Found problems: 85335
2016 Postal Coaching, 5
Find all nonnegative integers $k, n$ which satisfy $2^{2k+1} + 9\cdot 2^k + 5 = n^2.$
1974 Chisinau City MO, 77
Is it possible to simultaneously take away on eight three-ton vehicles $50$ stones, the weight of which is respectively equal to $416, 418, 420, .., 512, 514$ kg?
1996 Tournament Of Towns, (497) 4
Is it possible to tile space using a combination of regular tetrahedra and regular octahedra?
(A Belov)
1969 IMO Longlists, 22
$(FRA 5)$ Let $\alpha(n)$ be the number of pairs $(x, y)$ of integers such that $x+y = n, 0 \le y \le x$, and let $\beta(n)$ be the number of triples $(x, y, z)$ such that$ x + y + z = n$ and $0 \le z \le y \le x.$ Find a simple relation between $\alpha(n)$ and the integer part of the number $\frac{n+2}{2}$ and the relation among $\beta(n), \beta(n -3)$ and $\alpha(n).$ Then evaluate $\beta(n)$ as a function of the residue of $n$ modulo $6$. What can be said about $\beta(n)$ and $1+\frac{n(n+6)}{12}$? And what about $\frac{(n+3)^2}{6}$?
Find the number of triples $(x, y, z)$ with the property $x+ y+ z \le n, 0 \le z \le y \le x$ as a function of the residue of $n$ modulo $6.$What can be said about the relation between this number and the number $\frac{(n+6)(2n^2+9n+12)}{72}$?
2025 Sharygin Geometry Olympiad, 24
The insphere of a tetrahedron $ABCD$ touches the faces $ABC$, $BCD$, $CDA$, $DAB$ at $D^{\prime}$, $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ respectively. Denote by $S_{AB}$ the area of the triangle $AC^{\prime}B^{\prime}$. Define similarly $S_{AC}$, $S_{BC},$ $S_{AD}$, $S_{BD}$, $S_{CD}$. Prove that there exists a triangle with sidelengths $\sqrt{S_{AB}S_{CD}}$, $\sqrt{S_{AC}S_{BD}}$ , $\sqrt{S_{AD}S_{BC}}$.
Proposed by: S.Arutyunyan
1967 IMO Shortlist, 2
Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.
IV Soros Olympiad 1997 - 98 (Russia), 9.2
The student wrote on the board three natural numbers that are consecutive members of one arithmetic progression. Then he erased the commas separating the numbers, resulting in a seven-digit number. What is the largest number that could result?
2018 Taiwan APMO Preliminary, 3
Let $a,b$ be positive integers satisfying
$$\sqrt{\dfrac{ab}{2b^2-a}}=\dfrac{a+2b}{4b}$$.
Find $|10(a-5)(b-15)|+8$.
2017 BMT Spring, 3
Let $ABCDEF$ be a regular hexagon with side length $ 1$. Now, construct square $AGDQ$. What is the area of the region inside the hexagon and not the square?
1995 Romania Team Selection Test, 3
The altitudes of a triangle have integer length and its inradius is a prime number. Find all possible values of the sides of the triangle.
2004 May Olympiad, 2
Inside an $11\times 11$ square, Pablo drew a rectangle and extending its sides divided the square into $5$ rectangles, as shown in the figure.
[img]https://cdn.artofproblemsolving.com/attachments/5/a/7774da7085f283b3aae74fb5ff472572571827.gif[/img]
Sofía did the same, but she also managed to make the lengths of the sides of the $5$ rectangles be whole numbers between $1$ and $10$, all different. Show a figure like the one Sofia made.
2005 National Olympiad First Round, 2
Let $a_1, a_2, \dots, a_n$ be positive integers such that none of them is a multiple of $5$. What is the largest integer $n<2005$, such that $a_1^4 + a_2^4 + \cdots + a_n^4$ is divisible by $5$?
$
\textbf{(A)}\ 2000
\qquad\textbf{(B)}\ 2001
\qquad\textbf{(C)}\ 2002
\qquad\textbf{(D)}\ 2003
\qquad\textbf{(E)}\ 2004
$
1969 Dutch Mathematical Olympiad, 5
a) Prove that for $n = 2,3,4,...$ holds:
$$\sin a + \sin 2a + ...+ \sin (n-1)a=\frac{\cos a \left(\frac{a}{2}\right) - \cos \left(n-\frac{1}{2}\right) a}{2 \sin \left(\frac{a}{2}\right)}$$
b) A point on the circumference of a wheel, which, remaining in a vertical plane, rolls along a horizontal path, describes, at one revolution of the wheel, a curve having a length equal to four times the diameter of the wheel. Prove this by first considering tilting a regular $n$-gon.
[hide=original wording for part b]Een punt van de omtrek van een wiel dat, in een verticaal vlak blijvend, rolt over een horizontaal gedachte weg, beschrijft bij één omwenteling van het wiel een kromme die een lengte heeft die gelijk is aan viermaal de middellijn van het wiel.
Bewijs dit door eerst een rondkantelende regelmatige n-hoek te beschouwen.[/hide]
1982 AMC 12/AHSME, 25
The adjacent map is part of a city: the small rectangles are rocks, and the paths in between are streets. Each morning, a student walks from intersection A to intersection B, always walking along streets shown, and always going east or south. For variety, at each intersection where he has a choice, he chooses with probability $\frac{1}{2}$ whether to go east or south. Find the probability that through any given morning, he goes through $C$.
[asy]
defaultpen(linewidth(0.7)+fontsize(8));
size(250);
path p=origin--(5,0)--(5,3)--(0,3)--cycle;
path q=(5,19)--(6,19)--(6,20)--(5,20)--cycle;
int i,j;
for(i=0; i<5; i=i+1) {
for(j=0; j<6; j=j+1) {
draw(shift(6*i, 4*j)*p);
}}
clip((4,2)--(25,2)--(25,21)--(4,21)--cycle);
fill(q^^shift(18,-16)*q^^shift(18,-12)*q, black);
label("A", (6,19), SE);
label("B", (23,4), NW);
label("C", (23,8), NW);
draw((26,11.5)--(30,11.5), Arrows(5));
draw((28,9.5)--(28,13.5), Arrows(5));
label("N", (28,13.5), N);
label("W", (26,11.5), W);
label("E", (30,11.5), E);
label("S", (28,9.5), S);[/asy]
$\textbf {(A) } \frac{11}{32} \qquad \textbf {(B) } \frac 12 \qquad \textbf {(C) } \frac 47 \qquad \textbf {(D) } \frac{21}{32} \qquad \textbf {(E) } \frac 34$
2015 China Northern MO, 3
If $n=p_1^{a_1},p_2^{a_2}...p_s^{a_s}$ then $\phi (n)=n \left(1- \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right)...\left(1- \frac{1}{p_s}\right)$. Find the smallest positive integer $n$ such that $\phi (n)=\frac{2^5}{47}n.$
1986 IMO Longlists, 64
Let $(a_n)_{n\in \mathbb N}$ be the sequence of integers defined recursively by $a_1 = a_2 = 1, a_{n+2} = 7a_{n+1} - a_n - 2$ for $n \geq 1$. Prove that $a_n$ is a perfect square for every $n.$
PEN H Problems, 45
Show that there cannot be four squares in arithmetical progression.
1999 Romania National Olympiad, 3
Let $f:\mathbb{R} \to \mathbb{R}$ be a monotonic function and $a,b,c,d$ be real numbers with $a$ and $c$ nonzero. Prove that if the equalities [center]$\int\limits_x^{x+\sqrt{3}} f(t) \mathrm{d}t=ax+b$ and $\int\limits_x^{x+\sqrt{2}} f(t) \mathrm{d}t=cx+d$[/center] hold for every real number $x,$ then $f$ is a polynomial function of degree one.
2015 AMC 12/AHSME, 6
Back in 1930, Tillie had to memorize her multiplication tables from $0\times 0$ through $12\times 12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd?
$\textbf{(A) }0.21\qquad\textbf{(B) }0.25\qquad\textbf{(C) }0.46\qquad\textbf{(D) }0.50\qquad\textbf{(E) }0.75$
2010 ISI B.Math Entrance Exam, 9
Let $f(x)$ be a polynomial with integer co-efficients. Assume that $3$ divides the value $f(n)$ for each integer $n$. Prove that when $f(x)$ is divided by $x^3-x$ , the remainder is of the form $3r(x)$ where $r(x)$ is a polynomial with integer coefficients.
2018 Miklós Schweitzer, 10
In 3-dimensional hyperbolic space, we are given a plane $P$ and four distinct straight lines: the lines $a_1$ and $a_2$ are perpendicular to $P$; while the lines $r_1$ and $r_2$ do not intersect $P$, and their distances from $P$ are equal. Denote by $S_i$ the surface of revolution obtained by rotating $r_i$ around $a_i$. Show that the common points of $S_1$ and $S_2$ can be covered by two planes.
2008 Chile National Olympiad, 6
It is known that the number $\pi$ is transcendental, that is, it is not a root of any polynomial with integer coefficients. Using this fact, prove that the same is true for the number $\pi + \sqrt2$.
2008 National Olympiad First Round, 21
Let $ABC$ be a right triangle with $m(\widehat{A})=90^\circ$. Let $APQR$ be a square with area $9$ such that $P\in [AC]$, $Q\in [BC]$, $R\in [AB]$. Let $KLMN$ be a square with area $8$ such that $N,K\in [BC]$, $M\in [AB]$, and $L\in [AC]$. What is $|AB|+|AC|$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 10
\qquad\textbf{(C)}\ 12
\qquad\textbf{(D)}\ 14
\qquad\textbf{(E)}\ 16
$
2003 India IMO Training Camp, 9
Let $n$ be a positive integer and $\{A,B,C\}$ a partition of $\{1,2,\ldots,3n\}$ such that $|A|=|B|=|C|=n$. Prove that there exist $x \in A$, $y \in B$, $z \in C$ such that one of $x,y,z$ is the sum of the other two.
2000 All-Russian Olympiad, 4
Let $a_1, a_2, \cdots, a_n$ be a sequence of nonnegative integers. For $k=1,2,\cdots,n$ denote \[ m_k = \max_{1 \le l \le k} \frac{a_{k-l+1} + a_{k-l+2} + \cdots + a_k}{l}. \] Prove that for every $\alpha > 0$ the number of values of $k$ for which $m_k > \alpha$ is less than $\frac{a_1+a_2+ \cdots +a_n}{\alpha}.$