Bottle A contains 15% syrup. Bottle B contains 40% syrup. When these 2 bottles of syrup are mixed together, the syrup content is 30% and the total volume is 600 ml. How much syrup is in the bottle A at first?

Suppose that the polynomial \(f\left(x\right)=2x^5-9x^3+2x^2+9x-3\) has 5 solutions x_{1^{; X}2^{; x}3}; x₄; x₅. The other polynomial \(k\left(x\right)=x^2-4\). Find the value of \(P=k\left(x_1\right)\times k\left(x_2\right)\times k\left(x_3\right)\times k\left(x_4\right)\times k\left(x_5\right)\).

Let S (x ; y) = x² - y² with real numbers x and y then S (3;S ( 3 ; 4 )) = ...

The base of a right triangle measure x - 3 and x + 4. If the hypotenuse of the triangle is 2x - 3, what is the length of the hypotenuse?

A rectangle has length p cm and breadth q cm, where p and q are integers is p and q satisfy the equation pq + q = 13 + q². Find the maximum possible area of the rectangle.

A solution to the equation (x + a) (x + b) (x + c) + 5 = 0 is x = 1, where a, b, c are different integers. Find the value of a + b + c =...

Let # be defined by a # b = ab + a² + b². Let @ be defined by \(a@b=\dfrac{a}{3}-b\). 
Calculate (a # b)@ a for a = 15 and b = 6.

There are 2017 points on the plane. The area of any triangle with verticles does not exceed 1. Assume that in any case, all these points can be placed in a triangle whose area is a positive integer K. Find the least value of K.

Let N be the largest number of region that can be formed by drawing 2016 straight lines on a plane. Find the sum of all digits of N.

Given that  \(a^2-b^2=1\). Evaluate \(A=2\left(a^6-b^6\right)-3\left(a^4+b^4\right)\).

Let \(A,B\in Z^+\), \(a>b\), \(A\ne2B\). Find \(C\in Z\) such that \(\dfrac{a^3+b^3}{a^3+c^3}=\dfrac{a+b}{a+c}\)

Given \(f\left(x\right)=\dfrac{2x^2+3x+3}{2x-1}.\) If a and f(a) are both integer we say that f(a) is an element of the set A. What is the sum of all elements of the set A?

Given a quadrilateral ABCD with AB = 6cm, the diagonale AC intersects BD at O. Given that OA=8cm, OB = 4cm and OD= 6cm. Evaluate the measure of segment AD?

Suppose that the polynomial \(f\left(x\right)=x^5-x^4-4x^3+2x^2+4x+1\) has 5 solutions \(x_1;x_2;x_3;x_4;x_5\).The other polynomial \(K\left(x\right)=x^2-4\). Find the value of P=K(\(x_1\))×K(\(x_2\))×K(\(x_3\))×K(\(x_4\))×K(\(x_5\))

Suppose that the polynomial \(f\left(x\right)=2x^5-9x^3+2x^2+9x-3\) has 5 solutions \(x_1;x_2;x_3;x_4;x_5\). The other polynomial \(K\left(x\right)=x^2-4\). Find the value of \(P=K\left(X_1\right)\times K\left(X_2\right)\times K\left(X_3\right)\times K\left(X_4\right)\times K\left(X_5\right)\)

Let P(x) be the polynomial given by \(P\left(x\right)=\left(2+x+2x^3\right)^{15}\). Suppose that \(P\left(x\right)=a_0+a_1x+a_2x^2+...+a_{45}x^{45}\).The value of \(S=a_1-a_2+a_3-a_4+...-a_{44}+a_{45}\).

Find the number not equal to 0 such that triple of its square is equal to twice its cube.

Calculate: \(\dfrac{\left(4\times7+2\right)\left(6\times6+2\right)\left(8\times11+2\right)...\left(100\times103+2\right)}{\left(5\times8+2\right)\left(7\times10+2\right)\left(9\times12+2\right)...\left(99\times102+2\right)}=...\)

The minimum value of the expression A= 

Given that \(\dfrac{\overline{ab}}{b}=\dfrac{\overline{bc}}{c}=\dfrac{\overline{ca}}{a}\).

Then the value of \(M=\dfrac{a}{\overline{bc}}+\dfrac{b}{\overline{ab}}+\dfrac{c}{\overline{ab}}\) is: