Found problems: 85335
Let $I$ be the incentre and $O$ the circumcentre of a non-equilateral triangle $ABC$. Prove that $\angle AIO \le 90^{\circ}$ if and only if $2BC\le AB+AC$.
Two circles with radius one are drawn in the coordinate plane, one with center $(0,1)$ and the other with center $(2, y)$, for some real number y between $0$ and $1$. A third circle is drawn so as to be tangent to both of the other two circles as well as the $x$ axis. What is the smallest possible radius for this third circle?
Prove that $\sin x + \frac12 \sin 2x + \frac13 \sin 3x > 0$ for $0 < x < 180^o$.
Of the quadratic trinomials $x^2 + px + q$ where $p$; $q$ are integers and $1\leqslant p, q \leqslant 1997$, which are there more of: those having integer roots or those not having real roots?
[i]M. Evdokimov[/i]
Suppose $ A $ is a subset of $ n $-elements taken from $ 1,2,3,4,...,2009 $ such that the difference of any two numbers in $ A $ is not a prime number. Find the largest value of $ n $ and the set $ A $ with this number of elements.
The grading scale shown is used at Jones Junior High. The fifteen scores in Mr. Freeman's class were:
\[ \begin{tabular}[t]{lllllllll}89, & 72, & 54, & 97, & 77, & 92, & 85, & 74, & 75,\\ 63, & 84, & 78, & 71, & 80, & 90. & & &\\ \end{tabular} \]
In Mr. Freeman's class, what percent of the students received a grade of C?
\[ \boxed{\begin{tabular}[t]{cc}A: & 93-100\\ B: & 85-92\\ C: & 75-84\\ D: & 70-74\\ F: & 0-69\end{tabular}} \]
$ \text{(A)}\ 20\%\qquad\text{(B)}\ 25\%\qquad\text{(C)}\ 30\%\qquad\text{(D)}\ 33\frac{1}{3}\%\qquad\text{(E)}\ 40\% $
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\tfrac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive integers, and $r+s+t<1000$. Find $r+s+t$.
You have a 2 meter long string. You choose a point along the string uniformly at random and make a cut. You discard the shorter section. If you still have 0.5 meters or more of string, you repeat. You stop once you have less than 0.5 meters of string. On average, how many cuts will you make before stopping?
In one of the adjoining figures a square of side $2$ is dissected into four pieces so that $E$ and $F$ are the midpoints of opposite sides and $AG$ is perpendicular to $BF$. These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of height to base, $XY$ / $YZ$, in this rectangle is
[asy]
size(180);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=(0,1), B=(0,-1), C=(2,-1), D=(2,1), E=(1,-1), F=(1,1), G=(.8,.6);
pair X=(4,sqrt(5)), Y=(4,-sqrt(5)), Z=(4+2/sqrt(5),-sqrt(5)), W=(4+2/sqrt(5),sqrt(5)), T=(4,0), U=(4+2/sqrt(5),-4/sqrt(5)), V=(4+2/sqrt(5),1/sqrt(5));
draw(A--B--C--D--A^^B--F^^E--D^^A--G^^rightanglemark(A,G,F));
draw(X--Y--Z--W--X^^T--V--X^^Y--U);
label("A", A, NW);
label("B", B, SW);
label("C", C, SE);
label("D", D, NE);
label("E", E, S);
label("F", F, N);
label("G", G, E);
label("X", X, NW);
label("Y", Y, SW);
label("Z", Z, SE);
label("W", W, NE);
[/asy]
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 1+2\sqrt{3}\qquad\textbf{(C)}\ 2\sqrt{5}\qquad\textbf{(D)}\ \frac{8+4\sqrt{3}}{3}\qquad\textbf{(E)}\ 5 $
Which is greater: \[\frac{1^{-3}-2^{-3}}{1^{-2}-2^{-2}}-\frac{2^{-3}-3^{-3}}{2^{-2}-3^{-2}}+\frac{3^{-3}-4^{-3}}{3^{-2}-4^{-2}}-\cdots +\frac{2019^{-3}-2020^{-3}}{2019^{-2}-2020^{-2}}\]
or \[1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots +\frac{1}{5781}?\]
Each of ten boxes contains a di fferent number of pencils. No two pencils in the same box are of the same colour. Prove that one can choose one pencil from each box so that no two are of the same colour.
Find the least number $ c$ satisfyng the condition $\sum_{i=1}^n {x_i}^2\leq cn$
and all real numbers $x_1,x_2,...,x_n$ are greater than or equal to $-1$ such that $\sum_{i=1}^n {x_i}^3=0$
Let $n > 1$ be an integer and $X = \{1, 2, \cdots , n^2 \}$. If there exist $x, y$ such that $x^2\mid y$ in all subsets of $X$ with $k$ elements, find the least possible value of $k$.
Jeffrey rolls fair three six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
For what real values of $a$ the equation $\frac{2^{2x}}{2^{2x}+2^{x+1}+1}+a \frac{2^x}{2^x+1}+(a-1) = 0$ has a single root ?
Let $ M$ be a point on $ BC$ and $ N$ be a point on $ AB$ such that $ AM$ and $ CN$ are angle bisectors of the triangle $ ABC$. Given that $ \frac {\angle BNM}{\angle MNC} \equal{} \frac {\angle BMN}{\angle NMA}$, prove that the triangle $ ABC$ is isosceles.
A set of points in the plane is called rigid if each point is equidistant from the three (or more) points nearest to it.
(a) Does there exist a rigid set of $9$ points?
(b) Does there exist a rigid set of $11$ points?
Let $n>1$ be a natural number and $x_k{}$ be the residue of $n^2$ modulo $\lfloor n^2/k\rfloor+1$ for all natural $k{}$. Compute the sum \[\bigg\lfloor\frac{x_2}{1}\bigg\rfloor+\bigg\lfloor\frac{x_3}{2}\bigg\rfloor+\cdots+\left\lfloor\frac{x_n}{n-1}\right\rfloor.\]
Find all pairs of positive integers $(a, b)$ such that $a^b+b^a=a!+b^2+ab+1$.
Let $AX$ and $BZ$ be altitudes of the triangle $ABC$. Let $AY$ and $BT$ be its angle bisectors. It is given that angles $XAY$ and $ZBT$ are equal. Does this necessarily imply that $ABC$ is isosceles?
[i]The Jury[/i]
A $4\times4$ grid has its 16 cells colored arbitrarily in three colors. A [i]swap[/i] is an exchange between the colors of two cells. Prove or disprove that it always takes at most three swaps to produce a line of symmetry, regardless of the grid's initial coloring.
[i]Proposed by Matthew Babbitt[/i]
Suppose that $f:\{1, 2,\ldots ,1600\}\rightarrow\{1, 2,\ldots ,1600\}$ satisfies $f(1)=1$ and
\[f^{2005}(x)=x\quad\text{for}\ x=1,2,\ldots ,1600. \]
$(a)$ Prove that $f$ has a fixed point different from $1$.
$(b)$ Find all $n>1600$ such that any $f:\{1,\ldots ,n\}\rightarrow\{1,\ldots ,n\}$ satisfying the above condition has at least two fixed points.
1. Baron Munchhausen was told that some polynomial $P(x)=a_{n} x^{n}+\ldots+a_{1} x+a_{0}$ is such that $P(x)+P(-x)$ has exactly 45 distinct real roots. Baron doesn't know the value of $n$. Nevertheless he claims that he can determine one of the coefficients $a_{n}, \ldots, a_{1}, a_{0}$ (indicating its position and value). Isn't Baron mistaken?
Boris Frenkin
The number $123$ is shown on the screen of a computer. Each minute the computer adds $102$ to the number on the screen. The computer expert Misha may change the order of digits in the number on the screen whenever he wishes. Can he ensure that no four-digit number ever appears on the screen?
(F.L. Nazarov, Leningrad)
Let $N$ and $K$ be positive integers satisfying $1 \leq K \leq N$. A deck of $N$ different playing cards is shuffled by repeating the operation of reversing the order of $K$ topmost cards and moving these to the bottom of the deck. Prove that the deck will be back in its initial order after a number of operations not greater than $(2N/K)^2$.