Let $P(x)$ be a polynomial with integer coefficients such that $P(1) = 4$ and $P(2) = 11$. Find the remainder when $P(x)$ is divided by $(x-1)(x-2)$, evaluated at $x = 10$.
Algebra
Question · Easy
Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2 - 10x + 18 = 0$. Find the value of $\alpha^2 + \beta^2$.
If $x + 2y + 3z = 24$ and $3x + 2y + z = 16$, find the value of $5x + 6y + 7z$.
In an arithmetic progression, the sum of the first, third, and fifth terms is 15, and the sum of the second, fourth, and sixth terms is 24. Find the seventh term of this progression.
If $x^2 - 3x + 1 = 0$, find the value of $x^4 + \frac{1}{x^4}$.
Let $a$ and $b$ be the roots of the equation $x^2 - p x + 12 = 0$. If $a^2 + b^2 = 25$ and $p > 0$, find the value of $p^2$.
The sum of the first $n$ terms of a sequence is given by $S_n = 2n^2 + 3n$. Find the value of the 10th term, $a_{10}$.
For any positive real numbers $a$ and $b$ such that $a + b = 10$, find the maximum possible integer value of $ab$.
Let $\alpha, \beta, \gamma$ be the roots of the cubic polynomial $P(x) = x^3 - 5x^2 + 7x + 13$. Find the value of $(3-\alpha)(3-\beta)(3-\gamma)$.
Question · Medium
Let $\alpha, \beta, \gamma$ be the roots of $x^3 - 7x^2 + 5x - 1 = 0$. Find the value of $\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2}$.
If $x$ and $y$ are positive integers satisfying the system of equations $x + y + xy = 29$ and $x^2 + y^2 = 85$, find the value of $x + y$.
Let $a$ and $b$ be positive real numbers such that $3a + 2b = 10$. Find the maximum possible value of $a^3 b^2$.
Let $f: \mathbb{R} \to \mathbb{R}$ be a function satisfying $f(x+y) = f(x) + f(y) + 3xy(x+y)$ for all real numbers $x, y$. If $f(1) = 4$, find the value of $f(3)$.
Let $S = \sum_{n=1}^{\infty} \frac{n}{3^n}$. Find the value of $36S$.
Let $a, b, c$ be the roots of $x^3 - 3x^2 - 4x + 12 = 0$. Find the value of $a^3 + b^3 + c^3$.
Let $a, b, c$ be positive real numbers such that $a^2 + 2b^2 + 3c^2 = 24$. Find the maximum possible value of $a + 2b + 3c$.
Find the sum of all real values of $x$ that satisfy the equation $\sqrt{x^2 - 2x + 1} + \sqrt{x^2 - 10x + 25} = 8$.
Let $f: \mathbb{Z} \to \mathbb{Z}$ be a function such that $f(x+y) = f(x) + f(y) - 2$ for all integers $x, y$. If $f(1) = 5$, find the value of $f(10)$.
Let $P(x)$ be a quadratic polynomial such that $P(x) - P(x-1) = 4x - 3$ for all real $x$. If $P(0) = 5$, find the value of $P(5)$.
Let $a_1 = 1$, and for all $n \ge 1$, let $a_{n+1} = \frac{a_n}{1 + 3a_n}$. Find the value of $\frac{1}{a_{26}}$.
Let $f: \mathbb{Q} \to \mathbb{Q}$ be a function satisfying $f(x+y) + f(x-y) = 2f(x) + 2f(y)$ for all $x, y \in \mathbb{Q}$. If $f(1) = 3$, find the value of $f(5)$.
Question · Hard
Let $P(x) = x^4 - 15x^3 + ax^2 + bx + 64$ be a polynomial with positive real roots $r_1, r_2, r_3, r_4$. If $\frac{r_1}{1} + \frac{r_2}{2} + \frac{r_3}{4} + \frac{r_4}{8} = 4$, find the value of $a$.
Let $f: \mathbb{R}^+ \to \mathbb{R}^+$ be a function satisfying $f(f(x)) + 2f(x) = 15x$ for all $x > 0$. Find the value of $f(13)$.
Let $a, b, c$ be positive real numbers such that $a + b + c = 12$. Find the minimum possible value of $\frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b}$.
Let $a_1 = 1$, and for all $n \ge 1$, let $a_{n+1} = a_n + \frac{1}{a_n}$. Find the greatest integer less than or equal to $a_{100}$.
In \triangle ABC, the internal angle bisector of \angle B and the exterior angle bisector of \angle C intersect at point D. If \angle A = 56^\circ, find the measure of \angle BDC in degrees.
Geometry
A right-angled triangle has legs of length 15 and 20. Find the length of the inradius of this triangle.
Two concentric circles have radii 9 and 15. A chord of the larger circle is tangent to the smaller circle. Find the length of this chord.
In a cyclic quadrilateral ABCD, the opposite angles are \angle A = 3x + 10^\circ and \angle C = 5x - 30^\circ. Find the measure of \angle A in degrees.
In \triangle ABC, a line parallel to BC intersects the sides AB and AC at D and E respectively. If the ratio of the area of \triangle ADE to the area of the quadrilateral DBCE is 9:16, and the length of BC is 35, find the length of DE.
From an external point P, a tangent PT and a secant PAB are drawn to a circle. If PT = 12 and PA = 8, find the length of the segment AB.
In a right-angled triangle ABC with \angle A = 90^\circ, the altitude AD is drawn to the hypotenuse BC. If BD = 9 and CD = 16, find the length of the altitude AD.
In \triangle ABC, G is the centroid of the triangle. If the area of \triangle ABC is 72, find the area of \triangle ABG.
Let ABCD be a cyclic quadrilateral with side lengths AB = 25, BC = 60, CD = 39, and DA = 52. Find the length of the diagonal BD.
In \triangle ABC, D is a point on BC such that BD:DC = 2:1, and E is a point on AC such that AE:EC = 3:4. The line segments AD and BE intersect at F. If the ratio AF:FD can be written as a simplified fraction p/q, find the value of p + q.
Two chords AB and CD of a circle intersect at a point P inside the circle. If AP = 8, BP = 9, and CD = 18 with CP < DP, find the length of the segment DP.
In \triangle ABC, AB = 10, AC = 9, and \cos A = -\frac{3}{5}. Find the length of the side BC.
A circle is inscribed in a quadrilateral ABCD whose sides are AB = 12, BC = 15, and CD = 14. Find the length of the side DA.
In \triangle ABC, AD is the median to the side BC, and E is the midpoint of AD. The line segment BE is extended to intersect the side AC at F. If the area of \triangle ABC is 45, find the area of \triangle ABF.
In a right-angled triangle ABC with \angle B = 90^\circ, AB = 10 and BC = 15. A square BPQR is inscribed in the triangle such that P lies on AB, Q lies on AC, and R lies on BC. Find the side length of this square.
Let ABC be an equilateral triangle inscribed in a circle. Let P be a point on the minor arc BC of the circumcircle. If PB = 8 and PC = 17, find the length of the segment PA.
In \triangle ABC, the sides are AB = 8, AC = 5, and BC = 7. Find the measure of \angle A in degrees.
Let AB be a diameter of a circle with length 25. Let C and D be two points on the circle such that AC = 15 and BD = 7, and the chords AC and BD do not intersect. Find the length of the chord CD.
In \triangle ABC, AB = 12, BC = 18, and AC = 24. The internal bisector of \angle A meets BC at D. A line through D parallel to AB meets AC at E. Find the length of the segment AE.
In \triangle ABC, points D, E, and F lie on the sides BC, CA, and AB respectively such that BD:DC = 1:2, CE:EA = 1:2, and AF:FB = 1:2. If the area of \triangle ABC is 81, find the area of \triangle DEF.
In \triangle ABC, AB = 18 and AC = 32. Let AD be the internal angle bisector of \angle A with D on BC. The circumcircle of \triangle ABD intersects the side AC at E. If AE = 24, find the length of the side BC.
In \triangle ABC, D is a point on BC such that BD:DC = 3:1, and E is a point on AC such that AE:EC = 1:2. The line segments AD and BE intersect at P. The line CP intersects AB at F. If the area of \triangle ABC is 70, find the area of \triangle BPF.
In a circle, two chords AB = 48 and CD = 40 intersect at a point P inside the circle. The line through P perpendicular to AB passes through the midpoint of CD. If the distance from the center of the circle to the chord AB is 7, find the minimum possible length of the segment AP.
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