started bicycling along the same road from school to the park. Tanya ran at a constant rate of $2$ miles per hour faster than Patrick walked, Jose bicycled at a constant rate of $7$ miles per hour faster than Tanya ran, and all three arrived at the park at the same time. The distance from the school to the park is $\frac{m}{n}$ miles, where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . | 277 | 0.5 |
Find all positive integers $n$ for which the largest prime divisor of $n^2+3$ is equal to the least prime divisor of $n^4+6.$ | 3 | 0.833333 |
Find all positive integers $a,b,c$ satisfying $(a,b)=(b,c)=(c,a)=1$ and \[ \begin{cases} a^2+b\mid b^2+c b^2+c\mid c^2+a \end{cases} \] and none of prime divisors of $a^2+b$ are congruent to $1$ modulo $7$ | (1, 1, 1) | 0.75 |
Determine all functions $ f: \mathbb{N}_0 \rightarrow \mathbb{N}_0$ such that $ f(f(n))\plus{}f(n)\equal{}2n\plus{}6$ for all $ n \in \mathbb{N}_0$ . | f(n) = n + 2 | 0.166667 |
Given four points $O,\ A,\ B,\ C$ on a plane such that $OA=4,\ OB=3,\ OC=2,\ \overrightarrow{OB}\cdot \overrightarrow{OC}=3.$ Find the maximum area of $\triangle{ABC}$ . | 2\sqrt{7} + \frac{3\sqrt{3}}{2} | 0.083333 |
There are representants from $n$ different countries sit around a circular table ( $n\geq2$ ), in such way that if two representants are from the same country, then, their neighbors to the right are not from the same country. Find, for every $n$ , the maximal number of people that can be sit around the table. | n^2 | 0.166667 |
Let $N$ be the greatest integer multiple of $8,$ no two of whose digits are the same. What is the remainder when $N$ is divided by $1000?$ | 120 | 0.416667 |
$p(x)$ is the cubic $x^3 - 3x^2 + 5x$ . If $h$ is a real root of $p(x) = 1$ and $k$ is a real root of $p(x) = 5$ , find $h + k$ . | h + k = 2 | 0.416667 |
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$ ) Determine the smallest possible value of the probability of the event $X=0.$ | \frac{1}{3} | 0.416667 |
Harry Potter can do any of the three tricks arbitrary number of times: $i)$ switch $1$ plum and $1$ pear with $2$ apples $ii)$ switch $1$ pear and $1$ apple with $3$ plums $iii)$ switch $1$ apple and $1$ plum with $4$ pears
In the beginning, Harry had $2012$ of plums, apples and pears, each. Harry did some tricks and now he has $2012$ apples, $2012$ pears and more than $2012$ plums. What is the minimal number of plums he can have? | 2025 | 0.25 |
Given a square $ABCD,$ with $AB=1$ mark the midpoints $M$ and $N$ of $AB$ and $BC,$ respectively. A lasar beam shot from $M$ to $N,$ and the beam reflects of $BC,CD,DA,$ and comes back to $M.$ This path encloses a smaller area inside square $ABCD.$ Find this area. | \frac{1}{2} | 0.333333 |
In a ten-mile race, First beats Second by $2$ miles and First beats Third by $4$ miles. Determine by how many miles Second beats Third. | 2.5 | 0.75 |
Find the minimum value of the following function $f(x) $ defined at $0<x<\frac{\pi}{2}$ .
\[f(x)=\int_0^x \frac{d\theta}{\cos \theta}+\int_x^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta}\] | \ln\left(3 + 2\sqrt{2}\right) | 0.25 |
Let $a,b,c$ and $m$ be integers such that $0 \le m \le 26$ , and $a + b + c = (a - b)(b- c)(c - a) \equiv m \pmod{27}$ . Determine the value of $m$ . | 0 | 0.583333 |
Given $8$ coins, at most one of them is counterfeit. A counterfeit coin is lighter than a real coin. You have a free weight balance. What is the minimum number of weighings necessary to determine the identity of the counterfeit coin if it exists | 2 | 0.916667 |
The quartic (4th-degree) polynomial P(x) satisfies $P(1)=0$ and attains its maximum value of $3$ at both $x=2$ and $x=3$ . Compute $P(5)$ . | -24 | 0.75 |
What is the $33$ rd number after the decimal point of $(\sqrt{10} + 3)^{2001}$? | 0 | 0.083333 |
Let $ a$ , $ b$ , $ c$ , and $ d$ be real numbers with $ |a\minus{}b|\equal{}2$ , $ |b\minus{}c|\equal{}3$ , and $ |c\minus{}d|\equal{}4$ . Calculate the sum of all possible values of $ |a\minus{}d|$ . | 18 | 0.75 |
Find all positive integers $n$ such that $$ \phi(n) + \sigma(n) = 2n + 8. $$ | n = 343 \text{ and } n = 12 | 0.583333 |
The function $f$ defined by $\displaystyle f(x)= \frac{ax+b}{cx+d}$ . where $a,b,c$ and $d$ are nonzero real numbers, has the properties $f(19)=19, f(97)=97$ and $f(f(x))=x$ for all values except $\displaystyle \frac{-d}{c}$ . Find the unique number that is not in the range of $f$ . | 58 | 0.5 |
Let $ABCD$ be a convex quadrilateral with $AB=AD, m\angle A = 40^{\circ}, m\angle C = 130^{\circ},$ and $m\angle ADC - m\angle ABC = 20^{\circ}.$ Find the measure of the non-reflex angle $\angle CDB$ in degrees. | 35^\circ | 0.583333 |
Consider the sum
\[
S_n = \sum_{k = 1}^n \frac{1}{\sqrt{2k-1}} \, .
\]
Determine $\lfloor S_{4901} \rfloor$ . Recall that if $x$ is a real number, then $\lfloor x \rfloor$ (the *floor* of $x$ ) is the greatest integer that is less than or equal to $x$ . | 98 | 0.5 |
Find the maximal value of the following expression, if $a,b,c$ are nonnegative and $a+b+c=1$ .
\[ \frac{1}{a^2 -4a+9} + \frac {1}{b^2 -4b+9} + \frac{1}{c^2 -4c+9} \] | \frac{7}{18} | 0.916667 |
problem 1 :A sequence is defined by $ x_1 = 1, x_2 = 4$ and $ x_{n+2} = 4x_{n+1} -x_n$ for $n \geq 1$ . Find all natural numbers $m$ such that the number $3x_n^2 + m$ is a perfect square for all natural numbers $n$ | m = 1 | 0.666667 |
Let $ABCD$ and $ABEF$ be two squares situated in two perpendicular planes and let $O$ be the intersection of the lines $AE$ and $BF$ . If $AB=4$ compute:
a) the distance from $B$ to the line of intersection between the planes $(DOC)$ and $(DAF)$ ;
b) the distance between the lines $AC$ and $BF$ . | \frac{4\sqrt{3}}{3} | 0.166667 |
For any permutation $p$ of set $\{1, 2, \ldots, n\}$ , define $d(p) = |p(1) - 1| + |p(2) - 2| + \ldots + |p(n) - n|$ . Denoted by $i(p)$ the number of integer pairs $(i, j)$ in permutation $p$ such that $1 \leqq < j \leq n$ and $p(i) > p(j)$ . Find all the real numbers $c$ , such that the inequality $i(p) \leq c \cdot d(p)$ holds for any positive integer $n$ and any permutation $p.$ | 1 | 0.583333 |
Determine all non negative integers $k$ such that there is a function $f : \mathbb{N} \to \mathbb{N}$ that satisfies
\[ f^n(n) = n + k \]
for all $n \in \mathbb{N}$ | k = 0 | 0.833333 |
Using only the digits $2,3$ and $9$ , how many six-digit numbers can be formed which are divisible by $6$ ? | 81 | 0.25 |
Two people play the following game: there are $40$ cards numbered from $1$ to $10$ with $4$ different signs. At the beginning they are given $20$ cards each. Each turn one player either puts a card on the table or removes some cards from the table, whose sum is $15$ . At the end of the game, one player has a $5$ and a $3$ in his hand, on the table there's a $9$ , the other player has a card in his hand. What is it's value? | 8 | 0.5 |
If $f,g: \mathbb{R} \to \mathbb{R}$ are functions that satisfy $f(x+g(y)) = 2x+y $ $\forall x,y \in \mathbb{R}$ , then determine $g(x+f(y))$ . | \frac{x}{2} + y | 0.833333 |
Given two rockets of masses $m$ and $9m$ are in a negligible gravitational field. A constant force $F$ acts on the rocket of mass $m$ for a distance $d$ and imparts momentum $p$. Determine the momentum acquired by the rocket of mass $9m$ when the same constant force $F$ acts on it for the same distance $d$. | 3p | 0.333333 |
Given $ a_{i} \in \left\{0,1,2,3,4\right\}$ for every $ 0\le i\le 9$ and $6 \sum _{i = 0}^{9}a_{i} 5^{i} \equiv 1\, \, \left(mod\, 5^{10} \right)$ , find the value of $ a_{9} $. | 4 | 0.416667 |
Starting with an empty string, we create a string by repeatedly appending one of the letters $H$ , $M$ , $T$ with probabilities $\frac 14$ , $\frac 12$ , $\frac 14$ , respectively, until the letter $M$ appears twice consecutively. What is the expected value of the length of the resulting string? | 6 | 0.5 |
Given that the point O is the center of the circle circumscribed about triangle ABC, and ∠BOC = 120° and ∠AOB = 140°, find the degree measure of ∠ABC. | 50^\circ | 0.75 |
Martha writes down a random mathematical expression consisting of 3 single-digit positive integers with an addition sign " $+$ " or a multiplication sign " $\times$ " between each pair of adjacent digits. (For example, her expression could be $4 + 3\times 3$ , with value 13.) Each positive digit is equally likely, each arithmetic sign (" $+$ " or " $\times$ ") is equally likely, and all choices are independent. What is the expected value (average value) of her expression? | 50 | 0.25 |
Suppose that $x^2+px+q$ has two distinct roots $x=a$ and $x=b$ . Furthermore, suppose that the positive difference between the roots of $x^2+ax+b$ , the positive difference between the roots of $x^2+bx+a$ , and twice the positive difference between the roots of $x^2+px+q$ are all equal. Given that $q$ can be expressed in the form $\frac{m}{m}$ , where $m$ and $n$ are relatively prime positive integers, compute $m+n$ .
*2021 CCA Math Bonanza Lightning Round #4.1* | 21 | 0.333333 |
Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$ | 3 | 0.833333 |
Consider the following sequence $$ (a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots) $$ Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\lim_{n\to \infty}\frac{\displaystyle\sum_{k=1}^n a_k}{n^{\alpha}}=\beta$ .
(Proposed by Tomas Barta, Charles University, Prague) | \left(\frac{3}{2}, \frac{\sqrt{2}}{3}\right) | 0.166667 |
Let $0<a<1$ . Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ continuous at $x = 0$ such that $f(x) + f(ax) = x,\, \forall x \in \mathbb{R}$ | f(x) = \frac{x}{1 + a} | 0.333333 |
Find all polynomials $f(x)$ with integer coefficients and leading coefficient equal to 1, for which $f(0)=2010$ and for each irrational $x$ , $f(x)$ is also irrational. | f(x) = x + 2010 | 0.833333 |
Find all polynomials $P(x)$ with real coefficents, such that for all $x,y,z$ satisfying $x+y+z=0$ , the equation below is true: \[P(x+y)^3+P(y+z)^3+P(z+x)^3=3P((x+y)(y+z)(z+x))\] | P(x) = 0, \quad P(x) = x, \quad P(x) = -x | 0.083333 |
The $n^{th}$ derivative of a function $f(x)$ (if it exists) is denoted by $f^{(n)}(x) $ . Let $f(x)=\frac{e^x}{x}$ .
Suppose $f$ is differentiable infinitely many times in $(0,\infty) $ . Then find $\lim_{n \to \infty}\frac{f^{(2n)}1}{(2n)!}$ | 1 | 0.083333 |
Find all real numbers $x,y,z$ such that satisfied the following equalities at same time: $\sqrt{x^3-y}=z-1 \wedge \sqrt{y^3-z}=x-1\wedge \sqrt{z^3-x}=y-1$ | x = y = z = 1 | 0.083333 |
Determine all the possible non-negative integer values that are able to satisfy the expression: $\frac{(m^2+mn+n^2)}{(mn-1)}$
if $m$ and $n$ are non-negative integers such that $mn \neq 1$ . | 0, 4, 7 | 0.083333 |
Compute the expected sum of elements in a subset of $\{1, 2, 3, . . . , 2020\}$ (including the empty set) chosen uniformly at random. | 1020605 | 0.416667 |
A pyramid has a base which is an equilateral triangle with side length $300$ centimeters. The vertex of the pyramid is $100$ centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid toward the vertex at the top. When the mouse has walked a distance of $134$ centimeters, how many centimeters above the base of the pyramid is the mouse? | 67 | 0.666667 |
Let $f_1(x) = \frac{2}{3}-\frac{3}{3x+1}$ , and for $n \ge 2$ , define $f_n(x) = f_1(f_{n-1} (x))$ . The value of x that satisfies $f_{1001}(x) = x - 3$ can be expressed in the form $\frac{m}{n}$ ,
where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . | 8 | 0.25 |
Let $d$ be a real number. For each integer $m \geq 0,$ define a sequence $\left\{a_{m}(j)\right\}, j=0,1,2, \ldots$ by the condition
\begin{align*}
a_{m}(0)&=d / 2^{m},
a_{m}(j+1)&=\left(a_{m}(j)\right)^{2}+2 a_{m}(j), \quad j \geq 0.
\end{align*}
Evaluate $\lim _{n \rightarrow \infty} a_{n}(n).$ | e^d - 1 | 0.166667 |
Find all pairs of positive integers $(x, y)$ such that $\frac{xy^3}{x+y}$ is the cube of a prime. | (2, 14) | 0.166667 |
For each positive integer $n$ , define $a_n = 20 + n^2$ and $d_n = gcd(a_n, a_{n+1})$ . Find the set of all values that are taken by $d_n$ and show by examples that each of these values is attained. | \{1, 3, 9, 27, 81\} | 0.25 |
There exists a positive real number $x$ such that $ \cos (\arctan (x)) = x $ . Find the value of $x^2$ . | \frac{\sqrt{5} - 1}{2} | 0.166667 |
In how many different orders can the characters $P \ U \ M \ \alpha \ C$ be arranged such that the $M$ is to the left of the $\alpha$ and the $\alpha$ is to the left of the $C?$ | 20 | 0.833333 |
A book is published in three volumes, the pages being numbered from $1$ onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is $50$ more than that in the first volume, and the number pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is $1709$ . If $n$ is the last page number, what is the largest prime factor of $n$ ? | 17 | 0.916667 |
Let $n \geq 3$ be a positive integer. Find the smallest positive real $k$ , satisfying the following condition: if $G$ is a connected graph with $n$ vertices and $m$ edges, then it is always possible to delete at most $k(m-\lfloor \frac{n} {2} \rfloor)$ edges, so that the resulting graph has a proper vertex coloring with two colors. | k = \frac{1}{2} | 0.25 |
A function $f$ is defined on the set of all real numbers except $0$ and takes all real values except $1$ . It is also known that $\color{white}\ . \ \color{black}\ \quad f(xy)=f(x)f(-y)-f(x)+f(y)$ for any $x,y\not= 0$ and that $\color{white}\ . \ \color{black}\ \quad f(f(x))=\frac{1}{f(\frac{1}{x})}$ for any $x\not\in\{ 0,1\}$ . Determine all such functions $f$ . | f(x) = \frac{x - 1}{x} | 0.083333 |
For any positive integer $n$ , let $s(n)$ denote the number of ordered pairs $(x,y)$ of positive integers for which $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$ . Determine the set of positive integers for which $s(n) = 5$ | \{p^2 \mid p \text{ is a prime number}\} | 0.166667 |
A point-like mass moves horizontally between two walls on a frictionless surface with initial kinetic energy E . With every collision with the walls, the mass loses 1/2 of its kinetic energy to thermal energy. Determine the number of collisions necessary before the speed of the mass is reduced by a factor of 8 . | 6 | 0.916667 |
Let $f(x)=\tfrac{x+a}{x+b}$ for real numbers $x$ such that $x\neq -b$ . Compute all pairs of real numbers $(a,b)$ such that $f(f(x))=-\tfrac{1}{x}$ for $x\neq0$ . | (-1, 1) | 0.666667 |
Let $ p_1, p_2, p_3$ and $ p_4$ be four different prime numbers satisying the equations
$ 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162$
$ 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162$
Find all possible values of the product $ p_1p_2p_3p_4$ | 570 | 0.666667 |
Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$
with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$ | n = 3 | 0.416667 |
Find the largest $a$ for which there exists a polynomial $$ P(x) =a x^4 +bx^3 +cx^2 +dx +e $$ with real coefficients which satisfies $0\leq P(x) \leq 1$ for $-1 \leq x \leq 1.$ | 4 | 0.5 |
Let $ ABCD$ be a unit square (that is, the labels $ A, B, C, D$ appear in that order around the square). Let $ X$ be a point outside of the square such that the distance from $ X$ to $ AC$ is equal to the distance from $ X$ to $ BD$ , and also that $ AX \equal{} \frac {\sqrt {2}}{2}$ . Determine the value of $ CX^2$ . | \frac{5}{2} | 0.333333 |
$n$ balls are placed independently uniformly at random into $n$ boxes. One box is selected at random, and is found to contain $b$ balls. Let $e_n$ be the expected value of $b^4$ . Find $$ \lim_{n \to
\infty}e_n. $$ | 15 | 0.166667 |
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