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Find the positive constant $c_0$ such that the series \[ \displaystyle\sum_{n = 0}^{\infty} \dfrac {n!}{(cn)^n} \] converges for $c>c_0$ and diverges for $0<c<c_0$ . | \frac{1}{e} | 0.833333 |
Side $AC$ of right triangle $ABC$ is divided into $8$ equal parts. Seven line segments parallel to $BC$ are drawn to $AB$ from the points of division. If $BC = 10$, calculate the sum of the lengths of the seven line segments. | 35 | 0.666667 |
Find the number of integer solutions of the equation $x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$ | 0 | 0.75 |
Given the expression \[ \frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}, \] where $a$, $b$, and $c$ are non-zero real numbers, determine the set of all possible numerical values that can be formed by this expression. | \{-4, 0, 4\} | 0.666667 |
Let $n$ be a positive integer. Determine all positive real numbers $x$ satisfying $nx^2 +\frac{2^2}{x + 1}+\frac{3^2}{x + 2}+...+\frac{(n + 1)^2}{x + n}= nx + \frac{n(n + 3)}{2}$ | x = 1 | 0.916667 |
Given are positive reals $x_1, x_2,..., x_n$ such that $\sum\frac {1}{1+x_i^2}=1$ . Find the minimal value of the expression $\frac{\sum x_i}{\sum \frac{1}{x_i}}$ and find when it is achieved. | n-1 | 0.583333 |
If five squares of a $3 \times 3$ board initially colored white are chosen at random and blackened, what is the expected number of edges between two squares of the same color? | \frac{16}{3} | 0.166667 |
For how many primes $p$ less than $15$ , determine the number of integer triples $(m,n,k)$ such that
\[
\begin{array}{rcl}
m+n+k &\equiv& 0 \pmod p
mn+mk+nk &\equiv& 1 \pmod p
mnk &\equiv& 2 \pmod p.
\end{array} | 3 | 0.083333 |
$N$ students are seated at desks in an $m \times n$ array, where $m, n \ge 3$ . Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are $1020 $ handshakes, what is $N$ ? | 280 | 0.083333 |
Given a man with mass $m$ jumps off a high bridge with a bungee cord attached, falls through a maximum distance $H$ , and experiences a maximum tension in the bungee cord that is four times the man's weight, find the spring constant $k$ of the bungee cord. | \frac{8mg}{H} | 0.166667 |
Let $a, b, c$ be the side lengths of an non-degenerate triangle with $a \le b \le c$ . With $t (a, b, c)$ denote the minimum of the quotients $\frac{b}{a}$ and $\frac{c}{b}$ . Find all values that $t (a, b, c)$ can take. | \left[1, \frac{1 + \sqrt{5}}{2}\right) | 0.25 |
Brian writes down four integers $w > x > y > z$ whose sum is $44$. The pairwise positive differences of these numbers are $1,3,4,5,6,$ and $9$. Calculate the sum of the possible values for $w$. | 31 | 0.083333 |
Two isosceles triangles with sidelengths $x,x,a$ and $x,x,b$ ( $a \neq b$ ) have equal areas. Find $x$ . | \frac{\sqrt{a^2 + b^2}}{2} | 0.583333 |
It is given regular $n$ -sided polygon, $n \geq 6$ . How many triangles they are inside the polygon such that all of their sides are formed by diagonals of polygon and their vertices are vertices of polygon? | \frac{n(n-4)(n-5)}{6} | 0.25 |
In $\vartriangle ABC, AB=AC=14 \sqrt2 , D$ is the midpoint of $CA$ and $E$ is the midpoint of $BD$ . Suppose $\vartriangle CDE$ is similar to $\vartriangle ABC$ . Find the length of $BD$ . | 14 | 0.916667 |
Let $A$ , $M$ , and $C$ be nonnegative integers such that $A+M+C=10$ . Find the maximum value of $A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A$. | 69 | 0.833333 |
Determine the real numbers $x$ , $y$ , $z > 0$ for which $xyz \leq \min\left\{4(x - \frac{1}{y}), 4(y - \frac{1}{z}), 4(z - \frac{1}{x})\right\}$ | x = y = z = \sqrt{2} | 0.416667 |
Given that $\cos x = 0$ and $\cos (x + z) = \frac{1}{2}$, calculate the smallest possible positive value of $z$. | \frac{\pi}{6} | 0.25 |
What is the sum of all primes $p$ such that $7^p - 6^p + 2$ is divisible by 43? | 3 | 0.916667 |
Nonnegative real numbers $p_{1},\ldots,p_{n}$ and $q_{1},\ldots,q_{n}$ are such that $p_{1}+\cdots+p_{n}=q_{1}+\cdots+q_{n}$ Among all the matrices with nonnegative entries having $p_i$ as sum of the $i$ -th row's entries and $q_j$ as sum of the $j$ -th column's entries, find the maximum sum of the entries on the main diagonal. | \sum_{i=1}^n \min(p_i, q_i) | 0.666667 |
Given that the three height lengths are all roots of the polynomial $x^3 - 3.9 x^2 + 4.4 x - 1.2$, determine the length of the inradius of the triangle. | \frac{3}{11} | 0.5 |
Jamie, Linda, and Don bought bundles of roses at a flower shop, each paying the same price for each
bundle. Then Jamie, Linda, and Don took their bundles of roses to a fair where they tried selling their
bundles for a fixed price which was higher than the price that the flower shop charged. At the end of the
fair, Jamie, Linda, and Don donated their unsold bundles of roses to the fair organizers. Jamie had bought
20 bundles of roses, sold 15 bundles of roses, and made $60$ profit. Linda had bought 34 bundles of roses,
sold 24 bundles of roses, and made $69 profit. Don had bought 40 bundles of roses and sold 36 bundles of
roses. How many dollars profit did Don make? | 252 | 0.833333 |
We know that $201$ and $9$ give the same remainder when divided by $24$ . What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24$ ?
*2020 CCA Math Bonanza Lightning Round #1.1* | 1 | 0.833333 |
Let $\mathbb{N}_{\geqslant 1}$ be the set of positive integers.
Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers $m$ and $n$ :
\[\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) =
\mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right).\]
Note: if $a$ and $b$ are positive integers, $\mathrm{GCD}(a,b)$ is the largest positive integer that divides both $a$ and $b$ , and $\mathrm{LCM}(a,b)$ is the smallest positive integer that is a multiple of both $a$ and $b$ . | f(n) = n \text{ for all } n \in \mathbb{N}_{\geqslant 1} | 0.666667 |
We have an empty equilateral triangle with length of a side $l$ . We put the triangle, horizontally, over a sphere of radius $r$ . Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of $l$ and $r$ )? | \sqrt{r^2 + \frac{l^2}{4}} | 0.083333 |
$|5x^2-\tfrac25|\le|x-8|$ if and only if $x$ is in the interval $[a, b]$ . There are relatively prime positive integers $m$ and $n$ so that $b -a =\tfrac{m}{n}$ . Find $m + n$ . | 18 | 0.75 |
$99$ identical balls lie on a table. $50$ balls are made of copper, and $49$ balls are made of zinc. The assistant numbered the balls. Once spectrometer test is applied to $2$ balls and allows to determine whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What minimum number of tests is required to determine the material of each ball if all the tests should be performed today?
*Proposed by N. Vlasova, S. Berlov* | 98 | 0.083333 |
In triangle $ABC$ , find the smallest possible value of $$ |(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)| $$ | \frac{8\sqrt{3}}{9} | 0.583333 |
Let $n>4$ be a positive integer, which is divisible by $4$ . We denote by $A_n$ the sum of the odd positive divisors of $n$ . We also denote $B_n$ the sum of the even positive divisors of $n$ , excluding the number $n$ itself. Find the least possible value of the expression $$ f(n)=B_n-2A_n, $$ for all possible values of $n$ , as well as for which positive integers $n$ this minimum value is attained. | 4 | 0.25 |
A prime number $ q $ is called***'Kowai'***number if $ q = p^2 + 10$ where $q$ , $p$ , $p^2-2$ , $p^2-8$ , $p^3+6$ are prime numbers. WE know that, at least one ***'Kowai'*** number can be found. Find the summation of all ***'Kowai'*** numbers. | 59 | 0.833333 |
Find all real solutions of the system $$ \begin{cases} x_1 +x_2 +...+x_{2000} = 2000 x_1^4 +x_2^4 +...+x_{2000}^4= x_1^3 +x_2^3 +...+x_{2000}^3\end{cases} $$ | x_1 = x_2 = \cdots = x_{2000} = 1 | 0.333333 |
In how many ways can $17$ identical red and $10$ identical white balls be distributed into $4$ distinct boxes such that the number of red balls is greater than the number of white balls in each box? | 5720 | 0.25 |
Given that $n\geq 100$, find the least integer $n$ such that $77$ divides $1+2+2^2+2^3+\dots + 2^n$. | 119 | 0.833333 |
For each natural number $n\geq 2$ , solve the following system of equations in the integers $x_1, x_2, ..., x_n$ : $$ (n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2,\qquad \forall 1\le i\le n $$ where $$ S=x_1^2+x_2^2+\dots+x_n^2. $$ | x_i = 1 | 0.333333 |
A number $n$ is called multiplicatively perfect if the product of all the positive divisors of $n$ is $n^2$ . Determine the number of positive multiplicatively perfect numbers less than $100$ . | 33 | 0.166667 |
Let $S=\{1,2,3,...,12\}$ . How many subsets of $S$ , excluding the empty set, have an even sum but not an even product?
*Proposed by Gabriel Wu* | 31 | 0.416667 |
A natural number $n$ is called *perfect* if it is equal to the sum of all its natural divisors other than $n$ . For example, the number $6$ is perfect because $6 = 1 + 2 + 3$ . Find all even perfect numbers that can be given as the sum of two cubes positive integers. | 28 | 0.416667 |
Let $r$ , $s$ be real numbers, find maximum $t$ so that if $a_1, a_2, \ldots$ is a sequence of positive real numbers satisfying
\[ a_1^r + a_2^r + \cdots + a_n^r \le 2023 \cdot n^t \]
for all $n \ge 2023$ then the sum
\[ b_n = \frac 1{a_1^s} + \cdots + \frac 1{a_n^s} \]
is unbounded, i.e for all positive reals $M$ there is an $n$ such that $b_n > M$ . | t = 1 + \frac{r}{s} | 0.166667 |
Find $f(2)$ given that $f$ is a real-valued function that satisfies the equation $$ 4f(x)+\left(\frac23\right)(x^2+2)f\left(x-\frac2x\right)=x^3+1. $$ | \frac{19}{12} | 0.416667 |
Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz=1,$ $x+\frac{1}{z}=5,$ and $y+\frac{1}{x}=29.$ Then $z+\frac{1}{y}=\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 5 | 0.083333 |
Jenny places 100 pennies on a table, 30 showing heads and 70 showing tails. She chooses 40 of the pennies at random (all different) and turns them over. That is, if a chosen penny was showing heads, she turns it to show tails; if a chosen penny was showing tails, she turns it to show heads. At the end, what is the expected number of pennies showing heads? | 46 | 0.666667 |
Given two circles that are externally tangent and lines $\overline{PAB}$ and $\overline{PA'B'}$ are common tangents with $A$ and $A'$ on the smaller circle and $B$ and $B'$ on the larger circle, if $PA = AB = 4$, find the area of the smaller circle. | 2\pi | 0.083333 |
Let $ABC$ be an acute angled triangle and $CD$ be the altitude through $C$ . If $AB = 8$ and $CD = 6$ , find the distance between the midpoints of $AD$ and $BC$ . | 5 | 0.833333 |
A pair of positive integers $(m,n)$ is called *compatible* if $m \ge \tfrac{1}{2} n + 7$ and $n \ge \tfrac{1}{2} m + 7$ . A positive integer $k \ge 1$ is called *lonely* if $(k,\ell)$ is not compatible for any integer $\ell \ge 1$ . Find the sum of all lonely integers.
*Proposed by Evan Chen* | 91 | 0.5 |
The sides of a triangle have lengths $11$ , $15$ , and $k$ , where $k$ is an integer. For how many values of $k$ is the triangle obtuse? | 13 | 0.916667 |
An acute triangle $ABC$ is inscribed in a circle of radius 1 with centre $O;$ all the angles of $ABC$ are greater than $45^\circ.$ $B_{1}$ is the foot of perpendicular from $B$ to $CO,$ $B_{2}$ is the foot of perpendicular from $B_{1}$ to $AC.$
Similarly, $C_{1}$ is the foot of perpendicular from $C$ to $BO,$ $C_{2}$ is the foot of perpendicular from $C_{1}$ to $AB.$
The lines $B_{1}B_{2}$ and $C_{1}C_{2}$ intersect at $A_{3}.$ The points $B_{3}$ and $C_{3}$ are defined in the same way.
Find the circumradius of triangle $A_{3}B_{3}C_{3}.$ *Proposed by F.Bakharev, F.Petrov* | \frac{1}{2} | 0.833333 |
We call a path Valid if
i. It only comprises of the following kind of steps:
A. $(x, y) \rightarrow (x + 1, y + 1)$ B. $(x, y) \rightarrow (x + 1, y - 1)$ ii. It never goes below the x-axis.
Let $M(n)$ = set of all valid paths from $(0,0) $ , to $(2n,0)$ , where $n$ is a natural number.
Consider a Valid path $T \in M(n)$ .
Denote $\phi(T) = \prod_{i=1}^{2n} \mu_i$ ,
where $\mu_i$ =
a) $1$ , if the $i^{th}$ step is $(x, y) \rightarrow (x + 1, y + 1)$ b) $y$ , if the $i^{th} $ step is $(x, y) \rightarrow (x + 1, y - 1)$
Now Let $f(n) =\sum _{T \in M(n)} \phi(T)$ . Evaluate the number of zeroes at the end in the decimal expansion of $f(2021)$ | 0 | 0.416667 |
Given that for reals $a_1,\cdots, a_{2004},$ equation $x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\cdots +a_2x^2+a_1x+1=0$ has $2006$ positive real solution, find the maximum possible value of $a_1.$ | -2006 | 0.333333 |
Given $\triangle ABC$, $\cos(2A - B) + \sin(A + B) = 2$ and $AB = 4$. Find the length of $BC$. | 2 | 0.833333 |
If $$ \sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c} $$ where $a, b, c \in \mathbb{N}, b < c, gcd(b,c) =1 ,$ then what is the value of $a+ b ?$ | 80 | 0.583333 |
Find the positive integer $n$ such that $32$ is the product of the real number solutions of $x^{\log_2(x^3)-n} = 13$ | 15 | 0.916667 |
Given the equation $\sqrt{xy}-71\sqrt x + 30 = 0$, determine the number of pairs of positive integers $(x,y)$ satisfying the equation. | 8 | 0.833333 |
Let $a$ , $b$ , and $c$ be non-zero real number such that $\tfrac{ab}{a+b}=3$ , $\tfrac{bc}{b+c}=4$ , and $\tfrac{ca}{c+a}=5$ . There are relatively prime positive integers $m$ and $n$ so that $\tfrac{abc}{ab+bc+ca}=\tfrac{m}{n}$ . Find $m+n$ . | 167 | 0.083333 |
For how many integral values of $x$ can a triangle of positive area be formed having side lengths $\log_{2} x, \log_{4} x, 3$? | 59 | 0.916667 |
Find all functions $ f:R \implies R $ , such for all $x,y,z$ $f(xy)+f(xz)\geq f(x)f(yz) + 1$ | f(x) = 1 | 0.833333 |
In triangle $ABC$ , angles $A$ and $B$ measure 60 degrees and 45 degrees, respectively. The bisector of angle $A$ intersects $\overline{BC}$ at $T$ , and $AT=24.$ The area of triangle $ABC$ can be written in the form $a+b\sqrt{c},$ where $a$ , $b$ , and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$ | 291 | 0.083333 |
Let $O$ be the circumcenter of a triangle $ABC$ , and let $l$ be the line going through the midpoint of the side $BC$ and is perpendicular to the bisector of $\angle BAC$ . Determine the value of $\angle BAC$ if the line $l$ goes through the midpoint of the line segment $AO$ . | \angle BAC = 120^\circ | 0.083333 |
Price of some item has decreased by $5\%$ . Then price increased by $40\%$ and now it is $1352.06\$ $ cheaper than doubled original price. How much did the item originally cost? | 2018 | 0.833333 |
Given triangle $ABC$ with $AB=AC$, and a point $P$ strictly between $A$ and $B$ such that $AP=PC=CB$, find the measure of angle $A$. | \alpha = 36^\circ | 0.083333 |
Calculate the integral $$ \int \frac{dx}{\sin (x - 1) \sin (x - 2)} . $$ Hint: Change $\tan x = t$ . | \frac{1}{\sin 1} \ln \left| \frac{\sin(x-2)}{\sin(x-1)} \right| + C | 0.083333 |
$ f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$ satisfy the given conditions
$ a)$ $ f(0,0)\equal{}1$ , $ f(0,1)\equal{}1$ ,
$ b)$ $ \forall k \notin \left\{0,1\right\}$ $ f(0,k)\equal{}0$ and
$ c)$ $ \forall n \geq 1$ and $ k$ , $ f(n,k)\equal{}f(n\minus{}1,k)\plus{}f(n\minus{}1,k\minus{}2n)$
find the sum $ \displaystyle\sum_{k\equal{}0}^{\binom{2009}{2}}f(2008,k)$ | 2^{2008} | 0.583333 |
The residents of the local zoo are either rabbits or foxes. The ratio of foxes to rabbits in the zoo is $2:3$ . After $10$ of the foxes move out of town and half the rabbits move to Rabbitretreat, the ratio of foxes to rabbits is $13:10$ . How many animals are left in the zoo? | 690 | 0.833333 |
Suppose $m$ and $n$ are positive integers for which $\bullet$ the sum of the first $m$ multiples of $n$ is $120$ , and $\bullet$ the sum of the first $m^3$ multiples of $ n^3$ is $4032000$ .
Determine the sum of the first $m^2$ multiples of $n^2$ | 20800 | 0.833333 |
In the triangle $ABC$ it is known that $\angle A = 75^o, \angle C = 45^o$ . On the ray $BC$ beyond the point $C$ the point $T$ is taken so that $BC = CT$ . Let $M$ be the midpoint of the segment $AT$ . Find the measure of the $\angle BMC$ .
(Anton Trygub) | 45^\circ | 0.083333 |
Given two rays with a common endpoint O forming a 30° angle, point A lies on one ray and point B on the other ray. If AB = 1, find the maximum possible length of OB. | 2 | 0.916667 |
A clueless ant makes the following route: starting at point $ A $ goes $ 1$ cm north, then $ 2$ cm east, then $ 3$ cm south, then $ 4$ cm west, immediately $ 5$ cm north, continues $ 6$ cm east, and so on, finally $ 41$ cm north and ends in point $ B $ . Calculate the distance between $ A $ and $ B $ (in a straight line). | 29 | 0.416667 |
Find the number of the subsets $B$ of the set $\{1,2,\cdots, 2005 \}$ such that the sum of the elements of $B$ is congruent to $2006$ modulo $2048$ | 2^{1994} | 0.666667 |
Find all real solutions to $ x^3 \minus{} 3x^2 \minus{} 8x \plus{} 40 \minus{} 8\sqrt[4]{4x \plus{} 4} \equal{} 0$ | x = 3 | 0.916667 |
Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? (L-trominos are made up of three unit squares sharing a corner, as shown below.)
[asy]
draw((0,0) -- (2,0) -- (2,1) -- (0,1));
draw((0,0) -- (0,2) -- (1,2) -- (1,0));
draw((4,1) -- (6,1) -- (6,2) -- (4,2));
draw((4,2) -- (4,0) -- (5,0) -- (5,2));
draw((10,0) -- (8,0) -- (8,1) -- (10,1));
draw((9,0) -- (9,2) -- (10,2) -- (10,0));
draw((14,1) -- (12,1) -- (12,2) -- (14,2));
draw((13,2) -- (13,0) -- (14,0) -- (14,2));
[/asy]
*Proposed by Andrew Lin.* | 18 | 0.083333 |
Find all 4-digit numbers $n$ , such that $n=pqr$ , where $p<q<r$ are distinct primes, such that $p+q=r-q$ and $p+q+r=s^2$ , where $s$ is a prime number. | n = 5 \cdot 13 \cdot 31 = 2015 | 0.5 |
Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells. | k = 2n | 0.333333 |
Find all integer triples $(a, b, c)$ satisfying the equation $$ 5 a^2 + 9 b^2 = 13 c^2. $$ | (0, 0, 0) | 0.916667 |
Let $ABCD$ be a rectangle with $AB = 6$ and $BC = 6 \sqrt 3$ . We construct four semicircles $\omega_1$ , $\omega_2$ , $\omega_3$ , $\omega_4$ whose diameters are the segments $AB$ , $BC$ , $CD$ , $DA$ . It is given that $\omega_i$ and $\omega_{i+1}$ intersect at some point $X_i$ in the interior of $ABCD$ for every $i=1,2,3,4$ (indices taken modulo $4$ ). Compute the square of the area of $X_1X_2X_3X_4$ .
*Proposed by Evan Chen* | 243 | 0.166667 |
Broady The Boar is playing a boring board game consisting of a circle with $2021$ points on it, labeled $0$ , $1$ , $2$ , ... $2020$ in that order clockwise. Broady is rolling $2020$ -sided die which randomly produces a whole number between $1$ and $2020$ , inclusive.
Broady starts at the point labelled $0$ . After each dice roll, Broady moves up the same number of points as the number rolled (point $2020$ is followed by point $0$ ). For example, if they are at $0$ and roll a $5$ , they end up at $5$ . If they are at $2019$ and roll a $3$ , they end up at $1$ .
Broady continues rolling until they return to the point labelled $0$ . What is the expected number of times they roll the dice?
*2021 CCA Math Bonanza Lightning Round #2.3* | 2021 | 0.833333 |
Calculate the following indefinite integrals.
[1] $\int \frac{\sin x\cos x}{1+\sin ^ 2 x}dx$
[2] $\int x\log_{10} x dx$
[3] $\int \frac{x}{\sqrt{2x-1}}dx$
[4] $\int (x^2+1)\ln x dx$
[5] $\int e^x\cos x dx$ | \frac{e^x (\sin x + \cos x)}{2} + C | 0.083333 |
Given the sequence $(a_n)$ defined as $a_{n+1}-2a_n+a_{n-1}=7$ for every $n\geq 2$, where $a_1 = 1, a_2=5$. Find the value of $a_{17}$. | 905 | 0.583333 |
Let $\sigma (n)$ denote the sum and $\tau (n)$ denote the amount of natural divisors of number $n$ (including $1$ and $n$ ). Find the greatest real number $a$ such that for all $n>1$ the following inequality is true: $$ \frac{\sigma (n)}{\tau (n)}\geq a\sqrt{n} $$ | \frac{3 \sqrt{2}}{4} | 0.166667 |
How many primes $p$ are there such that $5p(2^{p+1}-1)$ is a perfect square? | 1 | 0.666667 |
Let $x_1 , x_2 ,\ldots, x_n$ be real numbers in $[0,1].$ Determine the maximum value of the sum of the $\frac{n(n-1)}{2}$ terms: $$ \sum_{i<j}|x_i -x_j |. $$ | \left\lfloor \frac{n^2}{4} \right\rfloor | 0.333333 |
Given a set $A$ which contains $n$ elements. For any two distinct subsets $A_{1}$ , $A_{2}$ of the given set $A$ , we fix the number of elements of $A_1 \cap A_2$ . Find the sum of all the numbers obtained in the described way. | n \left( 2^{2n-3} - 2^{n-2} \right) | 0.083333 |
Given the set $\{2000,2001,...,2010\}$, find the number of integers $n$ such that $2^{2n} + 2^n + 5$ is divisible by 7. | 4 | 0.833333 |
Let $\triangle ABC$ be a right-angled triangle and $BC > AC$ . $M$ is a point on $BC$ such that $BM = AC$ and $N$ is a point on $AC$ such that $AN = CM$ . Find the angle between $BN$ and $AM$ . | 45^\circ | 0.5 |
It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has $k$ weights of integral values. What is the minimum $k$ for which there exist weights that satisfy this condition? | 8 | 0.416667 |
How many lattice points are exactly twice as close to $(0,0)$ as they are to $(15,0)$ ? (A lattice point is a point $(a,b)$ such that both $a$ and $b$ are integers.) | 12 | 0.583333 |
Al and Bob play Rock Paper Scissors until someone wins a game. What is the probability that this happens on the sixth game? | \frac{2}{729} | 0.666667 |
For all real numbers $r$ , denote by $\{r\}$ the fractional part of $r$ , i.e. the unique real number $s\in[0,1)$ such that $r-s$ is an integer. How many real numbers $x\in[1,2)$ satisfy the equation $\left\{x^{2018}\right\} = \left\{x^{2017}\right\}?$ | 2^{2017} | 0.083333 |
Given the regular octagon $ABCDEFGH$ with its center at $J$, and each of the vertices and the center associated with the digits 1 through 9, with each digit used once, such that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal, determine the number of ways in which this can be done. | 1152 | 0.083333 |
Find all natural integers $n$ such that $(n^3 + 39n - 2)n! + 17\cdot 21^n + 5$ is a square. | n = 1 | 0.5 |
What is the minimum possible value of $(a + b + c + d)^2 + (e + f + g + h)^2$, where $a,b,c,d,e,f,g$, and $h$ are distinct elements in the set $\{ \minus{} 7, \minus{} 5, \minus{} 3, \minus{} 2, 2, 4, 6, 13\}$. | 34 | 0.333333 |
Find all triplets $(a,b,p)$ where $a,b$ are positive integers and $p$ is a prime number satisfying: $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$ | (2, 2, 2) | 0.75 |
Given distinct straight lines OA and OB. From a point in OA a perpendicular is drawn to OB; from the foot of this perpendicular a line is drawn perpendicular to OA. From the foot of this second perpendicular a line is drawn perpendicular to OB; and so on indefinitely. The lengths of the first and second perpendiculars are a and b, respectively. Then find the limit of the sum of the lengths of the perpendiculars as the number of perpendiculars grows beyond all bounds. | \frac{a^2}{a - b} | 0.083333 |
Let $n$ be a positive integer. Mariano divides a rectangle into $n^2$ smaller rectangles by drawing $n-1$ vertical lines and $n-1$ horizontal lines, parallel to the sides of the larger rectangle. On every step, Emilio picks one of the smaller rectangles and Mariano tells him its area. Find the least positive integer $k$ for which it is possible that Emilio can do $k$ conveniently thought steps in such a way that with the received information, he can determine the area of each one of the $n^2$ smaller rectangles. | 2n - 1 | 0.166667 |
Find all quadrupels $(a, b, c, d)$ of positive real numbers that satisfy the following two equations:
\begin{align*}
ab + cd &= 8,
abcd &= 8 + a + b + c + d.
\end{align*} | (2, 2, 2, 2) | 0.916667 |
Let $a,b,c,d,e$ be non-negative real numbers such that $a+b+c+d+e>0$. What is the least real number $t$ such that $a+c=tb$, $b+d=tc$, $c+e=td$. | \sqrt{2} | 0.166667 |
A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not inclued the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed $1000x.$ | 166 | 0.083333 |
Given a harmonic progression with the first three terms 3, 4, 6, find the value of $S_4$. | 25 | 0.833333 |
What is the largest integer $n$ that satisfies $(100^2-99^2)(99^2-98^2)\dots(3^2-2^2)(2^2-1^2)$ is divisible by $3^n$? | 49 | 0.25 |
Suppose that three prime numbers $p,q,$ and $r$ satisfy the equations $pq + qr + rp = 191$ and $p + q = r - 1$ . Find $p + q + r$ .
*Proposed by Andrew Wu* | 25 | 0.583333 |
A tetrahedron $ABCD$ satisfies the following conditions: the edges $AB,AC$ and $AD$ are pairwise orthogonal, $AB=3$ and $CD=\sqrt2$ . Find the minimum possible value of $$ BC^6+BD^6-AC^6-AD^6. $$ | 1998 | 0.583333 |
Consider a triangle $ABC$ with $\angle ACB=120^\circ$ . Let $A’, B’, C’$ be the points of intersection of the angular bisector through $A$ , $B$ and $C$ with the opposite side, respectively.
Determine $\angle A’C’B’$ . | \angle A'C'B' = 90^\circ | 0.166667 |
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