Found problems: 85335
Let $ p $ be prime. Denote by $ N (p) $ the number of integers $ x $ for which $ 1 \leq x \leq p $ and
$$
x ^ {x} \equiv 1 \quad (\bmod p)
$$Prove that there exist numbers $ c <1/2 $ and $ p_ {0}> 0 $ such that
$$
N (p) \leq p ^ {c}
$$if $ p \ge p_ {0} $.
All positive integers whose binary representations (excluding leading zeroes) have at least as many $1$’s as $0$’s are put in increasing order. Compute the number of digits in the binary representation of the $200$th number.
Let $B$ and $D$ be points on segments $[AE]$ and $[AF]$ respectively. Excircles of triangles $ABF$ and $ADE$ touching sides $BF$ and $DE$ is the same, and its center is $I$. $BF$ and $DE$ intersects at $C$. Let $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3, Q_4$ be the circumcenters of triangles $IAB, IBC, ICD, IDA, IAE, IEC, ICF, IFA$ respectively.
[b]a) [/b] Show that points $P_1, P_2, P_3, P_4$ concylic and points $Q_1, Q_2, Q_3, Q_4$ concylic.
[b]b) [/b] Denote centers of theese circles as $O_1$ and $O_2$. Prove that $O_1, O_2$ and $I$ are collinear.
Does there exist a function $f: \mathbb R \to \mathbb R $ satisfying the following conditions:
(i) for each real $y$ there is a real $x$ such that $f(x)=y$ , and
(ii) $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ?
[i]Proposed by Igor I. Voronovich, Belarus[/i]
A circle $K_1$ of radius $r_1 = 1\slash 2$ is inscribed in a semi-circle $H$ with diameter $AB$ and radius $1.$ A sequence of different circles $K_2, K_3, \ldots$ with radii $r_2, r_3, \ldots$ respectively are drawn so that for each $n\geq 1$, the circle $K_{n+1}$ is tangent to $H$, $K_n$ and $AB.$ Prove that $a_n = 1\slash r_n$ is an integer for each $n$, and that it is a perfect square for $n$ even and twice a perfect square for $n$ odd.
Two distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ such that each pair of points $A$ and $B$ has the same probability of being chosen. The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers. Find $m+n$.
[i]Ray Li.[/i]
Let \( 133\ldots 33 \) be a number with \( k \geq 2 \) digits, which we assume is prime. Prove that \( k(k + 2) \) is a multiple of 24. (For example, 133...33 is a prime number when \( k = 16\)
The circles $k_1$ and $k_2$ intersect at points $A$ and $B$, and $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ at the points $K ,O$ and $k_2$ at the points $L ,M$ so that $L$ lies between $K$ and $O$. The point $P$ is the projection of $L$ on the line $AB$. Prove that $KP$ is parallel to the median of triangle $ABM$ drawn from the vertex $M$.
Consider a circle $O_1$ with radius $R$ and a point $A$ outside the circle. It is known that $\angle BAC=60^\circ$, where $AB$ and $AC$ are tangent to $O_1$. We construct infinitely many circles $O_i$ $(i=1,2,\dots\>)$ such that for $i>1$, $O_i$ is tangent to $O_{i-1}$ and $O_{i+1}$, that they share the same tangent lines $AB$ and $AC$ with respect to $A$, and that none of the $O_i$ are larger than $O_1$. Find the total area of these circles.
I know this problem was easy, but it still appeared in the TST, and so I posted it. It was kind of a disappointment for me.
In a triangle with sides $a, b, c$ the side $a$ is the arithmetic mean of $b$ and $c$. Prove that:
a) $0^o \le A \le 60^o$.
b) The height relative to side $a$ is three times the inradius $r$.
c) The distance from the circumcenter to side $a$ is $R - r$, where $R$ is the circumradius.
Let positive integers $m,n$ satisfy $n=2^m-1$. $P_n =\{1,2,\cdots ,n\}$ is a set that contains $n$ points on an axis. A grasshopper on the axis can leap from one point to another adjacent point. Find the maximal value of $m$ satisfying following conditions:
(a) $x, y$ are two arbitrary points in $P_n$;
(b) starting at point $x$, the grasshopper leaps $2012$ times and finishes at point $y$; (the grasshopper is allowed to travel $x$ and $y$ more than once)
(c) there are even number ways for the grasshopper to do (b).
Prove that for all positive integers $n$, \[\lfloor \sqrt{n}+\sqrt{n+1}+\sqrt{n+2}\rfloor =\lfloor \sqrt{9n+8}\rfloor.\]
Let $k$ be a positive integer. Find all functions $f:\mathbb{N}\to \mathbb{N}$ satisfying the following two conditions:\\
• For infinitely many prime numbers $p$ there exists a positve integer $c$ such that $f(c)=p^k$.\\
• For all positive integers $m$ and $n$, $f(m)+f(n)$ divides $f(m+n)$.
Suppose that in a certain society, each pair of persons can be classified as either [i]amicable [/i]or [i]hostile[/i]. We shall say that each member of an amicable pair is a [i]friend[/i] of the other, and each member of a hostile pair is a [i]foe[/i] of the other. Suppose that the society has $\, n \,$ persons and $\, q \,$ amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include $\, q(1 - 4q/n^2) \,$ or fewer amicable pairs.
The diagram shows two intersecting line segments that form some of the sides of two squares with side lengths $3$ and $6$. Two line segments join vertices of these squares. Find the area of the region enclosed by the squares and segments.
Find all polynomials $P$ with integer coefficients which satisfy the property that, for any relatively prime integers $a$ and $b$, the sequence $\{P (an + b) \}_{n \ge 1}$ contains an infinite number of terms, any two of which are relatively prime.
Two real number sequences are guiven, one arithmetic $\left(a_n\right)_{n\in \mathbb {N}}$ and another geometric sequence $\left(g_n\right)_{n\in \mathbb {N}}$ none of them constant. Those sequences verifies $a_1=g_1\neq 0$, $a_2=g_2$ and $a_{10}=g_3$. Find with proof that, for every positive integer $p$, there is a positive integer $m$, such that $g_p=a_m$.
One hundred balls labelled $1$ to $100$ are to be put into two identical boxes so that each box contains at least one ball and the greatest common divisor of the product of the labels of all the balls in one box and the product of the labels of all the balls in the other box is $1$. Determine the number of ways that this can be done.
Let $A=\{1,2,\ldots,n\}$. For a permutation $P=(P(1), P(2), \ldots, P(n))$ of the elements of $A$, let $P(1)$ denote the first element of $P$. Find the number of all such permutations $P$ so that for all $i,j \in A$:
(a) if $i < j<P(1)$, then $j$ appears before $i$ in $P$; and
(b) if $P(1)<i<j$, then $i$ appears before $j$ in $P$.
How many zeros does $101^{100} - 1$ end with?
Let $\mathbf{Z}$ denote the set of all integers. Find all real numbers $c > 0$ such that there exists a labeling of the lattice points $ ( x, y ) \in \mathbf{Z}^2$ with positive integers for which:
[list]
[*] only finitely many distinct labels occur, and
[*] for each label $i$, the distance between any two points labeled $i$ is at least $c^i$.
[/list]
[i]Proposed by Ricky Liu[/i]
Let $n$ points be given on a circle, and let $nk + 1$ chords between these points be drawn, where $2k+1 < n$. Show that it is possible to select $k+1$ of the chords so that no two of them intersect.
Let $n$ denote the product of the first $2013$ primes. Find the sum of all primes $p$ with $20 \le p \le 150$ such that
(i) $\frac{p+1}{2}$ is even but is not a power of $2$, and
(ii) there exist pairwise distinct positive integers $a,b,c$ for which \[ a^n(a-b)(a-c) + b^n(b-c)(b-a) + c^n(c-a)(c-b) \] is divisible by $p$ but not $p^2$.
[i]Proposed by Evan Chen[/i]
How many distinct positive integers can be expressed in the form $ABCD - DCBA$, where $ABCD$ and $DCBA$ are 4-digit positive integers? (Here $A$, $B$, $C$ and $D$ are digits, possibly equal.)
Clarification: $A$ and $D$ can't be zero (because otherwise $ABCD$ or $DCBA$ wouldn't be a true 4-digit integer).
Suppose we have a $n$-gon. Some $n-3$ diagonals are coloured black and some other $n-3$ diagonals are coloured red (a side is not a diagonal), so that no two diagonals of the same colour can intersect strictly inside the polygon, although they can share a vertex. Find the maximum number of intersection points between diagonals coloured differently strictly inside the polygon, in terms of $n$.
[i]Proposed by Alexander Ivanov, Bulgaria[/i]