Found problems: 85335
1965 Miklós Schweitzer, 8
Let the continuous functions $ f_n(x), \; n\equal{}1,2,3,...,$ be defined on the interval $ [a,b]$ such that every point of $ [a,b]$ is a root of $ f_n(x)\equal{}f_m(x)$ for some $ n \not\equal{} m$. Prove that there exists a subinterval of $ [a,b]$ on which two of the functions are equal.
1997 Taiwan National Olympiad, 6
Show that every number of the form $2^{p}3^{q}$ , where $p,q$ are nonnegative integers, divides some number of the form $a_{2k}10^{2k}+a_{2k-2}10^{2k-2}+...+a_{2}10^{2}+a_{0}$, where $a_{2i}\in\{1,2,...,9\}$
1978 USAMO, 2
$ABCD$ and $A'B'C'D'$ are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for $O$.
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
real theta = -100, r = 0.3; pair D2 = (0.3,0.76);
string[] lbl = {'A', 'B', 'C', 'D'}; draw(unitsquare); draw(shift(D2)*rotate(theta)*scale(r)*unitsquare);
for(int i = 0; i < lbl.length; ++i) {
pair Q = dir(135-90*i), P = (.5,.5)+Q/2^.5;
label("$"+lbl[i]+"'$", P, Q);
label("$"+lbl[i]+"$",D2+rotate(theta)*(r*P), rotate(theta)*Q);
}[/asy]
2012 AMC 10, 18
Suppose that one of every $500$ people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a $2\%$ false positive rate; in other words, for such people, $98\%$ of the time the test will turn out negative, but $2\%$ of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let $p$ be the probability that a person who is chosen at random from the population and gets a positive test result actually has the disease. Which of the following is closest to $p$?
$ \textbf{(A)}\ \frac{1}{98}\qquad\textbf{(B)}\ \frac{1}{9}\qquad\textbf{(C)}\ \frac{1}{11}\qquad\textbf{(D)}\ \frac{49}{99}\qquad\textbf{(E)}\ \frac{98}{99}$
2009 Denmark MO - Mohr Contest, 5
Imagine a square scheme consisting of $n\times n$ fields with edge length $1$, where $n$ is an arbitrary positive integer. What is the maximum possible length of a route you can follow along the edges of the fields from point $A$ in the lower left corner to point $B$ in the upper right corner if you must never return to one point where you have been before? (The figure shows for $n = 5$ an example of a permitted route and an example of a not permitted route).
[img]https://cdn.artofproblemsolving.com/attachments/6/e/92931d87f11b9fb3120b8dccc2c37c35a04456.png[/img]
2005 District Olympiad, 4
Let $(A,+,\cdot)$ be a finite unit ring, with $n\geq 3$ elements in which there exist [b]exactly[/b] $\dfrac {n+1}2$ perfect squares (e.g. a number $b\in A$ is called a perfect square if and only if there exists an $a\in A$ such that $b=a^2$). Prove that
a) $1+1$ is invertible;
b) $(A,+,\cdot)$ is a field.
[i]Proposed by Marian Andronache[/i]
2018 AIME Problems, 13
Let \(\triangle ABC\) have side lengths \(AB=30\), \(BC=32\), and \(AC=34\). Point \(X\) lies in the interior of \(\overline{BC}\), and points \(I_1\) and \(I_2\) are the incenters of \(\triangle ABX\) and \(\triangle ACX\), respectively. Find the minimum possible area of \(\triangle AI_1I_2\) as \( X\) varies along \(\overline{BC}\).
2010 Malaysia National Olympiad, 4
A square $ABCD$ has side length $ 1$. A circle passes through the vertices of the square. Let $P, Q, R, S$ be the midpoints of the arcs which are symmetrical to the arcs $AB$, $BC$, $CD$, $DA$ when reflected on sides $AB$, $B$C, $CD$, $DA$, respectively. The area of square $PQRS$ is $a+b\sqrt2$, where $a$ and $ b$ are integers. Find the value of $a+b$.
[img]https://cdn.artofproblemsolving.com/attachments/4/3/fc9e1bd71b26cfd9ff076db7aa0a396ae64e72.png[/img]
2010 Contests, 1
Solve in the integers the diophantine equation
$$x^4-6x^2+1 = 7 \cdot 2^y.$$
1990 Chile National Olympiad, 6
Given a regular polygon with apothem $ A $ and circumradius $ R $. Find for a regular polygon of equal perimeter and with double number of sides, the apothem $ a $ and the circumcircle $ r $ in terms of $A,R$
2025 Turkey Team Selection Test, 9
Let \(n\) be a positive integer. For every positive integer $1 \leq k \leq n$ the sequence ${\displaystyle {\{ a_{i}+ki\}}_{i=1}^{n }}$ is defined, where $a_1,a_2, \dots ,a_n$ are integers. Among these \(n\) sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by \(n\)?
2015 Bosnia Herzegovina Team Selection Test, 2
Let $D$ be an arbitrary point on side $AB$ of triangle $ABC$. Circumcircles of triangles $BCD$ and $ACD$ intersect sides $AC$ and $BC$ at points $E$ and $F$, respectively. Perpendicular bisector of $EF$ cuts $AB$ at point $M$, and line perpendicular to $AB$ at $D$ at point $N$. Lines $AB$ and $EF$ intersect at point $T$, and the second point of intersection of circumcircle of triangle $CMD$ and line $TC$ is $U$. Prove that $NC=NU$
2018 Brazil Undergrad MO, 8
A student will take an exam in which they have to solve three chosen problems by chance of a list of $10$ possible problems. It will be approved if it correctly resolves two problems. Considering that the student can solve five of the problems on the list and not know how to solve others, how likely is he to pass the exam?
2009 Germany Team Selection Test, 2
Let $ \left(a_n \right)_{n \in \mathbb{N}}$ defined by $ a_1 \equal{} 1,$ and $ a_{n \plus{} 1} \equal{} a^4_n \minus{} a^3_n \plus{} 2a^2_n \plus{} 1$ for $ n \geq 1.$ Show that there is an infinite number of primes $ p$ such that none of the $ a_n$ is divisible by $ p.$
2023 BMT, 9
The boxes in the expression below are filled with the numbers $3$, $4$, $5$, $6$, $7$, and $8$, so that each number is used exactly once. What is the least possible value of the expression?
$$\square \times \square +\square \times \square -\square \times \square$$
2010 Saudi Arabia IMO TST, 3
Consider a circle of center $O$ and a chord $AB$ of it (not a diameter). Take a point $T$ on the ray $OB$. The perpendicular at $T$ onto $OB$ meets the chord $AB$ at $C$ and the circle at $D$ and $E$. Denote by $S$ the orthogonal projection of $T$ onto the chord $AB$. Prove that $AS \cdot BC = T E \cdot TD$.
PEN D Problems, 20
Show that $1994$ divides $10^{900}-2^{1000}$.
1982 IMO Shortlist, 1
The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ \[ f(m+n)-f(m)-f(n)=0 \text{ or } 1. \] Determine $f(1982)$.
2023 USAMTS Problems, 4
In this problem, a [i]simple polygon[/i] is a polygon that does not intersect itself and has no holes, and a [i]side[/i] of a polygon is a maximal set of collinear, consecutive line segments in the polygon. In particular, we allow two or more consecutive vertices in a simple polygon to be identical, and three or more consecutive vertices in a simple polygon to be collinear. By convention, polygons must have at least three sides. A simple polygon is [i]convex[/i] if every one of its interior angles is $180^\circ$ degrees or less. A simple polygon is concave if it is not [i]convex[/i].
Let P be the plane. Prove or disprove each of the following statements:
$(a)$ There exists a function $f : P \to P$ such that for all positive integers $n \geq 4$, if $v_1, v_2, \ldots , v_n$ are
the vertices of a simple concave $n$-sided polygon in some order, then $f(v_1), f(v_2), \ldots, f(v_n)$ are the
vertices of a simple convex polygon in some order (which may or may not have $n$ sides).
$(b)$ There exists a function $f : P \to P$ such that for all positive integers $n \geq 4$, if $v_1, v_2, \ldots , v_n$ are
the vertices of a simple convex $n$-sided polygon in some order, then $f(v_1), f(v_2), \ldots, f(v_n)$ are the
vertices of a simple concave polygon in some order (which may or may not have $n$ sides).
2006 Kurschak Competition, 2
Let $a,t,n$ be positive integers such that $a\le n$. Consider the subsets of $\{1,2,\dots,n\}$ whose any two elements differ by at least $t$. Prove that the number of such subsets not containing $a$ is at most $t^2$ times the number of those that do contain $a$.
2023 Yasinsky Geometry Olympiad, 2
Quadrilateral $ABCD$ is inscribed in a circle of radius $R$, and also circumscribed around a circle of radius $r$. It is known that $\angle ADB = 45^o$. Find the area of triangle $AIB$, where point $I$ is the center of the circle inscribed in $ABCD$.
(Hryhoriy Filippovskyi)
2019 IFYM, Sozopol, 2
Let $n$ be a natural number. At first the cells of a table $2n$ x $2n$ are colored in white. Two players $A$ and $B$ play the following game. First is $A$ who has to color $m$ arbitrary cells in red and after that $B$ chooses $n$ rows and $n$ columns and color their cells in black. Player $A$ wins, if there is at least one red cell on the board. Find the least value of $m$ for which $A$ wins no matter how $B$ plays.
1998 Mexico National Olympiad, 2
Rays $l$ and $m$ forming an angle of $a$ are drawn from the same point. Let $P$ be a fixed point on $l$. For each circle $C$ tangent to $l$ at $P$ and intersecting $m$ at $Q$ and $R$, let $T$ be the intersection point of the bisector of angle $QPR$ with $C$. Describe the locus of $T$ and justify your answer.
1989 USAMO, 1
For each positive integer $n$, let
\begin{eqnarray*} S_n &=& 1 + \frac 12 + \frac 13 + \cdots + \frac 1n, \\ T_n &=& S_1 + S_2 + S_3 + \cdots + S_n, \\ U_n &=& \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}. \end{eqnarray*} Find, with proof, integers $0 < a, b,c, d < 1000000$ such that $T_{1988} = a S_{1989} - b$ and $U_{1988} = c S_{1989} - d$.
2017 China National Olympiad, 4
Let $n \geq 2$ be a natural number. For any two permutations of $(1,2,\cdots,n)$, say $\alpha = (a_1,a_2,\cdots,a_n)$ and $\beta = (b_1,b_2,\cdots,b_n),$ if there exists a natural number $k \leq n$ such that
$$b_i = \begin{cases} a_{k+1-i}, & \text{ }1 \leq i \leq k; \\ a_i, & \text{} k < i \leq n, \end{cases}$$
we call $\alpha$ a friendly permutation of $\beta$.
Prove that it is possible to enumerate all possible permutations of $(1,2,\cdots,n)$ as $P_1,P_2,\cdots,P_m$ such that for all $i = 1,2,\cdots,m$, $P_{i+1}$ is a friendly permutation of $P_i$ where $m = n!$ and $P_{m+1} = P_1$.