Found problems: 85335
2012 Argentina National Olympiad, 4
For each natural number $n$ we denote $a_n$ as the greatest perfect square less than or equal to $n$ and $b_n$ as the least perfect square greater than $n$. For example $a_9=3^2$, $b_9=4^2$ and $a_{20}=4^2$, $b_{20}=5^2$. Calculate: $$\frac{1}{a_1b_1}+\frac{1}{a_2b_2}+\frac{1}{a_3b_3}+\ldots +\frac{1}{a_{600}b_{600}}$$
Kyiv City MO Juniors Round2 2010+ geometry, 2021.7.41
Point $C$ lies inside the right angle $AOB$. Prove that the perimeter of triangle $ABC$ is greater than $2 OC$.
1998 Swedish Mathematical Competition, 4
$ABCD$ is a quadrilateral with $\angle A = 90o$, $AD = a$, $BC = b$, $AB = h$, and area $\frac{(a+b)h}{2}$. What can we say about $\angle B$?
2017 Indonesia MO, 5
A polynomial $P$ has integral coefficients, and it has at least 9 different integral roots. Let $n$ be an integer such that $|P(n)| < 2017$. Prove that $P(n) = 0$.
2019 Romania EGMO TST, P2
Determine the digits $0\leqslant c\leqslant 9$ such that for any positive integer $k{}$ there exists a positive integer $n$ such that the last $k{}$ digits of $n^9$ are equal to $c{}.$
2022 IMO Shortlist, G2
In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.
2000 Slovenia National Olympiad, Problem 1
Find all prime numbers whose base $b$ representations (for some $b$) contain each of the digits $0,1,\ldots,b-1$ exactly once. (Digit $0$ may appear as the first digit.)
1977 IMO Longlists, 20
Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$
2014 CIIM, Problem 6
a) Let $\{x_n\}$ be a sequence with $x_n \in [0,1]$ for any $n$. Prove that there exists $C > 0$ such that for every positive integer $r$, there exists $m \geq 1$ and $n > m + r$ that satisfy $(n-m)|x_n-x_m| \leq C$.
b) Prove that for every $C > 0$, there exists a sequence $\{x_n\}$ with $x_n \in [0,1]$ for all $n$ and an integer $r$ such that, if $m \geq 1$ and $n > m+r$, then $(n-m)|x_n-x_m| > C.$
2024 Czech-Polish-Slovak Junior Match, 2
Among all triples $(a,b,c)$ of natural numbers satisfying
\[(a+14\sqrt{3})(b-14c\sqrt{3})=2024,\]
determine the one with the maximal value of $a$.
2014 District Olympiad, 3
Let $A=\{1,3,3^2,\ldots, 3^{2014}\}$. We obtain a partition of $A$ if $A$ is written as a disjoint union of nonempty subsets.
[list=a]
[*]Prove that there is no partition of $A$ such that the product of elements in each subset is a square.
[*]Prove that there exists a partition of $A$ such that the sum of elements in each subset is a square.[/list]
2022 MIG, 5
Jamie accidentally misinterprets the rules of the order of operations, and adds or subtracts before multiplying or dividing. What would be her result for the equation $4 + 3 \times 1 - 2$?
$\textbf{(A) }{-}7\qquad\textbf{(B) }{-}5\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }9$
2003 BAMO, 1
An integer is a perfect number if and only if it is equal to the sum of all of its divisors except itself.
For example, $28$ is a perfect number since $28 = 1 + 2 + 4 + 7 + 14$.
Let $n!$ denote the product $1\cdot 2\cdot 3\cdot ...\cdot n$, where $n$ is a positive integer.
An integer is a factorial if and only if it is equal to $n!$ for some positive integer $n$.
For example, $24$ is a factorial number since $24 = 4! = 1\cdot 2\cdot 3\cdot 4$.
Find all perfect numbers greater than $1$ that are also factorials.
1973 Dutch Mathematical Olympiad, 5
An infinite sequence of integers $a_1,a_2,a_3, ...$ is given with $a_1 = 0$ and further holds for every natural number $n$ that $a_{n+1} = a_n - n$ if $a_n \ge n$ and $a_{n+1} = a_n + n$ if $a_n < n$ .
(a) Prove that there are infinitely many numbers in the sequence equal to $0$.
(b) Express in terms of $k$ the ordinal number of the $k^e$ number from the sequence, which is equal to $0$.
2013 Purple Comet Problems, 28
Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ be the eight vertices of a $30 \times30\times30$ cube as shown. The two figures $ACFH$ and $BDEG$ are congruent regular tetrahedra. Find the volume of the intersection of these two tetrahedra.
[asy]
import graph; size(12.57cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
pen dotstyle = black;
real xmin = -3.79, xmax = 8.79, ymin = 0.32, ymax = 4.18; /* image dimensions */
pen ffqqtt = rgb(1,0,0.2); pen ffzzzz = rgb(1,0.6,0.6); pen zzzzff = rgb(0.6,0.6,1);
draw((6,3.5)--(8,1.5), zzzzff);
draw((7,3)--(5,1), blue);
draw((6,3.5)--(7,3), blue);
draw((6,3.5)--(5,1), blue);
draw((5,1)--(8,1.5), blue);
draw((7,3)--(8,1.5), blue);
draw((4,3.5)--(2,1.5), ffzzzz);
draw((1,3)--(2,1.5), ffqqtt);
draw((2,1.5)--(3,1), ffqqtt);
draw((1,3)--(3,1), ffqqtt);
draw((4,3.5)--(1,3), ffqqtt);
draw((4,3.5)--(3,1), ffqqtt);
draw((-3,3)--(-3,1), linewidth(1.6));
draw((-3,3)--(-1,3), linewidth(1.6));
draw((-1,3)--(-1,1), linewidth(1.6));
draw((-3,1)--(-1,1), linewidth(1.6));
draw((-3,3)--(-2,3.5), linewidth(1.6));
draw((-2,3.5)--(0,3.5), linewidth(1.6));
draw((0,3.5)--(-1,3), linewidth(1.6));
draw((0,3.5)--(0,1.5), linewidth(1.6));
draw((0,1.5)--(-1,1), linewidth(1.6));
draw((-3,1)--(-2,1.5));
draw((-2,1.5)--(0,1.5));
draw((-2,3.5)--(-2,1.5));
draw((1,3)--(1,1), linewidth(1.6));
draw((1,3)--(3,3), linewidth(1.6));
draw((3,3)--(3,1), linewidth(1.6));
draw((1,1)--(3,1), linewidth(1.6));
draw((1,3)--(2,3.5), linewidth(1.6));
draw((2,3.5)--(4,3.5), linewidth(1.6));
draw((4,3.5)--(3,3), linewidth(1.6));
draw((4,3.5)--(4,1.5), linewidth(1.6));
draw((4,1.5)--(3,1), linewidth(1.6));
draw((1,1)--(2,1.5));
draw((2,3.5)--(2,1.5));
draw((2,1.5)--(4,1.5));
draw((5,3)--(5,1), linewidth(1.6));
draw((5,3)--(6,3.5), linewidth(1.6));
draw((5,3)--(7,3), linewidth(1.6));
draw((7,3)--(7,1), linewidth(1.6));
draw((5,1)--(7,1), linewidth(1.6));
draw((6,3.5)--(8,3.5), linewidth(1.6));
draw((7,3)--(8,3.5), linewidth(1.6));
draw((7,1)--(8,1.5));
draw((5,1)--(6,1.5));
draw((6,3.5)--(6,1.5));
draw((6,1.5)--(8,1.5));
draw((8,3.5)--(8,1.5), linewidth(1.6));
label("$ A $",(-3.4,3.41),SE*labelscalefactor);
label("$ D $",(-2.16,4.05),SE*labelscalefactor);
label("$ H $",(-2.39,1.9),SE*labelscalefactor);
label("$ E $",(-3.4,1.13),SE*labelscalefactor);
label("$ F $",(-1.08,0.93),SE*labelscalefactor);
label("$ G $",(0.12,1.76),SE*labelscalefactor);
label("$ B $",(-0.88,3.05),SE*labelscalefactor);
label("$ C $",(0.17,3.85),SE*labelscalefactor);
label("$ A $",(0.73,3.5),SE*labelscalefactor);
label("$ B $",(3.07,3.08),SE*labelscalefactor);
label("$ C $",(4.12,3.93),SE*labelscalefactor);
label("$ D $",(1.69,4.07),SE*labelscalefactor);
label("$ E $",(0.60,1.15),SE*labelscalefactor);
label("$ F $",(2.96,0.95),SE*labelscalefactor);
label("$ G $",(4.12,1.67),SE*labelscalefactor);
label("$ H $",(1.55,1.82),SE*labelscalefactor);
label("$ A $",(4.71,3.47),SE*labelscalefactor);
label("$ B $",(7.14,3.10),SE*labelscalefactor);
label("$ C $",(8.14,3.82),SE*labelscalefactor);
label("$ D $",(5.78,4.08),SE*labelscalefactor);
label("$ E $",(4.6,1.13),SE*labelscalefactor);
label("$ F $",(6.93,0.96),SE*labelscalefactor);
label("$ G $",(8.07,1.64),SE*labelscalefactor);
label("$ H $",(5.65,1.90),SE*labelscalefactor);
dot((-3,3),dotstyle);
dot((-3,1),dotstyle);
dot((-1,3),dotstyle);
dot((-1,1),dotstyle);
dot((-2,3.5),dotstyle);
dot((0,3.5),dotstyle);
dot((0,1.5),dotstyle);
dot((-2,1.5),dotstyle);
dot((1,3),dotstyle);
dot((1,1),dotstyle);
dot((3,3),dotstyle);
dot((3,1),dotstyle);
dot((2,3.5),dotstyle);
dot((4,3.5),dotstyle);
dot((4,1.5),dotstyle);
dot((2,1.5),dotstyle);
dot((5,3),dotstyle);
dot((5,1),dotstyle);
dot((6,3.5),dotstyle);
dot((7,3),dotstyle);
dot((7,1),dotstyle);
dot((8,3.5),dotstyle);
dot((8,1.5),dotstyle);
dot((6,1.5),dotstyle); [/asy]
2018 MOAA, 10
Vincent is playing a game with Evil Bill. The game uses an infinite number of red balls, an infinite number of green balls, and a very large bag. Vincent first picks two nonnegative integers $g$ and $k$ such that $g < k \le 2016$, and Evil Bill places $g$ green balls and $2016 - g$ red balls in the bag, so that there is a total of $2016$ balls in the bag. Vincent then picks a ball of either color and places it in the bag. Evil Bill then inspects the bag. If the ratio of green balls to total balls in the bag is ever exactly $\frac{k}{2016}$ , then Evil Bill wins. If the ratio of green balls to total balls is greater than $\frac{k}{2016}$ , then Vincent wins. Otherwise, Vincent and Evil Bill repeat the previous two actions (Vincent picks a ball and Evil Bill inspects the bag). If $S$ is the sum of all possible values of $k$ that Vincent could choose and be able to win, determine the largest prime factor of $S$.
II Soros Olympiad 1995 - 96 (Russia), 10.7
Three straight lines $\ell_1$, $\ell_2$ and $\ell_3$, forming a triangle, divide the plane into $7$ parts. Each of the points $M_1$, $M_2$ and $M_3$ lies in one of the angles, vertical to some angle of the triangle. The distance from $M_1$ to straight lines $\ell_1$, $\ell_2$ and $\ell_3$ are equal to $7,3$ and $1$ respectively The distance from $M_2$ to the same lines are $4$, $1$ and $3$ respectively. For $M_3$ these distances are $3$, $5$ and $2$. What is the radius of the circle inscribed in the triangle?
[hide=second sentence in Russian]Каждая из точек М_1, М_2 и М_з лежит в одном из углов, вертикальном по отношению к какому-то углу треугольника.[/hide]
2002 Moldova National Olympiad, 4
Let $ ABCD$ be a convex quadrilateral and let $ N$ on side $ AD$ and $ M$ on side $ BC$ be points such that $ \dfrac{AN}{ND}\equal{}\dfrac{BM}{MC}$. The lines $ AM$ and $ BN$ intersect at $ P$, while the lines $ CN$ and $ DM$ intersect at $ Q$. Prove that if $ S_{ABP}\plus{}S_{CDQ}\equal{}S_{MNPQ}$, then either $ AD\parallel BC$ or $ N$ is the midpoint of $ DA$.
1990 AIME Problems, 5
Let $n$ be the smallest positive integer that is a multiple of $75$ and has exactly $75$ positive integral divisors, including $1$ and itself. Find $n/75$.
1957 AMC 12/AHSME, 17
A cube is made by soldering twelve $ 3$-inch lengths of wire properly at the vertices of the cube. If a fly alights at one of the vertices and then walks along the edges, the greatest distance it could travel before coming to any vertex a second time, without retracing any distance, is:
$ \textbf{(A)}\ 24\text{ in.}\qquad
\textbf{(B)}\ 12\text{ in.}\qquad
\textbf{(C)}\ 30\text{ in.}\qquad
\textbf{(D)}\ 18\text{ in.}\qquad
\textbf{(E)}\ 36\text{ in.}$
2019 Romania Team Selection Test, 2
Determine the largest natural number $ N $ having the following property: every $ 5\times 5 $ array consisting of pairwise distinct natural numbers from $ 1 $ to $ 25 $ contains a $ 2\times 2 $ subarray of numbers whose sum is, at least, $ N. $
[i]Demetres Christofides[/i] and [i]Silouan Brazitikos[/i]
2017 Brazil National Olympiad, 4.
[b]4.[/b] We see, in Figures 1 and 2, examples of lock screens from a cellphone that only works with a password that is not typed but drawn with straight line segments. Those segments form a polygonal line with vertices in a lattice. When drawing the pattern that corresponds to a password, the finger can't lose contact with the screen. Every polygonal line corresponds to a sequence of digits and this sequence is, in fact, the password. The tracing of the polygonal obeys the following rules:
[i]i.[/i] The tracing starts at some of the detached points which correspond to the digits from $1$ to $9$ (Figure 3).
[i]ii.[/i] Each segment of the pattern must have as one of its extremes (on which we end the tracing of the segment) a point that has not been used yet.
[i]iii.[/i] If a segment connects two points and contains a third one (its middle point), then the corresponding digit to this third point is included in the password. That does not happen if this point/digit has already been used.
[i]iv.[/i] Every password has at least four digits.
Thus, every polygonal line is associated to a sequence of four or more digits, which appear in the password in the same order that they are visited. In Figure 1, for instance, the password is 218369, if the first point visited was $2$. Notice how the segment connecting the points associated with $3$ and $9$ includes the points associated to digit $6$. If the first visited point were the $9$, then the password would be $963812$. If the first visited point were the $6$, then the password would be $693812$. In this case, the $6$ would be skipped, because it can't be repeated. On the other side, the polygonal line of Figure 2 is associated to a unique password.
Determine the smallest $n (n \geq 4)$ such that, given any subset of $n$ digits from $1$ to $9$, it's possible to elaborate a password that involves exactly those digits in some order.
2007 IMC, 1
Let $ f$ be a polynomial of degree 2 with integer coefficients. Suppose that $ f(k)$ is divisible by 5 for every integer $ k$. Prove that all coefficients of $ f$ are divisible by 5.
2020-2021 Winter SDPC, #5
Suppose that the positive divisors of a positive integer $n$ are $1=d_1<d_2<\ldots<d_k=n$, where $k \geq 5$. Given that $k \leq 1000$ and $n={d_2}^{d_3}{d_4}^{d_5}$, compute, with proof, all possible values of $k$.
2018 Kyiv Mathematical Festival, 1
A square of size $2\times2$ with one of its cells occupied by a tower is called a castle. What maximal number of castles one can place on a board of size $7\times7$ so that the castles have no common cells and all the towers stand on the diagonals of the board?