Found problems: 85335
1999 All-Russian Olympiad Regional Round, 9.7
Prove that every natural number is the difference of two natural numbers that have the same number of prime factors. (Each prime divisor is counted once, for example, the number $12$ has two prime factors: $2$ and $3$.)
2013 Math Prize for Girls Olympiad, 4
We are given a finite set of segments of the same line. Prove that we can color each segment red or blue such that, for each point $p$ on the line, the number of red segments containing $p$ differs from the number of blue segments containing $p$ by at most $1$.
1986 Canada National Olympiad, 5
Let $u_1$, $u_2$, $u_3$, $\dots$ be a sequence of integers satisfying the recurrence relation $u_{n + 2} = u_{n + 1}^2 - u_n$. Suppose $u_1 = 39$ and $u_2 = 45$. Prove that 1986 divides infinitely many terms of the sequence.
1990 AMC 12/AHSME, 20
$ABCD$ is a quadrilateral with right angles at $A$ and $C$. Points $E$ and $F$ are on $AC$, and $DE$ and $BF$ are perpendicular to $AC$. If $AE=3$, $DE=5$, and $CE=7$, then $BF=$
[asy]
draw((0,0)--(10,0)--(3,-5)--(0,0)--(6.5,3)--(10,0));
draw((6.5,0)--(6.5,3));
draw((3,0)--(3,-5));
dot((0,0));
dot((10,0));
dot((3,0));
dot((3,-5));
dot((6.5,0));
dot((6.5,3));
label("A", (0,0), W);
label("B", (6.5,3), N);
label("C", (10,0), E);
label("D", (3,-5), S);
label("E", (3,0), N);
label("F", (6.5,0), S);[/asy]
$\text{(A)} \ 3.6 \qquad \text{(B)} \ 4 \qquad \text{(C)} \ 4.2 \qquad \text{(D)} \ 4.5 \qquad \text{(E)} \ 5$
1998 All-Russian Olympiad Regional Round, 9.1
The lengths of the sides of a certain triangle and the diameter of the inscribed part circles are four consecutive terms of arithmetic progression. Find all such triangles.
2024 IFYM, Sozopol, 5
Find all functions \(f:\mathbb{R}^{+} \to \mathbb{R}^{+}\) such that
\[
f(x) > x \ \ \text{and} \ \ f(x-y+xy+f(y)) = f(x+y) + xf(y)
\]
for arbitrary positive real numbers \(x\) and \(y\).
2014 Taiwan TST Round 3, 1
In convex hexagon $ABCDEF$, $AB \parallel DE$, $BC \parallel EF$, $CD \parallel FA$, and \[ AB+DE = BC+EF = CD+FA. \] The midpoints of sides $AB$, $BC$, $DE$, $EF$ are $A_1$, $B_1$, $D_1$, $E_1$, and segments $A_1D_1$ and $B_1E_1$ meet at $O$. Prove that $\angle D_1OE_1 = \frac12 \angle DEF$.
2024 Nepal TST, P4
Vlad draws 100 rays in the Euclidean plane. David then draws a line $\ell$ and pays Vlad one pound for each ray that $\ell$ intersects. Naturally, David wants to pay as little as possible. What is the largest amount of money that Vlad can get from David?
[i]Proposed by Vlad Spătaru[/i]
1994 China Team Selection Test, 2
Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.
2005 Harvard-MIT Mathematics Tournament, 2
A plane curve is parameterized by $x(t)=\displaystyle\int_{t}^{\infty} \dfrac {\cos u}{u} \, \mathrm{d}u $ and $ y(t) = \displaystyle\int_{t}^{\infty} \dfrac {\sin u}{u} \, \mathrm{d}u $ for $ 1 \le t \le 2 $. What is the length of the curve?
2012 Vietnam National Olympiad, 2
Consider two odd natural numbers $a$ and $b$ where $a$ is a divisor of $b^2+2$ and $b$ is a divisor of $a^2+2.$ Prove that $a$ and $b$ are the terms of the series of natural numbers $\langle v_n\rangle$ defined by
\[v_1 = v_2 = 1; v_n = 4v_ {n-1}-v_{n-2} \ \ \text{for} \ n\geq 3.\]
2003 Croatia Team Selection Test, 2
Let $B$ be a point on a circle $k_1, A \ne B$ be a point on the tangent to the circle at $B$, and $C$ a point not lying on $k_1$ for which the segment $AC$ meets $k_1$ at two distinct points. Circle $k_2$ is tangent to line $AC$ at $C$ and to $k_1$ at point $D$, and does not lie in the same half-plane as $B$. Prove that the circumcenter of triangle $BCD$ lies on the circumcircle of $\vartriangle ABC$
2010 India IMO Training Camp, 6
Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to
\[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\]
2013 Dutch IMO TST, 2
Let $P$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$.Let $X,Y,Z$ be points on the interior of $AB,BC,CD$ respectively such that $\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2$. Suppose that $XY$ is tangent to the circumcircle of $\triangle CYZ$ and that $Y Z$ is tangent to the circumcircle of $\triangle BXY$.Show that $\angle APD=\angle XYZ$.
2020 ITAMO, 3
Let $a_1, a_2, \dots, a_{2020}$ and $b_1, b_2, \dots, b_{2020}$ be real numbers(not necessarily distinct). Suppose that the set of positive integers $n$ for which the following equation:
$|a_1|x-b_1|+a_2|x-b_2|+\dots+a_{2020}|x-b_{2020}||=n$ (1) has exactly two real solutions, is a finite set. Prove that the set of positive integers $n$ for which the equation (1) has at least one real solution, is also a finite set.
1993 Putnam, B6
Let $S$ be a set of three, not necessarily distinct, positive integers. Show that one can transform $S$ into a set containing $0$ by a finite number of applications of the following rule: Select two of the integers $x$ and $y$, where $x\leq y$ and replace them with $2x$ and $y-x.$
2001 National Olympiad First Round, 15
How many different solutions does the congruence $x^3+3x^2+x+3 \equiv 0 \pmod{25}$ have?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ 6
$
1999 Romania National Olympiad, 2
For a finite group $G$ we denote by $n(G)$ the number of elements of the group and by $s(G)$ the number of subgroups of it.
Decide whether the following statements are true or false.
a) For every $a>0$ the is a finite group $G$ with $\frac{n(G)}{s(G)}<a.$
b) For every $a>0$ the is a finite group $G$ with $\frac{n(G)}{s(G)}>a.$
2008 China Team Selection Test, 3
Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers $ a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots,$ such that inifinite sequence of positive integers $ a_{1},\frac {a_{1} \plus{} a_{2}}{2},a_{2},\frac {a_{2} \plus{} a_{3}}{2},a_{3},\frac {a_{3} \plus{} a_{4}}{2},\cdots$ has the same color.
2000 India National Olympiad, 6
For any natural numbers $n$, ( $n \geq 3$), let $f(n)$ denote the number of congruent integer-sided triangles with perimeter $n$. Show that
(i) $f(1999) > f (1996)$;
(ii) $f(2000) = f(1997)$.
1980 All Soviet Union Mathematical Olympiad, 290
There are several settlements on the bank of the Big Round Lake. Some of them are connected with the regular direct ship lines. Two settlements are connected if and only if two next (counterclockwise) to each ones are not connected. Prove that you can move from the arbitrary settlement to another arbitrary settlement, having used not more than three ships.
2019 Mathematical Talent Reward Programme, SAQ: P 5
Let a fi xed natural number m be given. Call a positive integer n to be an MTRP-number iff
[list]
[*] $n \equiv 1\ (mod\ m)$
[*] Sum of digits in decimal representation of $n^2$ is greater than equal to sum of digits in decimal
representation of $n$
[/list]
How many MTRP-numbers are there ?
2012 AIME Problems, 6
The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < n$. Find $n$.
2000 IMO Shortlist, 4
Find all triplets of positive integers $ (a,m,n)$ such that $ a^m \plus{} 1 \mid (a \plus{} 1)^n$.
1955 AMC 12/AHSME, 36
A cylindrical oil tank, lying horizontally, has an interior length of $ 10$ feet and an interior diameter of $ 6$ feet. If the rectangular surface of the oil has an area of $ 40$ square feet, the depth of the oil is:
$ \textbf{(A)}\ \sqrt{5} \qquad
\textbf{(B)}\ 2\sqrt{5} \qquad
\textbf{(C)}\ 3\minus{}\sqrt{5} \qquad
\textbf{(D)}\ 3\plus{}\sqrt{5} \\
\textbf{(E)}\ \text{either }3\minus{}\sqrt{5}\text{ or }3\plus{}\sqrt{5}$