Found problems: 85335
Find all polynomials $f$ such that $f$ has non-negative integer coefficients, $f(1)=7$ and $f(2)=2017$.
$N$ cards are arranged in a circle, with exactly one card face up and the rest face-down. In a turn, choose a proper divisor $k$ of $N$. You may begin at any card on the circle and flip every $k$-th card, counting clockwise, if and only if every $k$-th card begins the turn in the same orientation (either all face-up or all face-down).
For example, with $15$ cards, you may start at any position and flip the $3$rd, $6$th, $9$th, $12$th, and $15$th cards around the circle if they all begin the turn face up (or all face-down).
For what values of $N$ can all of the cards be flipped face-up in a finite number of turns?
Find the functions $ f(x),\ g(x)$ satisfying the following equations.
(1) $ f'(x) \equal{} 2f(x) \plus{} 10,\ f(0) \equal{} 0$
(2) $ \int_0^x u^3g(u)du \equal{} x^4 \plus{} g(x)$
Two circles intersect at points $A\neq B$. A line passing through $A{}$ intersects the circles again at $C$ and $D$. Let $E$ and $F$ be the midpoints of the arcs $\overarc{BC}$ and $\overarc{BD}$ which do not contain $A{}$ and let $M$ be the midpoint of the segment $CD$. Prove that $ME$ and $MF$ are perpendicular.
Marisa has two identical cubical dice labeled with the numbers $\{1, 2, 3, 4, 5, 6\}$. However, the two dice are not fair, meaning that they can land on each face with different probability. Marisa rolls the two dice and calculates their sum. Given that the sum is $2$ with probability $0.04$, and $12$ with probability $0.01$, the maximum possible probability of the sum being $7$ is $p$. Compute $\lfloor 100p \rfloor$.
What is the minimum number of movements that a horse must carry out on chess, on an $8\times 8$ board, to reach the upper right square starting at the lower left? Remember that the horse moves in the usual $L$-shaped manner.
If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, the the larger of two numbers is
$\text{(A)}\ 16 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 44 \qquad \text{(E)}\ 52$
Let $ABCD$ be an isosceles trapezoid with $AD = BC = 255$ and $AB = 128$. Let $M$ be the midpoint of $CD$ and let $N$ be the foot of the perpendicular from $A$ to $CD$. If $\angle MBC = 90^o$, compute $\tan\angle NBM$.
Determine the number of $8$-tuples $(\epsilon_1, \epsilon_2,...,\epsilon_8)$ such that $\epsilon_1, \epsilon_2, ..., 8 \in \{1,-1\}$ and $\epsilon_1 + 2\epsilon_2 + 3\epsilon_3 +...+ 8\epsilon_8$ is a multiple of $3$.
Let $ n $ and $ r $ be positive integers and $ A $ be a set of lattice points in the plane such that any open disc of radius $ r $ contains a point of $ A $. Show that
for any coloring of the points of $ A $ in $ n $ colors there exists four points of the same color which are the vertices of a rectangle.
Let $ABC$ be a triangle with $AB=c$ , $BC=a$ and $AC=b$. If $ x,y\in\mathbb{R}$ satisfy $ \frac{1}{x} +\frac{1}{y+z} = \frac{1}{a} $ , $ \frac{1}{y} +\frac{1}{x+z} = \frac{1}{b} $ , $ \frac{1}{z} +\frac{1}{y+x} = \frac{1}{c} $ . Prove that the following equality holds $ x(p-a) + y(p-b) + z(p-c) = 3r^2 + 12R*r , $ Where $p$ is semi-perimeter, $R$ is the circumradius and $r$ is the inradius.
Find the number of functions $f:\mathbb{Z}\mapsto\mathbb{Z}$ for which $f(h+k)+f(hk)=f(h)f(k)+1$, for all integers $h$ and $k$.
A square $ABCD$ is given. A point $P$ is chosen inside the triangle $ABC$ such that $\angle CAP = 15^\circ = \angle BCP$. A point $Q$ is chosen such that $APCQ$ is an isosceles trapezoid: $PC \parallel AQ$, and $AP=CQ, AP\nparallel CQ$. Denote by $N$ the midpoint of $PQ$. Find the angles of the triangle $CAN$.
Let $n\in\mathbb{N}^*$. $2n+3$ points on the plane are given so that no 3 lie on a line and no 4 lie on a circle. Is it possible to find 3 points so that the interior of the circle passing through them would contain exactly $n$ of the remaining points.
Let be two polynoms $ P,Q\in\mathbb{C} [X] $ with degree at least $ 1, $ and such that $ P $ has only simple roots. Prove that the following affirmations are equivalent:
$ \text{(i)} P\circ Q $ is divisible by $ P. $
$ \text{(ii)} $ The evaluation of $ Q $ at any root of $ P $ is a root of $ P. $
[i]Marcel Țena[/i]
Find all integer $n$ such that $n-5$ divide $n^2+n-27$.
Let $AA_1 , CC_1$ be the altitudes of triangle $ABC, B_0$ the common point of the altitude from $B$ and the circumcircle of $ABC$; and $Q$ the common point of the circumcircles of $ABC$ and $A_1C_1B_0$, distinct from $B_0$. Prove that $BQ$ is the symmedian of $ABC$.
[i]Proposed by D.Shvetsov[/i]
Let $ABCD$ be a convex quadrilateral. Lines $AB$ and $CD$ meet at $P$, and lines $AD$ and $BC$ meet at $Q$. Let $O$ be a point in
the interior of $ABCD$ such that $\angle BOP = \angle DOQ$. Prove that
$\angle AOB +\angle COD = 180$.
Let $ABC$ be a triangle with $AB = BC$. Prove that $\triangle ABC$ is an obtuse triangle if and only if the equation $$Ax^2 + Bx + C = 0$$ has two distinct real roots, where $A$, $B$, $C$, are the angles in radians.
Given non-zero real numbers $u,v,w,x,y,z$, how many different possibilities are there for the signs of these numbers if
$$(u+ix)(v+iy)(w+iz)=i?$$
Let $P$ be a polynomial with integer coefficients. Assume that there exists a positive integer $n$ with $P(n^2)=2022$. Prove that there cannot be a positive rational number $r$ with $P(r^2)=2024$.
A circle, its chord $AB$ and the midpoint $W$ of the minor arc $AB$ are given. Take an arbitrary point $C$ on the major arc $AB$. The tangent to the circle at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$ respectively. Lines $WX$ and WY meet AB at points $N$ and $M$ respectively. Prove that the length of segment $NM$ does not depend on point $C$.
(A. Zertsalov, D. Skrobot)
Let $n\ge 5$ be an integer. Consider $2n-1$ subsets $A_1, A_2, A_3, \ldots , A_{2n-1}$ of the set $\{ 1, 2, 3,\ldots , n \}$, these subsets have the property that each of them has $2$ elements (that is that is, for $1 \le i \le 2n-1$ it is true that $A_i$ has $2$ elements).
Show that it is always possible to select $n$ of these subsets in such a way that the union of these $n$ subsets has at most $\frac{2}{3}n + 1$ elements in total.
Circles with radii $R$ and $r$ ($R > r$) are externally tangent. Another common tangent of the circles in drawn.
This tangent and the circles bound a region inside which a circle as large as possible is drawn.
What is the radius of this circle?
Let $ABCD$ be a trapezoid with bases $AB,CD$ such that $CD=k \cdot AB$ ($0<k<1$). Point $P$ is such that $\angle PAB=\angle CAD$ and $\angle PBA=\angle DBC$. Prove that $PA+PB \leq \dfrac{1}{\sqrt{1-k^2}} \cdot AB$.