Found problems: 85335
2019 Romania National Olympiad, 1
Consider $A$, the set of natural numbers with exactly $2019$ natural divisors , and for each $n \in A$, denote $$S_n=\frac{1}{d_1+\sqrt{n}}+\frac{1}{d_2+\sqrt{n}}+...+\frac{1}{d_{2019}+\sqrt{n}}$$
where $d_1,d_2, .., d_{2019}$ are the natural divisors of $n$.
Determine the maximum value of $S_n$ when $n$ goes through the set $ A$.
1970 IMO Longlists, 41
Let a cube of side $1$ be given. Prove that there exists a point $A$ on the surface $S$ of the cube such that every point of $S$ can be joined to $A$ by a path on $S$ of length not exceeding $2$. Also prove that there is a point of $S$ that cannot be joined with $A$ by a path on $S$ of length less than $2$.
1999 Greece National Olympiad, 4
On a circle are given $n\ge 3$ points. At most, how many parts can the segments with the endpoints at these $n$ points divide the interior of the circle into?
2000 Mongolian Mathematical Olympiad, Problem 1
Let $\operatorname{rad}(k)$ denote the product of prime divisors of a natural number $k$ (define $\operatorname{rad}(1)=1$). A sequence $(a_n)$ is defined by setting $a_1$ arbitrarily, and $a_{n+1}=a_n+\operatorname{rad}(a_n)$ for $n\ge1$. Prove that the sequence $(a_n)$ contains arithmetic progressions of arbitrary length.
1998 All-Russian Olympiad Regional Round, 11.8
A sequence $a_1,a_2,\cdots$ of positive integers contains each positive integer exactly once. Moreover for every pair of distinct positive integer $m$ and $n$, $\frac{1}{1998} < \frac{|a_n- a_m|}{|n-m|} < 1998$, show that $|a_n - n | <2000000$ for all $n$.
2024 Turkey Team Selection Test, 1
In triangle $ABC$, the incenter is $I$ and the circumcenter is $O$. Let $AI$ intersects $(ABC)$ second time at $P$ .
The line passes through $I$ and perpendicular to $AI$ intersects $BC$ at $X$. The feet of the perpendicular from $X$ to $IO$ is $Y$. Prove that $A,P,X,Y$ cyclic.
1997 All-Russian Olympiad, 4
In an $m\times n$ rectangular grid, where m and n are odd integers, $1\times 2$ dominoes are initially placed so as to exactly cover all but one of the $1\times 1$ squares at one corner of the grid.
It is permitted to slide a domino towards the empty square, thus exposing another square.
Show that by a sequence of such moves, we can move the empty square to any corner of the rectangle.
[i]A. Shapovalov[/i]
1971 All Soviet Union Mathematical Olympiad, 144
Prove that for every natural $n$ there exists a number, containing only digits "$1$" and "$2$" in its decimal notation, that is divisible by $2^n$ ( $n$-th power of two ).
2024 Thailand TST, 1
For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.
2012 Online Math Open Problems, 30
Let $P(x)$ denote the polynomial
\[3\sum_{k=0}^{9}x^k + 2\sum_{k=10}^{1209}x^k + \sum_{k=1210}^{146409}x^k.\]Find the smallest positive integer $n$ for which there exist polynomials $f,g$ with integer coefficients satisfying $x^n - 1 = (x^{16} + 1)P(x) f(x) + 11\cdot g(x)$.
[i]Victor Wang.[/i]
1986 AMC 8, 15
Sale prices at the Ajax Outlet Store are $ 50 \%$ below original prices. On Saturdays an additional discount of $ 20 \%$ off the sale price is given. What is the Saturday price of a coat whose original price is $ \$180$?
\[ \textbf{(A)}\ \$54 \qquad
\textbf{(B)}\ \$72 \qquad
\textbf{(C)}\ \$90 \qquad
\textbf{(D)}\ \$108 \qquad
\textbf{(E)}\ \$110
\]
2017 Bundeswettbewerb Mathematik, 3
Given is a triangle with side lengths $a,b$ and $c$, incenter $I$ and centroid $S$.
Prove: If $a+b=3c$, then $S \neq I$ and line $SI$ is perpendicular to one of the sides of the triangle.
2023 Mongolian Mathematical Olympiad, 1
Find all functions $f : \mathbb{R} \to \mathbb{R}$ and $h : \mathbb{R}^2 \to \mathbb{R}$ such that \[f(x+y-z)^2=f(xy)+h(x+y+z, xy+yz+zx)\] for all real numbers $x,y,z$.
2002 Kazakhstan National Olympiad, 5
On the plane is given the acute triangle $ ABC $. Let $ A_1 $ and $ B_1 $ be the feet of the altitudes of $ A $ and $ B $ drawn from those vertices, respectively. Tangents at points $ A_1 $ and $ B_1 $ drawn to the circumscribed circle of the triangle $ CA_1B_1 $ intersect at $ M $. Prove that the circles circumscribed around the triangles $ AMB_1 $, $ BMA_1 $ and $ CA_1B_1 $ have a common point.
2001 Romania National Olympiad, 3
We consider the points $A,B,C,D$, not in the same plane, such that $AB\perp CD$ and $AB^2+CD^2=AD^2+BC^2$.
a) Prove that $AC\perp BD$.
b) Prove that if $CD<BC<BD$, then the angle between the planes $(ABC)$ and $(ADC)$ is greater than $60^{\circ}$.
2019 Auckland Mathematical Olympiad, 2
There are $2019$ segments $[a_1, b_1]$, $...$, $[a_{2019}, b_{2019}]$ on the line. It is known that any two of them intersect. Prove that they all have a point in common.
2017 CMIMC Team, 5
We have four registers, $R_1,R_2,R_3,R_4$, such that $R_i$ initially contains the number $i$ for $1\le i\le4$. We are allowed two operations:
[list]
[*] Simultaneously swap the contents of $R_1$ and $R_3$ as well as $R_2$ and $R_4$.
[*] Simultaneously transfer the contents of $R_2$ to $R_3$, the contents of $R_3$ to $R_4$, and the contents of $R_4$ to $R_2$. (For example if we do this once then $(R_1,R_2,R_3,R_4)=(1,4,2,3)$.)
[/list]
Using these two operations as many times as desired and in whatever order, what is the total number of possible outcomes?
2015 Rioplatense Mathematical Olympiad, Level 3, 2
Let $a , b , c$ positive integers, coprime. For each whole number $n \ge 1$, we denote by $s ( n )$ the number of elements in the set $\{ a , b , c \}$ that divide $n$. We consider $k_1< k_2< k_3<...$ .the sequence of all positive integers that are divisible by some element of $\{ a , b , c \}$. Finally we define the characteristic sequence of $( a , b , c )$ like the succession $ s ( k_1) , s ( k_2) , s ( k_3) , .... $ .
Prove that if the characteristic sequences of $( a , b , c )$ and $( a', b', c')$ are equal, then $a = a', b = b'$ and $c=c'$
2007 AMC 10, 10
Two points $ B$ and $ C$ are in a plane. Let $ S$ be the set of all points $ A$ in the plane for which $ \triangle ABC$ has area $ 1$. Which of the following describes $ S$?
$ \textbf{(A)}\ \text{two parallel lines}\qquad
\textbf{(B)}\ \text{a parabola}\qquad
\textbf{(C)}\ \text{a circle}\qquad
\textbf{(D)}\ \text{a line segment}\qquad
\textbf{(E)}\ \text{two points}$
Gheorghe Țițeica 2025, P3
Two regular pentagons $ABCDE$ and $AEKPL$ are given in space, such that $\angle DAK = 60^{\circ}$. Let $M$, $N$ and $S$ be the midpoints of $AE$, $CD$ and $EK$. Prove that:
[list=a]
[*] $\triangle NMS$ is a right triangle;
[*] planes $(ACK)$ and $(BAL)$ are perpendicular.
[/list]
[i]Ukraine Olympiad[/i]
1984 IMO Shortlist, 13
Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$
2021 IMO Shortlist, G5
Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$.
Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.
2014 BMT Spring, 1
For the team, power, and tournament rounds, BMT divided up the teams into $14$ rooms. You sign up to proctor all $3$ rounds, but you cannot proctor in the same room more than once.
How many ways can you be assigned for rooms for the $3$ rounds?
2022-IMOC, A5
Find all functions $f:\mathbb R\to \mathbb R$ such that \begin{align*} \left (x \left (f(x)-\dfrac{f(y)+f(z)}{2} \right) +y \left (f(y)-\dfrac{f(z)+f(x)}{2} \right ) +z\left (f(z)- \dfrac{f(x)+f(y)}{2} \right) \right )f(x+y+z)= \\ f(x^3)+f(y^3)+f(z^3)-3f(xyz) \end{align*} for all $x,y,z\in \mathbb R.$
1988 AIME Problems, 5
Let $m/n$, in lowest terms, be the probability that a randomly chosen positive divisor of $10^{99}$ is an integer multiple of $10^{88}$. Find $m + n$.