Found problems: 85335
PEN B Problems, 3
Show that for each odd prime $p$, there is an integer $g$ such that $1<g<p$ and $g$ is a primitive root modulo $p^n$ for every positive integer $n$.
2002 Moldova National Olympiad, 4
The circles $ C_1$ and $ C_2$ with centers $ O_1$ and $ O_2$ respectively are externally tangent. Their common tangent not intersecting the segment $ O_1O_2$ touches $ C_1$ at $ A$ and $ C_2$ at $ B$. Let $ C$ be the reflection of $ A$ in $ O_1O_2$ and $ P$ be the intersection of $ AC$ and $ O_1O_2$. Line $ BP$ meets $ C_2$ again at $ L$. Prove that line $ CL$ is tangent to the circle $ C_2$.
1987 Austrian-Polish Competition, 8
A circle of perimeter $1$ has been dissected into four equal arcs $B_1, B_2, B_3, B_4$. A closed smooth non-selfintersecting curve $C$ has been composed of translates of these arcs (each $B_j$ possibly occurring several times). Prove that the length of $C$ is an integer.
2004 Nordic, 2
Show that there exist strictly increasing infinite arithmetic sequence of integers which has no numbers in common with the Fibonacci sequence.
2013 Mid-Michigan MO, 7-9
[b]p1.[/b] A straight line is painted in two colors. Prove that there are three points of the same color such that one of them is located exactly at the midpoint of the interval bounded by the other two.
[b]p2.[/b] Find all positive integral solutions $x, y$ of the equation $xy = x + y + 3$.
[b]p3.[/b] Can one cut a square into isosceles triangles with angle $80^o$ between equal sides?
[b]p4.[/b] $20$ children are grouped into $10$ pairs: one boy and one girl in each pair. In each pair the boy is taller than the girl. Later they are divided into pairs in a different way. May it happen now that
(a) in all pairs the girl is taller than the boy;
(b) in $9$ pairs out of $10$ the girl is taller than the boy?
[b]p5.[/b] Mr Mouse got to the cellar where he noticed three heads of cheese weighing $50$ grams, $80$ grams, and $120$ grams. Mr. Mouse is allowed to cut simultaneously $10$ grams from any two of the heads and eat them. He can repeat this procedure as many times as he wants. Can he make the weights of all three pieces equal?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Kvant 2023, M2731
There are 2023 natural written in a row. The first number is 12, and each number starting from the third is equal to the product of the previous two numbers, or to the previous number increased by 4. What is the largest number of perfect squares that can be among the 2023 numbers?
[i]Based on the British Mathematical Olympiad[/i]
1952 AMC 12/AHSME, 33
A circle and a square have the same perimeter. Then:
$ \textbf{(A)}\ \text{their areas are equal} \qquad\textbf{(B)}\ \text{the area of the circle is the greater}$
$ \textbf{(C)}\ \text{the area of the square is the greater}$
$ \textbf{(D)}\ \text{the area of the circle is } \pi \text{ times the area of the square} \\
\qquad\textbf{(E)}\ \text{none of these}$
2016 HMNT, 13-15
13. How many functions $f : \{0, 1\}^3 \to \{0, 1\}$ satisfy the property that, for all ordered triples $(a_1, a_2, a_3)$ and $(b_1, b_2, b_3)$ such that $a_i \ge b_i$ for all $i$, $f(a_1, a_2, a_3) \ge f(b_1, b_2, b_3)$?
14. The very hungry caterpillar lives on the number line. For each non-zero integer $i$, a fruit sits on the point with coordinate $i$. The caterpillar moves back and forth; whenever he reaches a point with food, he eats the food, increasing his weight by one pound, and turns around. The caterpillar moves at a speed of $2^{-w}$ units per day, where $w$ is his weight. If the caterpillar starts off at the origin, weighing zero pounds, and initially moves in the positive $x$ direction, after how many days will he weigh $10$ pounds?
15. Let $ABCD$ be an isosceles trapezoid with parallel bases $AB = 1$ and $CD = 2$ and height $1$. Find the area of the region containing all points inside $ABCD$ whose projections onto the four sides of the trapezoid lie on the segments formed by $AB,BC,CD$ and $DA$.
2001 JBMO ShortLists, 4
The discriminant of the equation $x^2-ax+b=0$ is the square of a rational number and $a$ and $b$ are integers. Prove that the roots of the equation are integers.
2025 PErA, P6
Let $m$ and $n$ be positive integers. For a connected simple graph $G$ on $n$ vertices and $m$ edges, we consider the number $N(G)$ of orientations of (all of) its edges so that, in the resulting directed graph, every vertex has even outdegree.
Show that $N(G)$ only depends on $m$ and $n$, and determine its value.
1989 All Soviet Union Mathematical Olympiad, 491
Eight pawns are placed on a chessboard, so that there is one in each row and column. Show that an even number of the pawns are on black squares.
Oliforum Contest V 2017, 4
Let $p_n$ be the $n$-th prime, so that $p_1 = 2, p_2 = 3,...$ and de fine $$X_n = \{0\} \cup \{p_1,...,p_n\}$$ for each positive integer $n$. Find all $n$ for which there exist $A,B \subseteq N$ such that$ |A|,|B| \ge 2$ and
$$X_n = A + B$$, where $A + B :=\{a + b : a \in A; b \in B \}$ and $N := \{0,1, 2,...\}$.
(Salvatore Tringali)
1989 Balkan MO, 4
The elements of the set $F$ are some subsets of $\left\{1,2,\ldots ,n\right\}$ and satisfy the conditions:
i) if $A$ belongs to $F$, then $A$ has three elements;
ii)if $A$ and $B$ are distinct elements of $F$ , then $A$ and $B$ have at most one common element.
Let $f(n)$ be the greatest possible number of elements of $F$. Prove that $\frac{n^{2}-4n}{6}\leq f(n) \leq \frac{n^{2}-n}{6}$
2004 Singapore Team Selection Test, 3
Let $p \geq 5$ be a prime number. Prove that there exist at least 2 distinct primes $q_1, q_2$ satisfying $1 < q_i < p - 1$ and $q_i^{p-1} \not\equiv 1 \mbox{ (mod }p^2)$, for $i = 1, 2$.
2001 IMO Shortlist, 1
Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.
2007 China Team Selection Test, 3
Let $ n$ be a positive integer, let $ A$ be a subset of $ \{1, 2, \cdots, n\}$, satisfying for any two numbers $ x, y\in A$, the least common multiple of $ x$, $ y$ not more than $ n$. Show that $ |A|\leq 1.9\sqrt {n} \plus{} 5$.
1990 Brazil National Olympiad, 1
Show that a convex polyhedron with an odd number of faces has at least one face with an even number of edges.
2024 CMIMC Algebra and Number Theory, 4
For positive integer $n$, let $f(n)$ be the largest integer $k$ such that $k!\leq n$, let $g(n)=n-(f(n))!$, and for $j\geq 1$ let
$$g^j(n)=\underbrace{g(\dots(g(n))\dots)}_{\text{$j$ times}}.$$
Find the smallest positive integer $n$ such that $g^{j}(n)> 0$ for all $j<30$ and $g^{30}(n)=0$.
[i]Proposed by Connor Gordon[/i]
Ukrainian TYM Qualifying - geometry, 2011.5
The circle $\omega_0$ touches the line at point A. Let $R$ be a given positive number. We consider various circles $\omega$ of radius $R$ that touch a line $\ell$ and have two different points in common with the circle $\omega_0$. Let $D$ be the touchpoint of the circle $\omega_0$ with the line $\ell$, and the points of intersection of the circles $\omega$ and $\omega_0$ are denoted by $B$ and $C$ (Assume that the distance from point $B$ to the line $\ell$ is greater than the distance from point $C$ to this line). Find the locus of the centers of the circumscribed circles of all such triangles $ABD$.
1988 AMC 12/AHSME, 23
The six edges of a tetrahedron $ABCD$ measure $7$, $13$, $18$, $27$, $36$ and $41$ units. If the length of edge $AB$ is $41$, then the length of edge $CD$ is
$ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 36 $
2005 India IMO Training Camp, 1
Consider a $n$-sided polygon inscribed in a circle ($n \geq 4$). Partition the polygon into $n-2$ triangles using [b]non-intersecting[/b] diagnols. Prove that, irrespective of the triangulation, the sum of the in-radii of the triangles is a constant.
2010 IMO Shortlist, 6
Suppose that $f$ and $g$ are two functions defined on the set of positive integers and taking positive integer values. Suppose also that the equations $f(g(n)) = f(n) + 1$ and $g(f(n)) = g(n) + 1$ hold for all positive integers. Prove that $f(n) = g(n)$ for all positive integer $n.$
[i]Proposed by Alex Schreiber, Germany[/i]
2015 ISI Entrance Examination, 8
Find all the functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that
$$|f(x)-f(y)| = 2 |x - y| $$
2009 Tournament Of Towns, 3
Find all positive integers $a$ and $b$ such that $(a + b^2)(b + a^2) = 2^m$ for some integer $m.$
[i](6 points)[/i]
2000 Federal Competition For Advanced Students, Part 2, 2
A trapezoid $ABCD$ with $AB \parallel CD$ is inscribed in a circle $k$. Points $P$ and $Q$ are chose on the arc $ADCB$ in the order $A-P -Q-B$. Lines $CP$ and $AQ$ meet at $X$, and lines $BP$ and $DQ$ meet at $Y$. Show that points $P,Q,X, Y$ lie on a circle.