Found problems: 85335
2009 BMO TST, 3
For the give functions in $\mathbb{N}$:
[b](a)[/b] Euler's $\phi$ function ($\phi(n)$- the number of natural numbers smaller than $n$ and coprime with $n$);
[b](b)[/b] the $\sigma$ function such that the $\sigma(n)$ is the sum of natural divisors of $n$.
solve the equation $\phi(\sigma(2^x))=2^x$.
2019 MIG, 23
How many ordered pairs of integers $(x,y)$ satisfy $xy - 6y - 4x + 20 = 0$?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }6$
2023 IFYM, Sozopol, 7
The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at points $A_1$, $B_1$, and $C_1$. The line through the midpoints of segments $AB_1$ and $AC_1$ intersects the tangent at $A$ to the circumcircle of triangle $ABC$ at point $A_2$. Points $B_2$ and $C_2$ are defined similarly. Prove that points $A_2$, $B_2$, and $C_2$ lie on a line.
2010 Stanford Mathematics Tournament, 9
Suppose $xy-5x+2y=30$, where $x$ and $y$ are positive integers. Find the sum of all possible values of $x$
1967 Swedish Mathematical Competition, 5
$a_1, a_2, a_3, ...$ are positive reals such that $a_n^2 \ge a_1 + a_2 +... + a_{n-1}$.
Show that for some $C > 0$ we have $a_n \ge C n$ for all $n$.
2010 Romania National Olympiad, 3
Each of the small squares of a $50\times 50$ table is coloured in red or blue. Initially all squares are red. A [i]step[/i] means changing the colour of all squares on a row or on a column.
a) Prove that there exists no sequence of steps, such that at the end there are exactly $2011$ blue squares.
b) Describe a sequence of steps, such that at the end exactly $2010$ squares are blue.
[i]Adriana & Lucian Dragomir[/i]
2010 Saint Petersburg Mathematical Olympiad, 4
Natural number $N$ is given. Let $p_N$ - biggest prime, that $ \leq N$. On every move we replace $N$ by $N-p_N$. We repeat this until we get $0$ or $1$. If we get $1$ then $N$ is called as good, else is bad. For example, $95$ is good because we get $95 \to 6 \to 1$.
Prove that among numbers from $1$ to $1000000$ there are between one quarter and half good numbers
1980 Polish MO Finals, 4
Show that for every polynomial $W$ in three variables there exist polynomials $U$ and $V$ such that:
$$W(x,y,z) = U(x,y,z)+V(x,y,z),$$
$$U(x,y,z) = U(y,x,z),$$
$$V(x,y,z) = -V(x,z,y).$$
2018 Rio de Janeiro Mathematical Olympiad, 1
A natural number is a [i]factorion[/i] if it is the sum of the factorials of each of its decimal digits. For example, $145$ is a factorion because $145 = 1! + 4! + 5!$.
Find every 3-digit number which is a factorion.
2022 Novosibirsk Oral Olympiad in Geometry, 6
Anton has an isosceles right triangle, which he wants to cut into $9$ triangular parts in the way shown in the picture. What is the largest number of the resulting $9$ parts that can be equilateral triangles?
A more formal description of partitioning. Let triangle $ABC$ be given. We choose two points on its sides so that they go in the order $AC_1C_2BA_1A_2CB_1B_2$, and no two coincide. In addition, the segments $C_1A_2$, $A_1B_2$ and $B_1C_2$ must intersect at one point. Then the partition is given by segments $C_1A_2$, $A_1B_2$, $B_1C_2$, $A_1C_2$, $B_1A_2$ and $C_1B_2$.
[img]https://cdn.artofproblemsolving.com/attachments/0/5/5dd914b987983216342e23460954d46755d351.png[/img]
2010 Mediterranean Mathematics Olympiad, 3
Let $A'\in(BC),$ $B'\in(CA),C'\in(AB)$ be the points of tangency of the excribed circles of triangle $\triangle ABC$ with the sides of $\triangle ABC.$ Let $R'$ be the circumradius of triangle $\triangle A'B'C'.$ Show that \[
R'=\frac{1}{2r}\sqrt{2R\left(2R-h_{a}\right)\left(2R-h_{b}\right)\left(2R-h_{c}\right)}\]
where as usual, $R$ is the circumradius of $\triangle ABC,$ r is the inradius of $\triangle ABC,$ and $h_{a},h_{b},h_{c}$ are the lengths of altitudes of $\triangle ABC.$
1991 National High School Mathematics League, 3
Let $a$ be a positive integer, $a<100$, and $a^3+23$ is a multiple of $24$. Then, the number of such $a$ is
$\text{(A)}4\qquad\text{(B)}5\qquad\text{(C)}9\qquad\text{(D)}10$
2006 Stanford Mathematics Tournament, 10
Find the smallest positive $m$ for which there are at least 11 even and 11 odd positive integers $n$ so that $\tfrac{n^3+m}{n+2}$ is an integer.
2006 Estonia National Olympiad, 5
A pawn is placed on a square of a $ n \times n$ board. There are two types of legal
moves: (a) the pawn can be moved to a neighbouring square, which shares a common side with the current square; or (b) the pawn can be moved to a neighbouring square, which shares a common vertex, but not a common side with the current square. Any two consecutive moves must be of different type. Find all integers $ n \ge 2$, for which it is possible to choose an initial square and a sequence of moves such that the pawn visits each square exactly once (it is not required that the pawn returns to the initial square).
1960 IMO Shortlist, 6
Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let $V_1$ be the volume of the cone and $V_2$ be the volume of the cylinder.
a) Prove that $V_1 \neq V_2$;
b) Find the smallest number $k$ for which $V_1=kV_2$; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.
2007 Princeton University Math Competition, 1
Find the last three digits of
\[2008^{2007^{\cdot^{\cdot^{\cdot ^{2^1}}}}}.\]
2023 Romania National Olympiad, 3
We consider triangle $ABC$ and variables points $M$ on the half-line $BC$, $N$ on the half-line $CA$, and $P$ on the half-line $AB$, each start simultaneously from $B,C$ and respectively $A$, moving with constant speeds $ v_1, v_2, v_3 > 0 $, where $v_1$, $v_2$, and $v_3$ are expressed in the same unit of measure.
a) Given that there exist three distinct moments in which triangle $MNP$ is equilateral, prove that triangle $ABC$ is equilateral and that $v_1 = v_2 = v_3$.
b) Prove that if $v_1 = v_2 = v_3$ and there exists a moment in which triangle $MNP$ is equilateral, then triangle $ABC$ is also equilateral.
2016 CMIMC, 10
Let $f:\mathbb{N}\mapsto\mathbb{R}$ be the function \[f(n)=\sum_{k=1}^\infty\dfrac{1}{\operatorname{lcm}(k,n)^2}.\] It is well-known that $f(1)=\tfrac{\pi^2}6$. What is the smallest positive integer $m$ such that $m\cdot f(10)$ is the square of a rational multiple of $\pi$?
2011 Oral Moscow Geometry Olympiad, 3
A non-isosceles trapezoid $ABCD$ ($AB // CD$) is given. An arbitrary circle passing through points $A$ and $B$ intersects the sides of the trapezoid at points $P$ and $Q$, and the intersect the diagonals at points $M$ and $N$. Prove that the lines $PQ, MN$ and $CD$ are concurrent.
1980 VTRMC, 8
Let $z=x+iy$ be a complex number with $x$ and $y$ rational and with $|z| = 1.$
(a) Find two such complex numbers.
(b) Show that $|z^{2n}-1|=2|\sin n\theta|,$ where $z=e^{i\theta}.$
(c) Show that $|z^2n -1|$ is rational for every $n.$
2012 India IMO Training Camp, 2
Let $S$ be a nonempty set of primes satisfying the property that for each proper subset $P$ of $S$, all the prime factors of the number $\left(\prod_{p\in P}p\right)-1$ are also in $S$. Determine all possible such sets $S$.
2000 Moldova Team Selection Test, 4
Let $S{}$ be the set of nonnegative integers, which cointain only digits $0$ and $1$ in base $4$ numeral system.
a) Show that if $x\in S, y\in S, x\neq y,$ then $\frac{x+y}{2}\notin S$.
b) Let $T$ be a set of nonnegative integers such that $S\subset T, T\neq S$. Show that there exist $x\in T, y\in T, x\neq y,$ such that $\frac{x+y}{2} \in T$.
2022 Abelkonkurransen Finale, 2a
A triangle $ABC$ with circumcircle $\omega$ satisfies $|AB| > |AC|$. Points $X$ and $Y$ on $\omega$ are different from $A$, such that the line $AX$ passes through the midpoint of $BC$, $AY$ is perpendicular to $BC$, and $XY$ is parallel to $BC$. Find $\angle BAC$.
1998 ITAMO, 4
Let $ABCD$ be a trapezoid with the longer base $AB$ such that its diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of the triangle $ABC$ and $E$ be the intersection of the lines $OB$ and $CD$. Prove that $BC^2 = CD \cdot CE$.
2011 Abels Math Contest (Norwegian MO), 1
Let $n$ be the number that is produced by concatenating the numbers $1, 2,... , 4022$,
that is, $n = 1234567891011...40214022$.
a. Show that $n$ is divisible by $3$.
b. Let $a_1 = n^{2011}$, and let $a_i$ be the sum of the digits of $a_{i-1}$ for $i > 1$. Find $a_4$