Found problems: 85335
2020 Taiwan APMO Preliminary, P4
Let $(a,b)=(a_n,a_{n+1}),\forall n\in\mathbb{N}$ all be positive interger solutions that satisfies
$$1\leq a\leq b$$ and $$\dfrac{a^2+b^2+a+b+1}{ab}\in\mathbb{N}$$
And the value of $a_n$ is [b]only[/b] determined by the following recurrence relation:$ a_{n+2} = pa_{n+1} + qa_n + r$
Find $(p,q,r)$.
1998 IMO Shortlist, 2
Determine all pairs $(a,b)$ of real numbers such that $a \lfloor bn \rfloor =b \lfloor an \rfloor $ for all positive integers $n$. (Note that $\lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$.)
PEN A Problems, 52
Let $d$ be any positive integer not equal to 2, 5, or 13. Show that one can find distinct $a$ and $b$ in the set $\{2,5,13,d\}$ such that $ab - 1$ is not a perfect square.
2009 F = Ma, 7
A bird is flying in a straight line initially at $\text{10 m/s}$. It uniformly increases its speed to $\text{15 m/s}$ while covering a distance of $\text{25 m}$. What is the magnitude of the acceleration of the bird?
(A) $\text{5.0 m/s}^2$
(B) $\text{2.5 m/s}^2$
(C) $\text{2.0 m/s}^2$
(D) $\text{0.5 m/s}^2$
(E) $\text{0.2 m/s}^2$
1986 ITAMO, 6
Show that for any positive integer $n$ there exists an integer $m > 1$ such that $(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1}$.
2004 Bulgaria Team Selection Test, 3
Prove that among any $2n+1$ irrational numbers there are $n+1$ numbers such that the sum of any $k$ of them is irrational, for all $k \in \{1,2,3,\ldots, n+1 \}$.
1996 Baltic Way, 20
Is it possible to partition all positive integers into disjoint sets $A$ and $B$ such that
(i) no three numbers of $A$ form an arithmetic progression,
(ii) no infinite non-constant arithmetic progression can be formed by numbers of $B$?
1986 AIME Problems, 13
In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence HHTTHHHHTHHTTTT of 15 coin tosses we observe that there are two HH, three HT, four TH, and five TT subsequences. How many different sequences of 15 coin tosses will contain exactly two HH, three HT, four TH, and five TT subsequences?
1991 Arnold's Trivium, 92
Find the orders of the subgroups of the group of rotations of the cube, and find its normal subgroups.
2011 Iran MO (3rd Round), 1
prove that if graph $G$ is a tree, then there is a vertex that is common between all of the longest paths.
[i]proposed by Sina Rezayi[/i]
2017 Balkan MO Shortlist, C4
For any set of points $A_1, A_2,...,A_n$ on the plane, one defines $r( A_1, A_2,...,A_n)$ as the radius of the smallest circle that contains all of these points. Prove that if $n \ge 3$, there are indices $i,j,k$ such that $r( A_1, A_2,...,A_n)=r( A_i, A_j,A_k)$
1990 Romania Team Selection Test, 4
Let $M$ be a point on the edge $CD$ of a tetrahedron $ABCD$ such that the tetrahedra $ABCM$ and $ABDM$ have the same total areas. We denote by $\pi_{AB}$ the plane $ABM$. Planes $\pi_{AC},...,\pi_{CD}$ are analogously defined. Prove that the six planes $\pi_{AB},...,\pi_{CD}$ are concurrent in a certain point $N$, and show that $N$ is symmetric to the incenter $I$ with respect to the barycenter $G$.
1990 Tournament Of Towns, (259) 3
A cake is prepared for a dinner party to which only $p$ or $q$ persons will come ($p$ and $q$ are given co-prime integers). Find the minimum number of pieces (not necessarily equal) into which the cake must be cut in advance so that the cake may be equally shared between the persons in either case.
(D. Fomin, Leningrad)
1980 AMC 12/AHSME, 20
A box contains 2 pennies, 4 nickels, and 6 dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least 50 cents?
$\text{(A)} \ \frac{37}{924} \qquad \text{(B)} \ \frac{91}{924} \qquad \text{(C)} \ \frac{127}{924} \qquad \text{(D)} \ \frac{132}{924} \qquad \text{(E)} \ \text{none of these}$
1985 Tournament Of Towns, (085) 1
$a, b$ and $c$ are sides of a triangle, and $\gamma$ is its angle opposite $c$.
Prove that $c \ge (a + b) \sin \frac{\gamma}{2}$
(V. Prasolov )
2013 Purple Comet Problems, 29
You can tile a $2 \times5$ grid of squares using any combination of three types of tiles: single unit squares, two side by side unit squares, and three unit squares in the shape of an L. The diagram below shows the grid, the available tile shapes, and one way to tile the grid. In how many ways can the grid be tiled?
[asy]
import graph; size(15cm);
pen dps = linewidth(1) + fontsize(10); defaultpen(dps);
draw((-3,3)--(-3,1));
draw((-3,3)--(2,3));
draw((2,3)--(2,1));
draw((-3,1)--(2,1));
draw((-3,2)--(2,2));
draw((-2,3)--(-2,1));
draw((-1,3)--(-1,1));
draw((0,3)--(0,1));
draw((1,3)--(1,1));
draw((4,3)--(4,2));
draw((4,3)--(5,3));
draw((5,3)--(5,2));
draw((4,2)--(5,2));
draw((5.5,3)--(5.5,1));
draw((5.5,3)--(6.5,3));
draw((6.5,3)--(6.5,1));
draw((5.5,1)--(6.5,1));
draw((7,3)--(7,1));
draw((7,1)--(9,1));
draw((7,3)--(8,3));
draw((8,3)--(8,2));
draw((8,2)--(9,2));
draw((9,2)--(9,1));
draw((11,3)--(11,1));
draw((11,3)--(16,3));
draw((16,3)--(16,1));
draw((11,1)--(16,1));
draw((12,3)--(12,2));
draw((11,2)--(12,2));
draw((12,2)--(13,2));
draw((13,2)--(13,1));
draw((14,3)--(14,1));
draw((14,2)--(15,2));
draw((15,3)--(15,1));[/asy]
2016 Romania Team Selection Test, 2
Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$
1971 IMO Longlists, 51
Suppose that the sides $AB$ and $DC$ of a convex quadrilateral $ABCD$ are not parallel. On the sides $BC$ and $AD$, pairs of points $(M,N)$ and $(K,L)$ are chosen such that $BM=MN=NC$ and $AK=KL=LD$. Prove that the areas of triangles $OKM$ and $OLN$ are different, where $O$ is the intersection point of $AB$ and $CD$.
2007 Oral Moscow Geometry Olympiad, 5
Given triangle $ABC$. Points $A_1,B_1$ and $C_1$ are symmetric to its vertices with respect to opposite sides. $C_2$ is the intersection point of lines $AB_1$ and $BA_1$. Points$ A_2$ and $B_2$ are defined similarly. Prove that the lines $A_1 A_2, B_1 B_2$ and $C_1 C_2$ are parallel.
(A. Zaslavsky)
1979 Bulgaria National Olympiad, Problem 5
A convex pentagon $ABCDE$ satisfies $AB=BC=CA$ and $CD=DE=EC$. Let $S$ be the center of the equilateral triangle $ABC$ and $M$ and $N$ be the midpoints of $BD$ and $AE$, respectively. Prove that the triangles $SME$ and $SND$ are similar.
2015 Regional Olympiad of Mexico Center Zone, 2
In the triangle $ABC$, we have that $\angle BAC$ is acute. Let $\Gamma$ be the circle that passes through $A$ and is tangent to the side $BC$ at $C$. Let $M$ be the midpoint of $BC$ and let $D$ be the other point of intersection of $\Gamma$ with $AM$. If $BD$ cuts back to$ \Gamma$ at $E$, show that $AC$ is the bisector of $\angle BAE$.
2014 Junior Regional Olympiad - FBH, 2
Find value of $$\frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx}$$ if $x$, $y$ and $z$ are real numbers usch that $xyz=1$
1987 All Soviet Union Mathematical Olympiad, 442
It is known that, having $6$ weighs, it is possible to balance the scales with loads, which weights are successing natural numbers from $1$ to $63$. Find all such sets of weighs.
2019 IFYM, Sozopol, 6
Find all odd numbers $n\in \mathbb{N}$, for which the number of all natural numbers, that are no bigger than $n$ and coprime with $n$, divides $n^2+3$.
2023 Bulgarian Spring Mathematical Competition, 9.4
Given is a directed graph with $28$ vertices, such that there do not exist vertices $u, v$, such that $u \rightarrow v$ and $v \rightarrow u$. Every $16$ vertices form a directed cycle. Prove that among any $17$ vertices, we can choose $15$ which form a directed cycle.