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Higher Secondary P5
Let $x>1$ be an integer such that for any two positive integers $a$ and $b$ , if $x$ divides $ab$ then $x$ either divides $a$ or divides $b$ . Find with proof the number of positive integers that divide $x$ . | 2 | 0.166667 |
Tasha and Amy both pick a number, and they notice that Tasha's number is greater than Amy's number by 12 . They each square their numbers to get a new number and see that the sum of these new numbers is half of 169 . Finally, they square their new numbers and note that Tasha's latest number is now greater than Amy's by 5070 . What is the sum of their original numbers? | 5 | 0.75 |
A finite non-empty set of integers is called $3$ -*good* if the sum of its elements is divisible by $3$ . Find the number of $3$ -good subsets of $\{0,1,2,\ldots,9\}$ . | 351 | 0.083333 |
$4.$ Harry, Hermione, and Ron go to Diagon Alley to buy chocolate frogs. If Harry and Hermione spent one-fourth of their own money, they would spend $3$ galleons in total. If Harry and Ron spent one-fifth of their own money, they would spend $24$ galleons in total. Everyone has a whole number of galleons, and the number of galleons between the three of them is a multiple of $7$ . What are all the possible number of galleons that Harry can have? | 6 | 0.583333 |
Let $ f, g: \mathbb {R} \to \mathbb {R} $ function such that $ f (x + g (y)) = - x + y + 1 $ for each pair of real numbers $ x $ e $ y $ . What is the value of $ g (x + f (y) $ ? | -x + y - 1 | 0.75 |
Find the number of pairs of positive integers $a$ and $b$ such that $a\leq 100\,000$ , $b\leq 100\,000$ , and $$ \frac{a^3-b}{a^3+b}=\frac{b^2-a^2}{b^2+a^2}. $$ | 10 | 0.25 |
Determine the smallest real number $C$ such that the inequality \[ C(x_1^{2005} +x_2^{2005} + \cdots + x_5^{2005}) \geq x_1x_2x_3x_4x_5(x_1^{125} + x_2^{125}+ \cdots + x_5^{125})^{16} \] holds for all positive real numbers $x_1,x_2,x_3,x_4,x_5$ . | 5^{15} | 0.833333 |
What is the polynomial of smallest degree that passes through $(-2, 2), (-1, 1), (0, 2),(1,-1)$ , and $(2, 10)$ ? | x^4 + x^3 - 3x^2 - 2x + 2 | 0.416667 |
Let $n$ be a fixed positive integer and fix a point $O$ in the plane. There are $n$ lines drawn passing through the point $O$ . Determine the largest $k$ (depending on $n$ ) such that we can always color $k$ of the $n$ lines red in such a way that no two red lines are perpendicular to each other.
*Proposed by Nikola Velov* | \left\lceil \frac{n}{2} \right\rceil | 0.333333 |
Given that $(a_n)_{n \equal{} 1}^\infty$ is defined on real numbers with $a_n \not \equal{} 0$, $a_na_{n \plus{} 3} = a_{n \plus{} 2}a_{n \plus{} 5}$, and $a_1a_2 + a_3a_4 + a_5a_6 = 6$. Find the value of $a_1a_2 + a_3a_4 + \cdots + a_{41}a_{42}$. | 42 | 0.333333 |
A rock is dropped off a cliff of height $h$. As it falls, a camera takes several photographs, at random intervals. At each picture, you measure the distance the rock has fallen. Let the average (expected value) of all these distances be $kh$. If the number of photographs taken is huge, find $k$. | \frac{1}{3} | 0.583333 |
In a forest there are $5$ trees $A, B, C, D, E$ that are in that order on a straight line. At the midpoint of $AB$ there is a daisy, at the midpoint of $BC$ there is a rose bush, at the midpoint of $CD$ there is a jasmine, and at the midpoint of $DE$ there is a carnation. The distance between $A$ and $E$ is $28$ m; the distance between the daisy and the carnation is $20$ m. Calculate the distance between the rose bush and the jasmine. | 6 | 0.916667 |
Suppose that $n$ is s positive integer. Determine all the possible values of the first digit after the decimal point in the decimal expression of the number $\sqrt{n^3+2n^2+n}$ | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 | 0.583333 |
The sides $x$ and $y$ of a scalene triangle satisfy $x + \frac{2\Delta }{x}=y+ \frac{2\Delta }{y}$ , where $\Delta$ is the area of the triangle. If $x = 60, y = 63$ , what is the length of the largest side of the triangle? | 87 | 0.416667 |
**Evaluate**
$\int^{\pi/2}_0 \frac{\cos^4x + \sin x \cos^3 x + \sin^2x\cos^2x + \sin^3x\cos x}{\sin^4x + \cos^4x + 2\ sinx\cos^3x + 2\sin^2x\cos^2x + 2\sin^3x\cos x} dx$ | \frac{\pi}{4} | 0.75 |
By dividing $2023$ by a natural number $m$ , the remainder is $23$ . How many numbers $m$ are there with this property? | 12 | 0.333333 |
Let $ABC$ be a triangle such that $|AB|=7$, $|BC|=8$, $|AC|=6$. Let $D$ be the midpoint of side $[BC]$. If the circle through $A$, $B$ and $D$ cuts $AC$ at $E$, calculate $|AE|$. | \frac{2}{3} | 0.666667 |
Matt has somewhere between $1000$ and $2000$ pieces of paper he's trying to divide into piles of the same size (but not all in one pile or piles of one sheet each). He tries $2$ , $3$ , $4$ , $5$ , $6$ , $7$ , and $8$ piles but ends up with one sheet left over each time. How many piles does he need? | 41 | 0.25 |
The number $2017$ is prime. Given that $S=\sum_{k=0}^{62}\binom{2014}{k}$, find the remainder when $S$ is divided by $2017$. | 1024 | 0.5 |
The product of two of the four roots of the quartic equation $x^4 - 18x^3 + kx^2+200x-1984=0$ is $-32$ . Determine the value of $k$ . | 86 | 0.5 |
The points $A$ , $B$ , $C$ , $D$ , and $E$ lie in one plane and have the following properties: $AB = 12, BC = 50, CD = 38, AD = 100, BE = 30, CE = 40$ .
Find the length of the segment $ED$ . | 74 | 0.083333 |
Suppose $b > 1$ is a real number where $\log_5 (\log_5 b + \log_b 125) = 2$ . Find $log_5 \left(b^{\log_5 b}\right) + log_b \left(125^{\log_b 125}\right).$ | 619 | 0.416667 |
Let $f(xy) = \frac{f(x)}{y}$ for all positive real numbers $x$ and $y$. If $f(500) = 3$, find the value of $f(600)$. | \frac{5}{2} | 0.833333 |
Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$ . | 440 | 0.166667 |
Katie and Allie are playing a game. Katie rolls two fair six-sided dice and Allie flips two fair two-sided coins. Katie’s score is equal to the sum of the numbers on the top of the dice. Allie’s score is the product of the values of two coins, where heads is worth $4$ and tails is worth $2.$ What is the probability Katie’s score is strictly greater than Allie’s? | \frac{25}{72} | 0.166667 |
Let $x_{1}$ be a positive real number and for every integer $n \geq 1$ let $x_{n+1} = 1 + x_{1}x_{2}\ldots x_{n-1}x_{n}$ .
If $x_{5} = 43$ , what is the sum of digits of the largest prime factors of $x_{6}$ ? | 13 | 0.666667 |
Find all the real $a,b,c$ such that the equality $|ax+by+cz| + |bx+cy+az| + |cx+ay+bz| = |x|+|y|+|z|$ is valid for all the real $x,y,z$ . | (a, b, c) = (\pm 1, 0, 0), (0, \pm 1, 0), (0, 0, \pm 1) | 0.083333 |
A pet shop sells cats and two types of birds: ducks and parrots. In the shop, $\tfrac{1}{12}$ of animals are ducks, and $\tfrac{1}{4}$ of birds are ducks. Given that there are $56$ cats in the pet shop, how many ducks are there in the pet shop? | 7 | 0.75 |
Let $n>k>1$ be positive integers. Determine all positive real numbers $a_1, a_2, \ldots, a_n$ which satisfy $$ \sum_{i=1}^n \sqrt{\frac{k a_i^k}{(k-1) a_i^k+1}}=\sum_{i=1}^n a_i=n . $$ | a_i = 1 | 0.916667 |
Let $A_1, A_2, \ldots, A_n$ be an $n$ -sided regular polygon. If $\frac{1}{A_1 A_2} = \frac{1}{A_1 A_3} + \frac{1}{A_1A_4}$ , find $n$ . | 7 | 0.666667 |
Is $ \sqrt{2} $ the limit of a sequence of numbers of the form $ \sqrt[3]{n} - \sqrt[3]{m} $ , where $ n, m = 0, 1, 2, \cdots $ . | \sqrt{2} | 0.166667 |
For some constant $k$ the polynomial $p(x) = 3x^2 + kx + 117$ has the property that $p(1) = p(10)$ . Evaluate $p(20)$ . | 657 | 0.916667 |
Let $a_{n+1} = \frac{4}{7}a_n + \frac{3}{7}a_{n-1}$ and $a_0 = 1$ , $a_1 = 2$ . Find $\lim_{n \to \infty} a_n$ . | \frac{17}{10} | 0.916667 |
Let $M_1, M_2, . . ., M_{11}$ be $5-$ element sets such that $M_i \cap M_j \neq {\O}$ for all $i, j \in \{1, . . ., 11\}$ . Determine the minimum possible value of the greatest number of the given sets that have nonempty intersection. | 4 | 0.083333 |
Let $ABCD$ be a convex quadrilateral such that $AB + BC = 2021$ and $AD = CD$ . We are also given that $\angle ABC = \angle CDA = 90^o$ . Determine the length of the diagonal $BD$ . | \frac{2021 \sqrt{2}}{2} | 0.166667 |
Given a square with sides of length $10$ and a circle with radius $10$ centered at one of its vertices, calculate the area of the union of the regions enclosed by the square and the circle. | 100 + 75\pi | 0.833333 |
Given the polynomial $P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots(x^{1024}+1024)$, determine the exponent $a$ in the expression $2^a$ that is equal to the coefficient of $x^{2012}$. | 6 | 0.083333 |
Given that $BD$ is a median in triangle $ABC$, $CF$ intersects $BD$ at $E$ such that $BE=ED$, and $F$ is a point on $AB$, with $BF=5$, calculate the length of $BA$. | 15 | 0.416667 |
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $ 8: 7$ . Find the minimum possible value of their common perimeter. | 676 | 0.083333 |
Let ABC be a scalene triangle and AM is the median relative to side BC. The diameter circumference AM intersects for the second time the side AB and AC at points P and Q, respectively, both different from A. Assuming that PQ is parallel to BC, determine the angle measurement <BAC.
Any solution without trigonometry? | \angle BAC = 90^\circ | 0.333333 |
Circles $C_1$ and $C_2$ intersect at points $X$ and $Y$ . Point $A$ is a point on $C_1$ such that the tangent line with respect to $C_1$ passing through $A$ intersects $C_2$ at $B$ and $C$ , with $A$ closer to $B$ than $C$ , such that $2016 \cdot AB = BC$ . Line $XY$ intersects line $AC$ at $D$ . If circles $C_1$ and $C_2$ have radii of $20$ and $16$ , respectively, find $\sqrt{1+BC/BD}$ . | 2017 | 0.083333 |
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $ x, y,$ and $ r$ with $ |x| > |y|,$
\[ (x \plus{} y)^r \equal{} x^r \plus{} rx^{r \minus{} 1}y \plus{} \frac {r(r \minus{} 1)}2x^{r \minus{} 2}y^2 \plus{} \frac {r(r \minus{} 1)(r \minus{} 2)}{3!}x^{r \minus{} 3}y^3 \plus{} \cdots
\]
What are the first three digits to the right of the decimal point in the decimal representation of $ \left(10^{2002} \plus{} 1\right)^{10/7}?$ | 428 | 0.583333 |
Let the sequence $\{a_i\}^\infty_{i=0}$ be defined by $a_0 =\frac12$ and $a_n = 1 + (a_{n-1} - 1)^2$ . Find the product $$ \prod_{i=0}^\infty a_i=a_0a_1a_2\ldots $$ | \frac{2}{3} | 0.166667 |
Five rays $\overrightarrow{OA}$ , $\overrightarrow{OB}$ , $\overrightarrow{OC}$ , $\overrightarrow{OD}$ , and $\overrightarrow{OE}$ radiate in a clockwise order from $O$ forming four non-overlapping angles such that $\angle EOD = 2\angle COB$ , $\angle COB = 2\angle BOA$ , while $\angle DOC = 3\angle BOA$ . If $E$ , $O$ , $A$ are collinear with $O$ between $A$ and $E$ , what is the degree measure of $\angle DOB?$ | 90^\circ | 0.166667 |
The following is known about the reals $ \alpha$ and $ \beta$
$ \alpha^{3}-3\alpha^{2}+5\alpha-17=0$ and $ \beta^{3}-3\beta^{2}+5\beta+11=0$
Determine $ \alpha+\beta$ | 2 | 0.333333 |
Square $ABCD$ has side length $1$ ; circle $\Gamma$ is centered at $A$ with radius $1$ . Let $M$ be the midpoint of $BC$ , and let $N$ be the point on segment $CD$ such that $MN$ is tangent to $\Gamma$ . Compute $MN$ .
*2018 CCA Math Bonanza Individual Round #11* | \frac{5}{6} | 0.083333 |
When two stars are very far apart their gravitational potential energy is zero, and when they are separated by a distance $d$, the gravitational potential energy of the system is $U$. Determine the gravitational potential energy of the system when the stars are separated by a distance $2d$. | \frac{U}{2} | 0.416667 |
A boy's age is written after his father's age in a four-digit number. If the boy subtracts the absolute difference of their ages from this number, the result is $4,289$. What are the sum of their ages? | 59 | 0.75 |
Let $\{a_i\}_{i=0}^\infty$ be a sequence of real numbers such that \[\sum_{n=1}^\infty\dfrac {x^n}{1-x^n}=a_0+a_1x+a_2x^2+a_3x^3+\cdots\] for all $|x|<1$ . Find $a_{1000}$ .
*Proposed by David Altizio* | 16 | 0.583333 |
(1) Evaluate $ \int_{\minus{}\sqrt{3}}^{\sqrt{3}}( x^2\minus{}1)dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)^2dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\plus{}1)^2dx$ .
(2) If a linear function $ f(x)$ satifies $ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)f(x)dx\equal{}5\sqrt{3},\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\plus{}1)f(x)dx\equal{}3\sqrt{3}$ , then we have $ f(x)\equal{}\boxed{\ A\ }(x\minus{}1)\plus{}\boxed{\ B\ }(x\plus{}1)$ , thus we have $ f(x)\equal{}\boxed{\ C\ }$ . | f(x) = 2x - \frac{1}{2} | 0.333333 |
Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$ .
*Proposed by Evan Chen* | 1 | 0.916667 |
$x$ and $y$ are two distinct positive integers, calculate the minimum positive integer value of $(x + y^2)(x^2 - y)/(xy)$. | 14 | 0.333333 |
Katrine has a bag containing 4 buttons with distinct letters M, P, F, G on them (one letter per button). She picks buttons randomly, one at a time, without replacement, until she picks the button with letter G. What is the probability that she has at least three picks and her third pick is the button with letter M? | \frac{1}{12} | 0.166667 |
If the inequality $\frac {\sin ^{3} x}{\cos x} + \frac {\cos ^{3} x}{\sin x} \ge k$ holds for every $x\in \left(0,\frac {\pi }{2}\right)$, find the largest possible value of $k$. | 1 | 0.833333 |
Consider and odd prime $p$ . For each $i$ at $\{1, 2,..., p-1\}$ , let $r_i$ be the rest of $i^p$ when it is divided by $p^2$ .
Find the sum: $r_1 + r_2 + ... + r_{p-1}$ | \frac{p^2(p-1)}{2} | 0.083333 |
Determine the real values of $x$ such that the triangle with sides $5$ , $8$ , and $x$ is obtuse. | (3, \sqrt{39}) \cup (\sqrt{89}, 13) | 0.333333 |
**Problem 2**
Determine all pairs $(n, m)$ of positive integers satisfying the equation $$ 5^n = 6m^2 + 1\ . $$ | (2, 2) | 0.75 |
Let $A=\frac{2^2+3\cdot 2 + 1}{3! \cdot 4!} + \frac{3^2+3\cdot 3 + 1}{4! \cdot 5!} + \frac{4^2+3\cdot 4 + 1}{5! \cdot 6!} + \dots + \frac{10^2+3\cdot 10 + 1}{11! \cdot 12!}$ . Find the remainder when $11!\cdot 12! \cdot A$ is divided by $11$. | 10 | 0.166667 |
At a math competition, a team of $8$ students has $2$ hours to solve $30$ problems. If each problem needs to be solved by $2$ students, on average how many minutes can a student spend on a problem? | 16 | 0.083333 |
**Q13.** Determine the greatest value of the sum $M=11xy+3x+2012yz$ , where $x,y,z$ are non negative integers satisfying condition $x+y+z=1000.$ | 503000000 | 0.583333 |
The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play. If the Band can't play any two tiring songs in a row, how many ways can the band play its 8 songs? | 14400 | 0.916667 |
Let $O$ and $A$ be two points in the plane with $OA = 30$ , and let $\Gamma$ be a circle with center $O$ and radius $r$ . Suppose that there exist two points $B$ and $C$ on $\Gamma$ with $\angle ABC = 90^{\circ}$ and $AB = BC$ . Compute the minimum possible value of $\lfloor r \rfloor.$ | 12 | 0.083333 |
Let $ m$ a positive integer and $ p$ a prime number, both fixed. Define $ S$ the set of all $ m$ -uple of positive integers $ \vec{v} \equal{} (v_1,v_2,\ldots,v_m)$ such that $ 1 \le v_i \le p$ for all $ 1 \le i \le m$ . Define also the function $ f(\cdot): \mathbb{N}^m \to \mathbb{N}$ , that associates every $ m$ -upla of non negative integers $ (a_1,a_2,\ldots,a_m)$ to the integer $ \displaystyle f(a_1,a_2,\ldots,a_m) \equal{} \sum_{\vec{v} \in S} \left(\prod_{1 \le i \le m}{v_i^{a_i}} \right)$ .
Find all $ m$ -uple of non negative integers $ (a_1,a_2,\ldots,a_m)$ such that $ p \mid f(a_1,a_2,\ldots,a_m)$ .
*(Pierfrancesco Carlucci)* | (a_1, a_2, \ldots, a_m) | 0.666667 |
Given that $1 < (x-2)^2 < 25$, find the sum of all integer solutions for x. | 12 | 0.833333 |
A given rectangle $ R$ is divided into $mn$ small rectangles by straight lines parallel to its sides. (The distances between the parallel lines may not be equal.) What is the minimum number of appropriately selected rectangles’ areas that should be known in order to determine the area of $ R$ ? | m + n - 1 | 0.833333 |
When a block of wood with a weight of 30 N is completely submerged under water the buoyant force on the block of wood from the water is 50 N, calculate the fraction of the block that will be visible above the surface of the water when the block is floating. | \frac{2}{5} | 0.583333 |
A rectangle with sides $a$ and $b$ has an area of $24$ and a diagonal of length $11$ . Find the perimeter of this rectangle. | 26 | 0.75 |
Let $a,b$ be constant numbers such that $0<a<b.$ If a function $f(x)$ always satisfies $f'(x) >0$ at $a<x<b,$ for $a<t<b$ find the value of $t$ for which the following the integral is minimized.
\[ \int_a^b |f(x)-f(t)|x\ dx. \] | t = \sqrt{\frac{a^2 + b^2}{2}} | 0.25 |
Let $f$ be a non-constant polynomial such that \[ f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} \] for all nonzero real numbers $x$ . Find the sum of all possible values of $f(1)$ .
*Proposed by Ahaan S. Rungta* | 6039 | 0.75 |
Let $a$ and $b$ be two real numbers and let $M(a,b)=\max\left \{ 3a^{2}+2b; 3b^{2}+2a\right \}$ . Find the values of $a$ and $b$ for which $M(a,b)$ is minimal. | a = -\frac{1}{3}, b = -\frac{1}{3} | 0.083333 |
In how many ways, can we draw $n-3$ diagonals of a $n$ -gon with equal sides and equal angles such that: $i)$ none of them intersect each other in the polygonal. $ii)$ each of the produced triangles has at least one common side with the polygonal. | \frac{1}{n-1} \binom{2n-4}{n-2} | 0.916667 |
A set $S$ has $7$ elements. Several $3$ -elements subsets of $S$ are listed, such
that any $2$ listed subsets have exactly $1$ common element. What is the maximum number of subsets that can be listed? | 7 | 0.916667 |
Given that $ 1 \minus{} y$ is used as an approximation to the value of $ \frac {1}{1 \plus{} y}$, find the ratio of the error made to the correct value. | y^2 | 0.75 |
Hugo, Evo, and Fidel are playing Dungeons and Dragons, which requires many twenty-sided dice. Attempting to slay Evo's *vicious hobgoblin +1 of viciousness,* Hugo rolls $25$ $20$ -sided dice, obtaining a sum of (alas!) only $70$ . Trying to console him, Fidel notes that, given that sum, the product of the numbers was as large as possible. How many $2$ s did Hugo roll? | 5 | 0.25 |
In an $n$ -by- $m$ grid, $1$ row and $1$ column are colored blue, the rest of the cells are white. If precisely $\frac{1}{2010}$ of the cells in the grid are blue, how many values are possible for the ordered pair $(n,m)$ | 96 | 0.75 |
The minimum value of $x(x+4)(x+8)(x+12)$ in real numbers. | -256 | 0.916667 |
In the future, each country in the world produces its Olympic athletes via cloning and strict training
programs. Therefore, in the fi nals of the 200 m free, there are two indistinguishable athletes from each
of the four countries. How many ways are there to arrange them into eight lanes? | 2520 | 0.833333 |
A set $S$ consists of $k$ sequences of $0,1,2$ of length $n$ . For any two sequences $(a_i),(b_i)\in S$ we can construct a new sequence $(c_i)$ such that $c_i=\left\lfloor\frac{a_i+b_i+1}2\right\rfloor$ and include it in $S$ . Assume that after performing finitely many such operations we obtain all the $3n$ sequences of $0,1,2$ of length $n$ . Find the least possible value of $k$ . | n + 1 | 0.083333 |
Let $N = 123456789101112\dots4344$ be the $79$ -digit number obtained that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$ ? | 9 | 0.75 |
Given the polynomial $x^5+x^4-4x^3-7x^2-7x-2$, find the sum of its distinct real roots. | 0 | 0.083333 |
Let $a, b, c, d$ be the roots of the quartic polynomial $f(x) = x^4 + 2x + 4$ . Find the value of $$ \frac{a^2}{a^3 + 2} + \frac{b^2}{b^3 + 2} + \frac{c^2}{c^3 + 2} + \frac{d^2}{d^3 + 2}. $$ | \frac{3}{2} | 0.083333 |
Let $F$ be a finite field having an odd number $m$ of elements. Let $p(x)$ be an irreducible (i.e. nonfactorable) polynomial over $F$ of the form $$ x^2+bx+c, ~~~~~~ b,c \in F. $$ For how many elements $k$ in $F$ is $p(x)+k$ irreducible over $F$ ? | n = \frac{m-1}{2} | 0.583333 |
Points A and B lie on a circle centered at O, and ∠AOB = 60°. A second circle is internally tangent to the first and tangent to both OA and OB. What is the ratio of the area of the smaller circle to that of the larger circle? | \frac{1}{9} | 0.25 |
How many ways are there to partition $7$ students into the groups of $2$ or $3$ ? | 105 | 0.916667 |
A pen costs $\mathrm{Rs.}\, 13$ and a note book costs $\mathrm{Rs.}\, 17$ . A school spends exactly $\mathrm{Rs.}\, 10000$ in the year $2017-18$ to buy $x$ pens and $y$ note books such that $x$ and $y$ are as close as possible (i.e., $|x-y|$ is minimum). Next year, in $2018-19$ , the school spends a little more than $\mathrm{Rs.}\, 10000$ and buys $y$ pens and $x$ note books. How much **more** did the school pay? | 40 | 0.083333 |
Let be a real number $ a\in \left[ 2+\sqrt 2,4 \right] . $ Find $ \inf_{\stackrel{z\in\mathbb{C}}{|z|\le 1}} \left| z^2-az+a \right| . $ | 1 | 0.833333 |
Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m -3n)^2 = 2019$ . Find $3m + n$ . | 46 | 0.416667 |
Consider the set $M=\{1,2,...,n\},n\in\mathbb N$ . Find the smallest positive integer $k$ with the following property: In every $k$ -element subset $S$ of $M$ there exist two elements, one of which divides the other one. | k = \left\lceil \frac{n}{2} \right\rceil + 1 | 0.25 |
Given $f(x)$ as the minimum of the numbers $4x+1$, $x+2$, and $-2x+4$, calculate the maximum value of $f(x)$. | \frac{8}{3} | 0.333333 |
A triangle $ABC$ with $AC=20$ is inscribed in a circle $\omega$ . A tangent $t$ to $\omega$ is drawn through $B$ . The distance $t$ from $A$ is $25$ and that from $C$ is $16$ .If $S$ denotes the area of the triangle $ABC$ , find the largest integer not exceeding $\frac{S}{20}$ | 10 | 0.25 |
Determine the largest constant $K\geq 0$ such that $$ \frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2 $$ holds for all positive real numbers $a,b,c$ such that $ab+bc+ca=abc$ .
*Proposed by Orif Ibrogimov (Czech Technical University of Prague).* | K = 18 | 0.333333 |
Suppose $P(x)$ is a quadratic polynomial with integer coefficients satisfying the identity \[P(P(x)) - P(x)^2 = x^2+x+2016\] for all real $x$ . What is $P(1)$ ? | 1010 | 0.416667 |
The triangle ABC has sides AB = 137, AC = 241, and BC =200. There is a point D, on BC, such that both incircles of triangles ABD and ACD touch AD at the same point E. Determine the length of CD.
[asy]
pair A = (2,6);
pair B = (0,0);
pair C = (10,0);
pair D = (3.5,0) ;
pair E = (3.1,2);
draw(A--B);
draw(B--C);
draw(C--A);
draw (A--D);
dot ((3.1,1.7));
label ("E", E, dir(45));
label ("A", A, dir(45));
label ("B", B, dir(45));
label ("C", C, dir(45));
label ("D", D, dir(45));
draw(circle((1.8,1.3),1.3));
draw(circle((4.9,1.7),1.75));
[/asy] | 152 | 0.166667 |
Let $M$ be the midpoint of side $AC$ of the triangle $ABC$ . Let $P$ be a point on the side $BC$ . If $O$ is the point of intersection of $AP$ and $BM$ and $BO = BP$ , determine the ratio $\frac{OM}{PC}$ . | \frac{1}{2} | 0.916667 |
A basket is called "*Stuff Basket*" if it includes $10$ kilograms of rice and $30$ number of eggs. A market is to distribute $100$ Stuff Baskets. We know that there is totally $1000$ kilograms of rice and $3000$ number of eggs in the baskets, but some of market's baskets include either more or less amount of rice or eggs. In each step, market workers can select two baskets and move an arbitrary amount of rice or eggs between selected baskets. Starting from an arbitrary situation, what's the minimum number of steps that workers provide $100$ Stuff Baskets? | 99 | 0.166667 |
You have $8$ friends, each of whom lives at a different vertex of a cube. You want to chart a path along the cube’s edges that will visit each of your friends exactly once. You can start at any vertex, but you must end at the vertex you started at, and you cannot travel on any edge more than once. How many different paths can you take? | 96 | 0.75 |
Let $P(x)$ be a polynomial with integer coefficients and roots $1997$ and $2010$ . Suppose further that $|P(2005)|<10$ . Determine what integer values $P(2005)$ can get. | P(2005) = 0 | 0.916667 |
$ABCDE$ is a regular pentagon. What is the degree measure of the acute angle at the intersection of line segments $AC$ and $BD$ ? | 72^\circ | 0.083333 |
Let $ x_1, x_2, \ldots, x_n$ be positive real numbers with sum $ 1$ . Find the integer part of:
$ E\equal{}x_1\plus{}\dfrac{x_2}{\sqrt{1\minus{}x_1^2}}\plus{}\dfrac{x_3}{\sqrt{1\minus{}(x_1\plus{}x_2)^2}}\plus{}\cdots\plus{}\dfrac{x_n}{\sqrt{1\minus{}(x_1\plus{}x_2\plus{}\cdots\plus{}x_{n\minus{}1})^2}}$ | 1 | 0.916667 |
Given the sequence $\{a_k\}$ where $a_1 = 1$ and $a_{m+n} = a_m + a_n + mn$ for all positive integers $m$ and $n$, find the value of $a_{12}$. | 78 | 0.916667 |
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