2017 InIMC keystage3-IWYMIC individual

An equal number of novels and textbooks are in hard covers; $\frac25$ of the novels and $\frac34$ of the textbooks are in hard covers. What fraction of the total number of books is in hard cover?

A farmer picks 2017 apples with an average weight of 100 grams. The average weight of all the apples heavier than 100 grams is 122 grams while the average weight of all the apples lighter than 100 grams is 77 grams. At least how many apples weighing exactly 100 grams did the farmer pick?

The sum of three sides of a rectangle is 2017 cm while the sum of the fourth side and the diagonal is also 2017 cm. Find the length, in cm, of the diagonal of the rectangle.

Answer format: enter an exact expression; sqrt(n) denotes a square root.

Let $a,b,c,d$ be real numbers such that $0\le a\le b\le c\le d$ and $c+d=a^2+b^2+c^2+d^2=1$. Find the maximum value of $a+b$.

Find the least possible value of the fraction $\frac{a^2+b^2+c^2}{ab+bc}$ where $a,b$ and $c$ are positive real numbers.

Answer format: enter an exact expression; sqrt(n) denotes a square root.

An octagon which has side lengths 3, 3, 11, 11, 15, 15, 15 and 15 cm is inscribed in a circle. What is the area, in $\text{cm}^2$, of the octagon?

If $x$ and $y$ are real numbers such that $4x^2+y^2=4x-2y+7$, find the maximum value of $5x+6y$.

In triangle $ABC$, points $E$ and $D$ are on side $AC$ and point $F$ is on side $BC$ such that $AE=ED=DC$ and $BF:FC=2:3$. $AF$ intersects $BD$ and $BE$ at points $P$ and $Q$, respectively. Find the ratio of the area $EDPQ$ to the area of $ABC$.

The sum of the non-negative real numbers $x_1,x_2,\ldots,x_8$ is 8. Find the largest possible value of the expression $x_1x_2+x_2x_3+x_3x_4+\cdots+x_7x_8$.

Let $ABC$ be an isosceles triangle with $AB=AC$ and $\angle BAC=100^\circ$. A point $P$ inside the triangle $ABC$ satisfies that $\angle CBP=35^\circ$ and $\angle PCB=30^\circ$. Find the measure, in degrees, of angle $\angle BAP$.

If $xyz=-1$ and $a=x+\frac1x$, $b=y+\frac1y$, $c=z+\frac1z$, calculate $\sqrt{a^2+b^2+c^2+abc}$.

Mal, Num, and Pin each have distinct number of marbles. Five times the sum of the product of the number of marbles of any two of them equals to seven times the product of the number of the marbles the three of them have. Find the largest possible sum of their marbles.

Let $x$ and $y$ be non-negative integers such that $2^6+2^x+2^{3y}$ is a perfect square and the expression should be less than 10,000. Find the maximum value of $x+y$.

Let $ABC$ be a triangle such that $\angle B=16^\circ$ and $\angle C=28^\circ$. Let $P$ be a point on $BC$ such that $\angle BAP=44^\circ$ and let $Q$ be a point on $AB$ such that $\angle QCB=14^\circ$. Find, in degrees, $\angle PQC$.

Let $f(x)$ and $g(x)$ be distinct quadratic polynomials such that the leading coefficients of both polynomials are equal to 1 and
$$f(1)+f(2017)+f(2017^2)=g(1)+g(2017)+g(2017^2).$$
Find $x$ if $f(x)=g(x)$.