Found problems: 85335
Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
Solve the system \begin{eqnarray*} \\ (x+y)(x+y+z) &=& 18 \\ (y+z)(x+y+z) &=& 30 \\ (x+z)(x+y+z) &=& 2A \end{eqnarray*} in terms of the parameter $A$.
You are given $n \ge 2$ distinct positive integers. Let's call a pair of these integers [i]elegant[/i] if their sum is an integer power of $2$. For every $n$ find the largest possible number of elegant pairs.
[i]Proposed by Oleksiy Masalitin[/i]
We are given an infinite set of points in the plane such that any two of them have a distance of at most one. Prove that all the axes of symmetry of this set are concurrent, provided that there are at least two of them.
[i]Proposed by David Anghel[/i]
Consider a triangle $ABC$ with $BC = a, CA = b, AB = c.$ Let $D$ be the midpoint of $BC$ and $E$ be the intersection of the bisector of $A$ with $BC$ . The circle through $A, D, E$ meets $AC, AB$ again at $F, G$ respectively. Let $H\not = B$ be a point on $AB$ with $BG = GH$ . Prove that triangles $EBH$ and $ABC$ are similar and find the ratio of their areas.
In a mathematics test number of participants is $N < 40$. The passmark is fixed at $65$. The test results are
the following:
The average of all participants is $66$, that of the promoted $71$ and that of the repeaters $56$.
However, due to an error in the wording of a question, all scores are increased by $5$. At this point
the average of the promoted participants becomes $75$ and that of the non-promoted $59$.
(a) Find all possible values of $N$.
(b) Find all possible values of $N$ in the case where, after the increase, the average of the promoted had become $79$ and that of non-promoted $47$.
Carlos and Yue play the following game: First Carlos writes a $+$ sign or a $-$ sign in front of each of the $50$ numbers $1,2,\cdots,50$.
Then, in turns, each one chooses a number from the sequence obtained; Start by choosing Yue. If the absolute value of the sum of the $25$ numbers that Carlos chose is greater than or equal to the absolute value of the sum of the $25$ numbers that Yue chose, Carlos wins. In the other case, Yue wins.
Determine which of the two players can develop a strategy that will ensure victory, no matter how well their opponent plays, and describe said strategy.
Let $\omega$ be the circumcircle of an acute-angled triangle $ABC$. The lines tangent to $\omega$ at the points $A$ and $B$ meet at $K$. The line passing through $K$ and parallel to $BC$ intersects the side $AC$ at $S$. Prove that $BS=CS$
In a $10 \times 10$ table some cells(at least one) are marked such that in every $3 \times 3$ subtable an even number of cells are marked.
What is the minimal possible amount of marked cells?
For all positive real numbers $x, y, z$, show that $ \frac{x(2x-y)}{y(2z+x)}+\frac{y(2y-z)}{z(2x+y)}+\frac{z(2z-x)}{x(2y+z)} \geq 1$ is true.
Circumcircle of $\triangle ABC$ is $\odot O$, incentre of $\triangle ABC$ is $I$. $\angle B=60^{\circ}.\angle A<\angle C$. Bisector of outer angle $\angle A$ intersects $\odot O$ at $E$. Prove:
[b](a)[/b] $IO=AE$.
[b](b)[/b] The radius of $\odot O$ is $R$, then $2R<IO+IA+IC<(1+\sqrt3)R$.
Point $P$ is outside circle $C$ on the plane. At most how many points on $C$ are $3 \text{cm}$ from $P$?
$\text{(A)} \ 1 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ 4 \qquad \text{(E)} \ 8$
A line $l$ intersects the segment $AB$ perpendicular to $C$. Three circles are drawn successively with $AB, AC$ and $BC$ as the diameter. The largest circle intersects $l$ in $D$. The segments $DA$ and $DB$ still intersect the two smaller circles in $E$ and $F$.
a. Prove that quadrilateral $CFDE$ is a rectangle.
b. Prove that the line through $E$ and $F$ touches the circles with diameters $AC$ and $BC$ in $E$ and $F$.
[asy]
unitsize (2.5 cm);
pair A, B, C, D, E, F, O;
O = (0,0);
A = (-1,0);
B = (1,0);
C = (-0.3,0);
D = intersectionpoint(C--(C + (0,1)), Circle(O,1));
E = (C + reflect(A,D)*(C))/2;
F = (C + reflect(B,D)*(C))/2;
draw(Circle(O,1));
draw(Circle((A + C)/2, abs(A - C)/2));
draw(Circle((B + C)/2, abs(B - C)/2));
draw(A--B);
draw(interp(C,D,-0.4)--D);
draw(A--D--B);
dot("$A$", A, W);
dot("$B$", B, dir(0));
dot("$C$", C, SE);
dot("$D$", D, NW);
dot("$E$", E, SE);
dot("$F$", F, SW);
[/asy]
Circles $ \mathcal{C}_1$ and $ \mathcal{C}_2$ intersect at $ A$ and $ B$. Let $ M\in AB$. A line through $ M$ (different from $ AB$) cuts circles $ \mathcal{C}_1$ and $ \mathcal{C}_2$ at $ Z,D,E,C$ respectively such that $ D,E\in ZC$. Perpendiculars at $ B$ to the lines $ EB,ZB$ and $ AD$ respectively cut circle $ \mathcal{C}_2$ in $ F,K$ and $ N$. Prove that $ KF\equal{}NC$.
Let $C$ be a cube. By connecting the centres of the faces of $C$ with lines we form an octahedron $O$. By connecting the centers of each face of $O$ with lines we get a smaller cube $C'$. What is the ratio between the side length of $C$ and the side length of $C'$?
Let $ABC$ be a non isosceles triangle inscribed in a circle $(O)$ and $BE, CF$ are two angle bisectors intersect at $I$ with $E$ belongs to segment $AC$ and $F$ belongs to segment $AB$. Suppose that $BE, CF$ intersect $(O)$ at $M,N$ respectively. The line $d_1$ passes through $M$ and perpendicular to $BM$ intersects $(O)$ at the second point $P,$ the line $d_2$ passes through $N$ and perpendicular to $CN$ intersect $(O)$ at the second point $Q$. Denote $H, K$ are two midpoints of $MP$ and $NQ$ respectively.
1. Prove that triangles $IEF$ and $OKH$ are similar.
2. Suppose that S is the intersection of two lines $d_1$ and $d_2$. Prove that $SO$ is perpendicular to $EF$.
Amelia has $27$ unit cubes. She selects one and paints one of its faces. She then randomly glues all $27$ cubes together to form a $3 \times 3 \times 3$ cube (with all possible arrangements of the unit cubes being equally likely). Compute the probability that the resulting cube appears unpainted.
Let $a,b,c$ be the sides of a triangle, $\alpha ,\beta ,\gamma$ the corresponding angles and $r$ the inradius. Prove that $$a\cdot sin\alpha+b\cdot sin\beta+c\cdot sin\gamma\geq 9r$$
Let $ABCD$ be a parallelogram with area 160. Let diagonals $AC$ and $BD$ intersect at $E$. Point $P$ is on $\overline{AE}$ such that $EC = 4EP$. If line $DP$ intersects $AB$ at $F$, find the area of $BFPC$.
[i]Proposed by Andy Xu[/i]
Consider an $n\times n$ matrix whose entry at the intersection of the $i$-th row and the $j-$th column equals $i+ j -1$. What is the largest possible value of the product of $n$ entries of the matrix, no two of which are in the same row or column?
Let $a_0$ be a positive integer and $a_{n + 1} =\sqrt{a_n^2 + 1}$, for all $n \ge 0$.
1) Prove that for all $a_0$ the sequence contains infinitely many integers and infinitely many irrational numbers.
2) Is there an $a_0$ for which $a_{2010}$ is an integer?
Prove that if $$\frac{p}{q}=1-\frac{1}{2} + \frac{1}{3}- \frac{1}{4} + ... -\frac{1}{1334} + \frac{1}{1335}$$ where $p, q \in Z_+$ then $p$ is divisible by $2003$.
[b]1.[/b] Let $k$ be an arbitrary cardinality. Show that there exists a tournament $T_k = (V_n , E_n)$ such that for any coloring $f: E_n \to k$ of the edge set $E_n$, there are three different vertices $x_0 , x_1 , x_2 \in V_n$ such that
$x_0 x_1 , x_1 x_2 , x_2 x_0 \in E_n$
and
$\left | \{ f(x_0 x_1), f(x_1 x_2), f(x_2 x_0)\} \right |\leq 2$
(A [i]tounament[/i] is a directed graph such that for any vertices $x, y \in V_n, x \neq y$ exactly one of the relations $xy \in E_n$ holds.) ([b]C.19[/b]) [A. Hajnal]
How many positive integers $ N$ less than $ 1000$ are there such that the equation $ x^{\lfloor x\rfloor} \equal{} N$ has a solution for $ x$? (The notation $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$.)