Found problems: 85335
1987 Bulgaria National Olympiad, Problem 4
The sequence $(x_n)_{n\in\mathbb N}$ is defined by $x_1=x_2=1$, $x_{n+2}=14x_{n+1}-x_n-4$ for each $n\in\mathbb N$. Prove that all terms of this sequence are perfect squares.
Russian TST 2015, P2
In the isosceles triangle $ABC$ where $AB = AC$, the point $I{}$ is the center of the inscribed circle. Through the point $A{}$ all the rays lying inside the angle $BAC$ are drawn. For each such ray, we denote by $X{}$ and $Y{}$ the points of intersection with the arc $BIC$ and the straight line $BC$ respectively. The circle $\gamma$ passing through $X{}$ and $Y{}$, which touches the arc $BIC$ at the point $X{}$ is considered. Prove that all the circles $\gamma$ pass through a fixed point.
2013 BMT Spring, P2
If $f(x)=x^n-7x^{n-1}+17x^{n-2}+a_{n-3}x^{n-3}+\ldots+a_0$ is a real-valued function of degree $n>2$ with all real roots, prove that no root has value greater than $4$ and at least one root has value less than $0$ or greater than $2$.
2017 Mid-Michigan MO, 10-12
[b]p1.[/b] In the group of five people any subgroup of three persons contains at least two friends. Is it possible to divide these five people into two subgroups such that all members of any subgroup are friends?
[b]p2.[/b] Coefficients $a,b,c$ in expression $ax^2+bx+c$ are such that $b-c>a$ and $a \ne 0$. Is it true that equation $ax^2+bx+c=0$ always has two distinct real roots?
[b]p3.[/b] Point $D$ is a midpoint of the median $AF$ of triangle $ABC$. Line $CD$ intersects $AB$ at point $E$. Distances $|BD|=|BF|$. Show that $|AE|=|DE|$.
[b]p4.[/b] Real numbers $a,b$ satisfy inequality $a+b^5>ab^5+1$. Show that $a+b^7>ba^7+1$.
[b]p5.[/b] A positive number was rounded up to the integer and got the number that is bigger than the original one by $28\%$. Find the original number (find all solutions).
[b]p6.[/b] Divide a $5\times 5$ square along the sides of the cells into $8$ parts in such a way that all parts are different.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1983 All Soviet Union Mathematical Olympiad, 366
Given a point $O$ inside triangle $ABC$ . Prove that $$S_A * \overrightarrow{OA} + S_B * \overrightarrow{OB} + S_C * \overrightarrow{OC} = \overrightarrow{0}$$
where $S_A, S_B, S_C$ denote areas of triangles $BOC, COA, AOB$ respectively.
2022 JBMO Shortlist, G4
Given is an equilateral triangle $ABC$ and an arbitrary point, denoted by $E$, on the line segment $BC$. Let $l$ be the line through $A$ parallel to $BC$ and let $K$ be the point on $l$ such that $KE$ is perpendicular to $BC$. The circle with centre $K$ and radius $KE$ intersects the sides $AB$ and $AC$ at $M$ and $N$, respectively. The line perpendicular to $AB$ at $M$ intersects $l$ at $D$, and the line perpendicular to $AC$ at $N$ intersects $l$ at $F$. Show that the point of intersection of the angle bisectors of angles $MDA$ and $NFA$ belongs to the line $KE$.
2024 Silk Road, 4
Let $a_1, a_2, \ldots$ be a strictly increasing sequence of positive integers, such that for any positive integer $n$, $a_n$ is not representable in the for $\sum_{i=1}^{n-1}c_ia_i$ for $c_i \in \{0, 1\}$. For every positive integer $m$, let $f(m)$ denote the number of $a_i$ that are at most $m$. Show that for any positive integers $m, k$, we have that $$f(m) \leq a_k+\frac{m} {k+1}.$$
2024 JHMT HS, 4
Let $N_3$ be the answer to problem 3.
Compute the sum of all real solutions $x$ to the equation
\[ 50^x+72^x+(N_3)^x=800^x. \]
2012 All-Russian Olympiad, 1
$101$ wise men stand in a circle. Each of them either thinks that the Earth orbits Jupiter or that Jupiter orbits the Earth. Once a minute, all the wise men express their opinion at the same time. Right after that, every wise man who stands between two people with a different opinion from him changes his opinion himself. The rest do not change. Prove that at one point they will all stop changing opinions.
MathLinks Contest 3rd, 3
An integer $z$ is said to be a [i]friendly [/i] integer if $|z|$ is not the square of an integer. Determine all integers $n$ such that there exists an infinite number of triplets of distinct friendly integers $(a, b, c)$ such that $n = a+b+c$ and $abc$ is the square of an odd integer.
2023 Yasinsky Geometry Olympiad, 2
Let $BC$ and $BD$ be the tangent lines to the circle with diameter $AC$. Let $E$ be the second point of intersection of line $CD$ and the circumscribed circle of triangle $ABC$. Prove that $CD= 2DE$.
(Matthew Kurskyi)
2015 Peru IMO TST, 6
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
[i]Proposed by Hong Kong[/i]
2006 Bulgaria Team Selection Test, 1
[b]Problem 1. [/b]In the cells of square table are written the numbers $1$, $0$ or $-1$ so that in every line there is exactly one $1$, amd exactly one $-1$. Each turn we change the places of two columns or two rows. Is it possible, from any such table, after finite number of turns to obtain its opposite table (two tables are opposite if the sum of the numbers written in any two corresponding squares is zero)?
[i] Emil Kolev[/i]
PEN K Problems, 5
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(m)+f(n))=m+n.\]
2024 Ecuador NMO (OMEC), 4
Danielle writes a sign '+' or '-' in each of the next $64$ spaces:
$$\_\_1 \_\_2 \_\_3 \_\_4 \text{ }.... \text{ }\_\_63 \_\_64=2024$$
such that the equality holds. Find the largest number of negative signs Danielle can use.
2000 Brazil Team Selection Test, Problem 3
Consider an equilateral triangle with every side divided by $n$ points into $n+1$ equal parts. We put a marker on every of the $3n$ division points. We draw lines parallel to the sides of the triangle through the division points, and this way divide the triangle into $(n+1)^2$ smaller ones.
Consider the following game: if there is a small triangle with exactly one vertex unoccupied, we put a marker on it and simultaneously take markers from the two its occupied vertices. We repeat this operation as long as it is possible.
(a) If $n\equiv1\pmod3$, show that we cannot manage that only one marker remains.
(b) If $n\equiv0$ or $n\equiv2\pmod3$, prove that we can finish the game leaving exactly one marker on the triangle.
1999 Belarusian National Olympiad, 8
Let $n$ be an integer greater than 2. A positive integer is said to be [i]attainable [/i]if it is 1 or can be obtained from 1 by a sequence of operations with the following properties:
1.) The first operation is either addition or multiplication.
2.) Thereafter, additions and multiplications are used alternately.
3.) In each addition, one can choose independently whether to add 2 or $n$
4.) In each multiplication, one can choose independently whether to multiply by 2 or by $n$.
A positive integer which cannot be so obtained is said to be [i]unattainable[/i].
[b]a.)[/b] Prove that if $n\geq 9$, there are infinitely many unattainable positive integers.
[b]b.)[/b] Prove that if $n=3$, all positive integers except 7 are attainable.
2006 Iran MO (3rd Round), 1
$n$ is a natural number. $d$ is the least natural number that for each $a$ that $gcd(a,n)=1$ we know $a^{d}\equiv1\pmod{n}$. Prove that there exist a natural number that $\mbox{ord}_{n}b=d$
2002 USAMTS Problems, 1
Some unit cubes are stacked atop a flat 4 by 4 square. The figures show views of the stacks from two different sides. Find the maximum and minimum number of cubes that could be in the stacks. Also give top views of a maximum arrangement and a minimum arrangement with each stack marked with its height.
[asy]
string s = "1010101010111111";
defaultpen(linewidth(0.7));
for(int x=0;x<4;++x) {
for(int y=0;y<4;++y) {
if(hex(substr(s,4*(3-y)+x,1))==1) {
draw((x,y)--(x,y+1)--(x+1,y+1)--(x+1,y)--cycle);
}
}}
label("South View",(2,4),N);
s = "0101110111111111";
for(int x=0;x<4;++x) {
for(int y=0;y<4;++y) {
if(hex(substr(s,4*(3-y)+x,1))==1) {
x=x+5;
draw((x,y)--(x,y+1)--(x+1,y+1)--(x+1,y)--cycle);
x=x-5;
}
}}
label("East View",(7,4),N);[/asy]
1996 IMO Shortlist, 1
Let $ ABC$ be a triangle, and $ H$ its orthocenter. Let $ P$ be a point on the circumcircle of triangle $ ABC$ (distinct from the vertices $ A$, $ B$, $ C$), and let $ E$ be the foot of the altitude of triangle $ ABC$ from the vertex $ B$. Let the parallel to the line $ BP$ through the point $ A$ meet the parallel to the line $ AP$ through the point $ B$ at a point $ Q$. Let the parallel to the line $ CP$ through the point $ A$ meet the parallel to the line $ AP$ through the point $ C$ at a point $ R$. The lines $ HR$ and $ AQ$ intersect at some point $ X$. Prove that the lines $ EX$ and $ AP$ are parallel.
1978 Romania Team Selection Test, 8
For any set $ A $ we say that two functions $ f,g:A\longrightarrow A $ are [i]similar,[/i] if there exists a bijection $ h:A\longrightarrow A $ such that $ f\circ h=h\circ g. $
[b]a)[/b] If $ A $ has three elements, construct a finite, arbitrary number functions, having as domain and codomain $ A, $ that are two by two similar, and every other function with the same domain and codomain as the ones determined is similar to, at least, one of them.
[b]b)[/b] For $ A=\mathbb{R} , $ show that the functions $ \sin $ and $ -\sin $ are similar.
2003 Romania National Olympiad, 3
Prove that the midpoints of the altitudes of a triangle are collinear if and only if the triangle is right.
[i]Dorin Popovici[/i]
2014 USAMTS Problems, 2:
Find all triples $(x, y, z)$ such that $x, y, z, x - y, y - z, x - z$ are all prime positive integers.
1986 Bundeswettbewerb Mathematik, 2
A triangle has sides $a, b,c$, radius of the incircle $r$ and radii of the excircles $r_a, r_b, r_c$: Prove that:
a) The triangle is right-angled if and only if: $r + r_a + r_b + r_c = a + b + c$.
b) The triangle is right-angled if and only if: $r^2 + r^2_a + r^2_b + r^2_c = a^2 + b^2 + c^2$.
2020 Macedonia Additional BMO TST, 2
Let $ABCD$ be a convex quadrilateral. On the sides $AB$ and $CD$ there are interior points $K$ and $L$, respectively, such that $\angle BAL = \angle CDK$. Prove that the following statements are equivalent:
$i) \angle BLA= \angle CKD$
$ii) AD \parallel BC $