In a right triangle with legs \(a\) and \(b\), compute the length of the median from the right angle.
In triangle ABC, prove that the centroid divides each median in a 2:1 ratio.
Compute the sum of the series \(1 + 2 + 4 + 8 + \dots + 2^n\).
O is the centre of the circle. Lengths of AB and BC are both 10cm. The area of the quadrilateral OABC is 120cm^2. Calculate the radius of the circle
About modules, reminders and a lot of prime numbers
Given a triangle with perimeter 30, prove that the sum of the altitudes is less than 20.
Solve the equation \(x^2 + (k-1)x + k = 0\) in terms of \(k\). (solve for \(x\))
In triangle ABC, medians intersect at point G. Prove that G divides each median in the ratio 2:1.
Let \(a, b, c\) be positive real numbers. Prove that:$$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} \ge \frac{3}{2}$$
Prove that $$7^n - 1$$ is divisible by 6 for all $$n \in \mathbb{N}$$.
Prove that for concurrent lines from the vertices of a triangle to the opposite sides, the product of the ratios of the divided segments equals one.
Find the area of the triangle with vertices \((1,2)\), \((4,5)\), and \((6,1)\)
In how many ways can \(n\) identical candies be distributed among \(k\) children?
Prove that for a point on the circumcircle of a triangle, the feet of the perpendiculars to the three sides are collinear, forming the Simson line.
Use mathematical induction to prove the formula for the sum of the first n cubes: $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$.
Prove that in any triangle, the sum of the lengths of any two sides is greater than the length of the third side.
Find all prime factors of $$ 2^{64}-1$$.
Prove that the area of a triangle with sides a, b, c can be computed as the square root of s(s-a)(s-b)(s-c), where s is the semiperimeter.
Solve the system $$\begin{cases} 2x + 3y = 7 \\ 5x - y = 8. \end{cases}$$
Prove that the incenter of a triangle is equidistant from all sides and lies at the intersection of the angle bisectors.
Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
A triangle has side lengths 5, 12, and 13. Compute its area.
Prove that for points lying on the sides of a triangle and collinear, the product of the ratios of the segments equals one.
Compute the sum of the first 16 terms of the geometric sequence \(2, 6, 18, \dots\).
In a circle, an inscribed angle subtending an arc is equal to half the measure of the corresponding central angle.
Determine whether \(2^{10} + 3^{5}\) is divisible by 7.
Find the area of the triangle with vertices (1,2), (4,5), and (6,1).
Prove that each excenter of a triangle is the intersection point of two external angle bisectors and the remaining internal bisector, and is the center of an excircle tangent to one side and the extensions of the other two.
Prove that the centroid, circumcenter, and orthocenter of a non-equilateral triangle lie on a single line called the Euler line.
Prove that $$\sin(A + B) = \sin A \cos B + \cos A \sin B$$.
From a group of 17 men and 23 women, how many 3-person committees can be formed that include at least one woman?
Let \(a, b, c\) be positive real numbers. Prove that: i) \(a^2 + b^2 + c^2 \ge ab + bc + ca\) ii) \(a^3 + b^3 + c^3 + ab^2 + bc^2 + ca^2 \ge 2(a^2b + b^2c + c^2a)\)
In triangle ABC, prove that the median from vertex A is shorter than the sum of half the other two sides.
Prove that the sum of the interior angles of any triangle is equal to $$180^\circ$$.
Prove that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.
Use mathematical induction to prove the formula for the sum of the first n squares: $$1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$$.
In a circle, prove that the angle subtended by a diameter is always a right angle.
Prove that the sum of the first n squares is a perfect square only for n = 1 and n = 24. That is, show that $$1^2 + 2^2 + \dots + n^2 = m^2$$ has solutions only for these values of n.
Prove that the area of a triangle with sides a, b, c can be computed as $$\sqrt{s(s-a)(s-b)(s-c)}$$, where s is the semiperimeter.
How many shortest paths are there from (0,0) to (m,n) in a grid if you can only move right or up?
Three fair dice are rolled. What is the probability that their sum is a prime number?
Prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
Compute the sum of squares of the first 1000 positive integers: $$1^2 + 2^2 + \dots + 1000^2.$$
Prove that in a cyclic quadrilateral, the sum of the opposite angles is 180 degrees.
Use mathematical induction to prove the formula for the nth triangular number: $$T_n = \frac{n(n+1)}{2}$$.
Given triangle ABC, the perpendicular bisectors of its sides intersect at point O. Prove that O is equidistant from A, B, and C.
Given a triangle with concurrent cevians, prove that the product of ratios of divided sides equals one.
Factor the polynomial \(x^3 - 3x^2 + 3x - 1\).
Prove that in a cyclic quadrilateral, the sum of opposite angles is 180 degrees.
Prove that in any triangle, the ratio of a side to the sine of its opposite angle is constant for all three sides.
Prove that in any triangle, the square of a side equals the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the included angle.
Prove formulas just using mathematical induction
In triangle ABC, the internal angle bisector of angle A meets side BC at point D. Prove that $$\frac{BD}{DC} = \frac{AB}{AC}$$.
Prove that the midpoints of the sides, the feet of the altitudes, and the midpoints between vertices and orthocenter of a triangle lie on a circle called the nine-point circle.
Let \(a, b, c\) be positive real numbers. Prove that:$$\frac{\sqrt{2a+b} + \sqrt{2b+c} + \sqrt{2c+a}}{\sqrt{a+b+c}} \le 3$$
Solve the inequality \(x^2 - 4x + 3 \le 0\).
