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The polynomial $f(x)=1-x+x^2-x^3+\cdots-x^{19}+x^{20}$ is written into the form $g(y)=a_0+a_1y+a_2y^2+\cdots+a_{20}y^{20}$ , where $y=x-4$ , then $a_0+a_1+\cdots+a_{20}=$ ________. | \frac{1 + 5^{21}}{6} | 0.5 |
What is the greatest integer not exceeding the number $\left( 1 + \frac{\sqrt 2 + \sqrt 3 + \sqrt 4}{\sqrt 2 + \sqrt 3 + \sqrt 6 + \sqrt 8 + 4}\right)^{10}$? | 32 | 0.083333 |
The triangle $K_2$ has as its vertices the feet of the altitudes of a non-right triangle $K_1$ . Find all possibilities for the sizes of the angles of $K_1$ for which the triangles $K_1$ and $K_2$ are similar. | A = 60^\circ, B = 60^\circ, C = 60^\circ | 0.083333 |
An infinite sequence of positive real numbers is defined by $a_0=1$ and $a_{n+2}=6a_n-a_{n+1}$ for $n=0,1,2,\cdots$ . Find the possible value(s) of $a_{2007}$ . | 2^{2007} | 0.166667 |
Given $x^2 - x - 1$ is a factor of $ax^3 + bx^2 + 1$, find the value of $b$. | -2 | 0.916667 |
For how many integers $n$ does the equation system
\[\begin{array}{rcl}
2x+3y &=& 7 \\
5x + ny &=& n^2
\end{array}\]
have a solution over integers. | 8 | 0.083333 |
Let $n$ be a positive integer. We call $(a_1,a_2,\cdots,a_n)$ a *good* $n-$ tuple if $\sum_{i=1}^{n}{a_i}=2n$ and there doesn't exist a set of $a_i$ s such that the sum of them is equal to $n$ . Find all *good* $n-$ tuple.
(For instance, $(1,1,4)$ is a *good* $3-$ tuple, but $(1,2,1,2,4)$ is not a *good* $5-$ tuple.) | (2, 2, \ldots, 2) | 0.25 |
Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers | n = 3 | 0.75 |
Let $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and $S = \{(a,b)\in X\times X:x^2+ax+b \text{ and }x^3+bx+a \text{ have at least a common real zero .}\}$ How many elements are there in $S$ ? | 21 | 0.083333 |
A club with $x$ members is organized into four committees in accordance with these two rules: each member belongs to two and only two committees, and each pair of committees has one and only one member in common. Determine the possible values of $x$. | 6 | 0.916667 |
Let $\lfloor x \rfloor$ denote the largest integer less than or equal to $x$ , and let $\{x\}$ denote the fractional part of $x$ . For example, $\lfloor \pi \rfloor=3$ , and $\{\pi\}=0.14159\dots$ , while $\lfloor 100 \rfloor=100$ and $\{100\}=0$ . If $n$ is the largest solution to the equation $\frac{\lfloor n \rfloor}{n}=\frac{2015}{2016}$ , compute $\{n\}$ . | \frac{2014}{2015} | 0.666667 |
$\textbf{Problem 1.}$ ****There are less than $400$ marbles.** If they are distributed among $3$ childrens, there is one left over if they are distributed among $7$ children, there are 2 left over. Finally if they are distributed among $5$ children, there are none left over.
What is the largest number of the marbels you have? | 310 | 0.666667 |
The vertex $ A$ of the acute triangle $ ABC$ is equidistant from the circumcenter $ O$ and the orthocenter $ H.$ Determine all possible values for the measure of angle $ A.$ | 60^\circ | 0.916667 |
Let $a,b\in\mathbb{R},~a>1,~b>0.$ Find the least possible value for $\alpha$ such that : $$ (a+b)^x\geq a^x+b,~(\forall)x\geq\alpha. $$ | 1 | 0.833333 |
Find, with proof, a polynomial $f(x,y,z)$ in three variables, with integer coefficients, such that for all $a,b,c$ the sign of $f(a,b,c)$ (that is, positive, negative, or zero) is the same as the sign of $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ . | f(x, y, z) = x^3 + 2y^3 + 4z^3 - 6xyz | 0.083333 |
The sequence $\{c_{n}\}$ is determined by the following equation. \[c_{n}=(n+1)\int_{0}^{1}x^{n}\cos \pi x\ dx\ (n=1,\ 2,\ \cdots).\] Let $\lambda$ be the limit value $\lim_{n\to\infty}c_{n}.$ Find $\lim_{n\to\infty}\frac{c_{n+1}-\lambda}{c_{n}-\lambda}.$ | 1 | 0.333333 |
$f$ is a function on the set of complex numbers such that $f(z)=1/(z*)$ , where $z*$ is the complex conjugate of $z$ . $S$ is the set of complex numbers $z$ such that the real part of $f(z)$ lies between $1/2020$ and $1/2018$ . If $S$ is treated as a subset of the complex plane, the area of $S$ can be expressed as $m× \pi$ where $m$ is an integer. What is the value of $m$ ? | 2019 | 0.916667 |
Among all polynomials $P(x)$ with integer coefficients for which $P(-10) = 145$ and $P(9) = 164$ , compute the smallest possible value of $|P(0)|.$ | 25 | 0.416667 |
The perpendicular bisectors of the sides $AB$ and $CD$ of the rhombus $ABCD$ are drawn. It turned out that they divided the diagonal $AC$ into three equal parts. Find the altitude of the rhombus if $AB = 1$ . | \frac{\sqrt{3}}{2} | 0.333333 |
Given that $(1+\sin t)(1+\cos t)=5/4$ and \[ (1-\sin t)(1-\cos t)=\frac mn-\sqrt{k}, \] where $k, m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$ | 27 | 0.833333 |
Let $ x,y$ are real numbers such that $x^2+2cosy=1$ . Find the ranges of $x-cosy$ . | [-1, 1 + \sqrt{3}] | 0.916667 |
If $a$ and $b$ are the roots of $x^2 - 2x + 5$ , what is $|a^8 + b^8|$ ? | 1054 | 0.666667 |
Given the equation $x = |2x - |60 - 2x||$, find the sum of all the solutions. | 92 | 0.916667 |
Find the positive integer $k$ such that the roots of $x^3 - 15x^2 + kx -1105$ are three distinct collinear points in the complex plane. | 271 | 0.333333 |
Find all possible values of integer $n > 3$ such that there is a convex $n$ -gon in which, each diagonal is the perpendicular bisector of at least one other diagonal.
Proposed by Mahdi Etesamifard | n = 4 | 0.416667 |
There are 2012 backgammon checkers with one side being black and the other side being white. These checkers are arranged in a line such that no two consecutive checkers are the same color. At each move, two checkers are chosen, and their colors are reversed along with those of the checkers between them. Determine the minimum number of moves required to make all checkers the same color. | 1006 | 0.916667 |
Positive integers a, b, c, d, and e satisfy the equations $$ (a + 1)(3bc + 1) = d + 3e + 1 $$ $$ (b + 1)(3ca + 1) = 3d + e + 13 $$ $$ (c + 1)(3ab + 1) = 4(26-d- e) - 1 $$ Find $d^2+e^2$ . | 146 | 0.166667 |
Let $ABCD$ be a rectangle in which $AB + BC + CD = 20$ and $AE = 9$ where $E$ is the midpoint of the side $BC$ . Find the area of the rectangle. | 19 | 0.166667 |
Let $f(n)$ be the smallest prime which divides $n^4+1$. Determine the remainder when the sum $f(1)+f(2)+\cdots+f(2014)$ is divided by 8$. | 5 | 0.083333 |
A circle with a radius of $16$ feet has a rhombus formed by two radii and two chords. Calculate the area of the rhombus. | 128\sqrt{3} | 0.083333 |
Mr. Squash bought a large parking lot in Utah, which has an area of $600$ square meters. A car needs $6$ square meters of parking space while a bus needs $30$ square meters of parking space. Mr. Squash charges $\$ 2.50 $ per car and $ \ $7.50$ per bus, but Mr. Squash can only handle at most $60$ vehicles at a time. Find the ordered pair $(a,b)$ where $a$ is the number of cars and $b$ is the number of buses that maximizes the amount of money Mr. Squash makes.
*Proposed by Nathan Cho* | (50, 10) | 0.833333 |
Find the smallest positive integer $n$ such that the $73$ fractions $\frac{19}{n+21}, \frac{20}{n+22},\frac{21}{n+23},...,\frac{91}{n+93}$ are all irreducible. | 95 | 0.666667 |
Determine all intergers $n\geq 2$ such that $a+\sqrt{2}$ and $a^n+\sqrt{2}$ are both rational for some real number $a$ depending on $n$ | n = 2 | 0.5 |
For any finite set $S$ , let $f(S)$ be the sum of the elements of $S$ (if $S$ is empty then $f(S)=0$ ). Find the sum over all subsets $E$ of $S$ of $\dfrac{f(E)}{f(S)}$ for $S=\{1,2,\cdots,1999\}$ . | 2^{1998} | 0.916667 |
If $\phi$ is the Golden Ratio, we know that $\frac1\phi = \phi - 1$ . Define a new positive real number, called $\phi_d$ , where $\frac1{\phi_d} = \phi_d - d$ (so $\phi = \phi_1$ ). Given that $\phi_{2009} = \frac{a + \sqrt{b}}{c}$ , $a, b, c$ positive integers, and the greatest common divisor of $a$ and $c$ is 1, find $a + b + c$ . | 4038096 | 0.583333 |
$k$ marbles are placed onto the cells of a $2024 \times 2024$ grid such that each cell has at most one marble and there are no two marbles are placed onto two neighboring cells (neighboring cells are defined as cells having an edge in common).
a) Assume that $k=2024$ . Find a way to place the marbles satisfying the conditions above, such that moving any placed marble to any of its neighboring cells will give an arrangement that does not satisfy both the conditions.
b) Determine the largest value of $k$ such that for all arrangements of $k$ marbles satisfying the conditions above, we can move one of the placed marble onto one of its neighboring cells and the new arrangement satisfies the conditions above. | k = 2023 | 0.166667 |
Choose a permutation of $ \{1,2, ..., 20\}$ at random. Let $m$ be the amount of numbers in the permutation larger than all numbers before it. Find the expected value of $2^m$ .
*Proposed by Evan Chang (squareman), USA* | 21 | 0.416667 |
Given that square $ABCD$ is divided into four rectangles by $EF$ and $GH$, where $EF$ is parallel to $AB$ and $GH$ is parallel to $BC$, $\angle BAF = 18^\circ$, $EF$ and $GH$ meet at point $P$, and the area of rectangle $PFCH$ is twice that of rectangle $AGPE$, find the value of $\angle FAH$ in degrees. | 45^\circ | 0.083333 |
Calculate the following indefinite integral.
[1] $\int \frac{e^{2x}}{(e^x+1)^2}dx$
[2] $\int \sin x\cos 3x dx$
[3] $\int \sin 2x\sin 3x dx$
[4] $\int \frac{dx}{4x^2-12x+9}$
[5] $\int \cos ^4 x dx$ | \frac{3}{8} x + \frac{1}{4} \sin 2x + \frac{1}{32} \sin 4x + C | 0.333333 |
Let $ABC$ be a triangle with $AC = 28$ , $BC = 33$ , and $\angle ABC = 2\angle ACB$ . Compute the length of side $AB$ .
*2016 CCA Math Bonanza #10* | 16 | 0.25 |
For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. Determine the number of positive integer divisors of the number $81n^4$. | 325 | 0.75 |
On a clock, there are two instants between $12$ noon and $1 \,\mathrm{PM}$ , when the hour hand and the minute hannd are at right angles. The difference *in minutes* between these two instants is written as $a + \dfrac{b}{c}$ , where $a, b, c$ are positive integers, with $b < c$ and $b/c$ in the reduced form. What is the value of $a+b+c$ ? | 51 | 0.833333 |
Find the ordered triple of natural numbers $(x,y,z)$ such that $x \le y \le z$ and $x^x+y^y+z^z = 3382.$ [i]Proposed by Evan Fang | (1, 4, 5) | 0.25 |
Given that there are 5 points in the plane no three of which are collinear, draw all the segments whose vertices are these points, and determine the minimum number of new points made by the intersection of the drawn segments. | 5 | 0.666667 |
A line parallel to the side $AC$ of a triangle $ABC$ with $\angle C = 90$ intersects side $AB$ at $M$ and side $BC$ at $N$ , so that $CN/BN = AC/BC = 2/1$ . The segments $CM$ and $AN$ meet at $O$ . Let $K$ be a point on the segment $ON$ such that $MO+OK = KN$ . The bisector of $\angle ABC$ meets the line through $K$ perpendicular to $AN$ at point $T$ .
Determine $\angle MTB$ . | \angle MTB = 90^\circ | 0.333333 |
Let $a, b, c$ be integers not all the same with $a, b, c\ge 4$ that satisfy $$ 4abc = (a + 3) (b + 3) (c + 3). $$ Find the numerical value of $a + b + c$ . | 16 | 0.416667 |
The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$ . Determine $f (2014)$ . $N_0=\{0,1,2,...\}$ | 671 | 0.333333 |
A convex polyhedron has $m$ triangular faces (there can be faces of other kind too). From each vertex there are exactly 4 edges. Find the least possible value of $m$ . | 8 | 0.75 |
Determine the largest natural number $m$ such that for each non negative real numbers $a_1 \ge a_2 \ge ... \ge a_{2014} \ge 0$ , it is true that $$ \frac{a_1+a_2+...+a_m}{m}\ge \sqrt{\frac{a_1^2+a_2^2+...+a_{2014}^2}{2014}} $$ | m = 44 | 0.333333 |
Let $k$ be a real number such that the product of real roots of the equation $$ X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0 $$ is $-2013$ . Find the sum of the squares of these real roots. | 4027 | 0.166667 |
Given that $\left(1+\sqrt{2}\right)^{2012}=a+b\sqrt{2}$, where $a$ and $b$ are integers, find the greatest common divisor of $b$ and $81$. | 3 | 0.166667 |
There are 22 black and 3 blue balls in a bag. Ahmet chooses an integer $ n$ in between 1 and 25. Betül draws $ n$ balls from the bag one by one such that no ball is put back to the bag after it is drawn. If exactly 2 of the $ n$ balls are blue and the second blue ball is drawn at $ n^{th}$ order, Ahmet wins, otherwise Betül wins. Calculate the value of n that maximizes the chance for Ahmet to win. | 13 | 0.083333 |
For any set $S$ of five points in the plane, no three of which are collinear, let $M(S)$ and $m(S)$ denote the greatest and smallest areas, respectively, of triangles determined by three points from $S$ . What is the minimum possible value of $M(S)/m(S)$ ? | \frac{1 + \sqrt{5}}{2} | 0.083333 |
Let $m$ be the smallest positive integer such that $m^2+(m+1)^2+\cdots+(m+10)^2$ is the square of a positive integer $n$ . Find $m+n$ | 95 | 0.166667 |
If $a$ and $b$ are positive integers such that $a \cdot b = 2400,$ find the least possible value of $a + b.$ | 98 | 0.916667 |
Find the greatest real number $C$ such that, for all real numbers $x$ and $y \neq x$ with $xy = 2$ it holds that
\[\frac{((x + y)^2 - 6)((x - y)^2 + 8)}{(x-y)^2}\geq C.\]
When does equality occur? | 18 | 0.166667 |
If $a$ and $b$ are complex numbers such that $a^2 + b^2 = 5$ and $a^3 + b^3 = 7$ , then their sum, $a + b$ , is real. The greatest possible value for the sum $a + b$ is $\tfrac{m+\sqrt{n}}{2}$ where $m$ and $n$ are integers. Find $n.$ | 57 | 0.75 |
Find all pairs $(x, y)$ of real numbers satisfying the system : $\begin{cases} x + y = 3
x^4 - y^4 = 8x - y \end{cases}$ | (2, 1) | 0.916667 |
It is possible to arrange eight of the nine numbers $2, 3, 4, 7, 10, 11, 12, 13, 15$ in the vacant squares of the $3$ by $4$ array shown on the right so that the arithmetic average of the numbers in each row and in each column is the same integer. Exhibit such an arrangement, and specify which one of the nine numbers must be left out when completing the array.
[asy]
defaultpen(linewidth(0.7));
for(int x=0;x<=4;++x)
draw((x+.5,.5)--(x+.5,3.5));
for(int x=0;x<=3;++x)
draw((.5,x+.5)--(4.5,x+.5));
label(" $1$ ",(1,3));
label(" $9$ ",(2,2));
label(" $14$ ",(3,1));
label(" $5$ ",(4,2));[/asy] | 10 | 0.25 |
Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$ , then $n+c$ divides $a^n+b^n+n$ .
*Proposed by Evan Chen* | (1, 1, 2) | 0.083333 |
Let $S$ be a real number. It is known that however we choose several numbers from the interval $(0, 1]$ with sum equal to $S$ , these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5.
Find the greatest possible value of $S$ . | 5.5 | 0.083333 |
$ABCD$ is a square of side 1. $P$ and $Q$ are points on $AB$ and $BC$ such that $\widehat{PDQ} = 45^{\circ}$ . Find the perimeter of $\Delta PBQ$ . | 2 | 0.833333 |
A tree has $10$ pounds of apples at dawn. Every afternoon, a bird comes and eats $x$ pounds of apples. Overnight, the amount of food on the tree increases by $10\%$ . What is the maximum value of $x$ such that the bird can sustain itself indefinitely on the tree without the tree running out of food? | \frac{10}{11} | 0.666667 |
Find all real numbers $x, y, z$ that satisfy the following system $$ \sqrt{x^3 - y} = z - 1 $$ $$ \sqrt{y^3 - z} = x - 1 $$ $$ \sqrt{z^3 - x} = y - 1 $$ | (1, 1, 1) | 0.916667 |
Let $p$ be a prime number. $T(x)$ is a polynomial with integer coefficients and degree from the set $\{0,1,...,p-1\}$ and such that $T(n) \equiv T(m) (mod p)$ for some integers m and n implies that $ m \equiv n (mod p)$ . Determine the maximum possible value of degree of $T(x)$ | p-2 | 0.083333 |
$n$ students take a test with $m$ questions, where $m,n\ge 2$ are integers. The score given to every question is as such: for a certain question, if $x$ students fails to answer it correctly, then those who answer it correctly scores $x$ points, while those who answer it wrongly scores $0$ . The score of a student is the sum of his scores for the $m$ questions. Arrange the scores in descending order $p_1\ge p_2\ge \ldots \ge p_n$ . Find the maximum value of $p_1+p_n$ . | m(n-1) | 0.333333 |
Denote by $S(n)$ the sum of the digits of the positive integer $n$ . Find all the solutions of the equation $n(S(n)-1)=2010.$ | 402 | 0.5 |
Determine the largest integer $n\geq 3$ for which the edges of the complete graph on $n$ vertices
can be assigned pairwise distinct non-negative integers such that the edges of every triangle have numbers which form an arithmetic progression. | n = 4 | 0.166667 |
Given the polynomial $a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$, where $n$ is a positive integer or zero, and $a_0$ is a positive integer. The remaining $a$'s are integers or zero. Set $h=n+a_0+|a_1|+|a_2|+\cdots+|a_n|$. Determine the number of polynomials with $h=3$. | 5 | 0.5 |
For real numbers $ x_{i}>1,1\leq i\leq n,n\geq 2,$ such that:
$ \frac{x_{i}^2}{x_{i}\minus{}1}\geq S\equal{}\displaystyle\sum^n_{j\equal{}1}x_{j},$ for all $ i\equal{}1,2\dots, n$
find, with proof, $ \sup S.$ | \frac{n^2}{n - 1} | 0.083333 |
Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x\le 2y\le 60$ and $y\le 2x\le 60.$ | 480 | 0.083333 |
$2023$ players participated in a tennis tournament, and any two players played exactly one match. There was no draw in any match, and no player won all the other players. If a player $A$ satisfies the following condition, let $A$ be "skilled player".**(Condition)** For each player $B$ who won $A$ , there is a player $C$ who won $B$ and lost to $A$ .
It turned out there are exactly $N(\geq 0)$ skilled player. Find the minimum value of $N$ . | 3 | 0.083333 |
Let $ ABC$ be a right angled triangle of area 1. Let $ A'B'C'$ be the points obtained by reflecting $ A,B,C$ respectively, in their opposite sides. Find the area of $ \triangle A'B'C'.$ | 3 | 0.166667 |
Given that $ 2^{2004}$ is a $ 604$ -digit number whose first digit is $ 1$ , how many elements of the set $ S \equal{} \{2^0,2^1,2^2, \ldots,2^{2003}\}$ have a first digit of $ 4$ ? | 194 | 0.666667 |
Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assuming the Earth is a perfect sphere with center $C$, determine the degree measure of $\angle ACB$. | 120^\circ | 0.75 |
Suppose $a$ , $b$ , $c$ , and $d$ are non-negative integers such that
\[(a+b+c+d)(a^2+b^2+c^2+d^2)^2=2023.\]
Find $a^3+b^3+c^3+d^3$ .
*Proposed by Connor Gordon* | 43 | 0.333333 |
Let $V$ be a rectangular prism with integer side lengths. The largest face has area $240$ and the smallest face has area $48$ . A third face has area $x$ , where $x$ is not equal to $48$ or $240$ . What is the sum of all possible values of $x$ ? | 260 | 0.333333 |
Ana & Bruno decide to play a game with the following rules.:
a) Ana has cards $1, 3, 5,7,..., 2n-1$ b) Bruno has cards $2, 4,6, 8,...,2n$ During the first turn and all odd turns afterwards, Bruno chooses one of his cards first and reveals it to Ana, and Ana chooses one of her cards second. Whoever's card is higher gains a point. During the second turn and all even turns afterwards, Ana chooses one of her cards first and reveals it to Bruno, and Bruno chooses one of his cards second. Similarly, whoever's card is higher gains a point. During each turn, neither player can use a card they have already used on a previous turn. The game ends when all cards have been used after $n$ turns. Determine the highest number of points Ana can earn, and how she manages to do this. | \left\lfloor \frac{n}{2} \right\rfloor | 0.25 |
Given the numbers $a_1,a_2,a_3,a_4,a_5,a_6 \in \{-1,0,1 \}$, find the total count of values of the expression $a_1\cdot 5^1+a_2\cdot 5^2+a_3\cdot 5^3+a_4\cdot 5^4+a_5\cdot 5^5+a_6\cdot 5^6$ that are negative. | 364 | 0.25 |
Suppose $ P(x) \equal{} a_nx^n\plus{}\cdots\plus{}a_1x\plus{}a_0$ be a real polynomial of degree $ n > 2$ with $ a_n \equal{} 1$ , $ a_{n\minus{}1} \equal{} \minus{}n$ , $ a_{n\minus{}2} \equal{}\frac{n^2 \minus{} n}{2}$ such that all the roots of $ P$ are real. Determine the coefficients $ a_i$ . | a_i = (-1)^{n-i} \binom{n}{i} | 0.333333 |
Let $f:\mathbb{N} \rightarrow \mathbb{N},$ $f(n)=n^2-69n+2250$ be a function. Find the prime number $p$ , for which the sum of the digits of the number $f(p^2+32)$ is as small as possible. | 3 | 0.333333 |
Let $u$ and $v$ be integers satisfying $0<v<u.$ Let $A=(u,v),$ let $B$ be the reflection of $A$ across the line $y=x,$ let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the y-axis. The area of pentagon $ABCDE$ is 451. Find $u+v.$ | 21 | 0.083333 |
Find the minimum real $x$ that satisfies $$ \lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots $$ | \sqrt[3]{3} | 0.583333 |
For $ n \in \mathbb{N}$ , let $ f(n)\equal{}1^n\plus{}2^{n\minus{}1}\plus{}3^{n\minus{}2}\plus{}...\plus{}n^1$ . Determine the minimum value of: $ \frac{f(n\plus{}1)}{f(n)}.$ | \frac{8}{3} | 0.833333 |
Suppose that $x, y, z$ are three distinct prime numbers such that $x + y + z = 49$. Find the maximum possible value for the product $xyz$. | 4199 | 0.083333 |
Given acute triangle $\triangle ABC$ in plane $P$ , a point $Q$ in space is defined such that $\angle AQB = \angle BQC = \angle CQA = 90^\circ.$ Point $X$ is the point in plane $P$ such that $QX$ is perpendicular to plane $P$ . Given $\angle ABC = 40^\circ$ and $\angle ACB = 75^\circ,$ find $\angle AXC.$ | 140^\circ | 0.25 |
Let $n>2$ be an integer. In a country there are $n$ cities and every two of them are connected by a direct road. Each road is assigned an integer from the set $\{1, 2,\ldots ,m\}$ (different roads may be assigned the same number). The *priority* of a city is the sum of the numbers assigned to roads which lead to it. Find the smallest $m$ for which it is possible that all cities have a different priority. | m = 3 | 0.166667 |
Find the smallest positive real $\alpha$ , such that $$ \frac{x+y} {2}\geq \alpha\sqrt{xy}+(1 - \alpha)\sqrt{\frac{x^2+y^2}{2}} $$ for all positive reals $x, y$ . | \alpha = \frac{1}{2} | 0.333333 |
Let $S$ be the set of all partitions of $2000$ (in a sum of positive integers). For every such partition $p$ , we define $f (p)$ to be the sum of the number of summands in $p$ and the maximal summand in $p$ . Compute the minimum of $f (p)$ when $p \in S .$ | 90 | 0.166667 |
$a,b,c,x,y,z$ are positive real numbers and $bz+cy=a$ , $az+cx=b$ , $ay+bx=c$ . Find the least value of following function $f(x,y,z)=\frac{x^2}{1+x}+\frac{y^2}{1+y}+\frac{z^2}{1+z}$ | \frac{1}{2} | 0.333333 |
Find all polynomials $P(x)$ with real coefficients satisfying: $P(2017) = 2016$ and $$ (P(x)+1)^2=P(x^2+1). $$ | P(x) = x - 1 | 0.833333 |
Given $\triangle ABC$ with $\angle A = 15^{\circ}$ , let $M$ be midpoint of $BC$ and let $E$ and $F$ be points on ray $BA$ and $CA$ respectively such that $BE = BM = CF$ . Let $R_1$ be the radius of $(MEF)$ and $R_{2}$ be
radius of $(AEF)$ . If $\frac{R_1^2}{R_2^2}=a-\sqrt{b+\sqrt{c}}$ where $a,b,c$ are integers. Find $a^{b^{c}}$ | 256 | 0.083333 |
Real numbers $x, y$ satisfy the inequality $x^2 + y^2 \le 2$ . Orove that $xy + 3 \ge 2x + 2y$ | xy + 3 \ge 2x + 2y | 0.833333 |
Evaluate $ \int_0^1 (1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{n \minus{} 1})\{1 \plus{} 3x \plus{} 5x^2 \plus{} \cdots \plus{} (2n \minus{} 3)x^{n \minus{} 2} \plus{} (2n \minus{} 1)x^{n \minus{} 1}\}\ dx.$ | n^2 | 0.333333 |
For every positive integeer $n>1$ , let $k(n)$ the largest positive integer $k$ such that there exists a positive integer $m$ such that $n = m^k$ .
Find $$ lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n} $$ | 1 | 0.583333 |
How many primes $p$ are there such that the number of positive divisors of $p^2+23$ is equal to 14? | 1 | 0.666667 |
A circle of radius $ 1$ is surrounded by $ 4$ circles of radius $ r$. What is the value of $ r$? | \sqrt{2} + 1 | 0.583333 |
Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$ | 12 | 0.833333 |
Let $ABC$ be a triangle such that midpoints of three altitudes are collinear. If the largest side of the triangle is $10$, determine the largest possible area of the triangle. | 25 | 0.666667 |
Find the number of distinct integral solutions of $ x^{4} \plus{}2x^{3} \plus{}3x^{2} \minus{}x\plus{}1\equiv 0\, \, \left(mod\, 30\right)$ where $ 0\le x<30$. | 1 | 0.833333 |
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