Found problems: 85335
1979 Dutch Mathematical Olympiad, 2
Solve in $N$:
$$\begin{cases} a^3=b^3+c^3+12a \\ a^2=5(b+c) \end{cases}$$
2017 Saudi Arabia JBMO TST, 6
Find all pairs of prime numbers $(p, q)$ such that $p^2 + 5pq + 4q^2$ is a perfect square.
1953 Moscow Mathematical Olympiad, 246
a) On a plane, $11$ gears are arranged so that the teeth of the first gear mesh with the teeth of the second gear, the teeth of the second gear with those of the third gear, etc., and the teeth of the last gear mesh with those of the first gear. Can the gears rotate?
b) On a plane, $n$ gears are arranged so that the teeth of the first gear mesh with the teeth of the second gear, the teeth of the second gear with those of the third gear, etc., and the teeth of the last gear mesh with those of the first gear. Can the gears rotate?
1982 USAMO, 5
$A,B$, and $C$ are three interior points of a sphere $S$ such that $AB$ and $AC$ are perpendicular to the diameter of $S$ through $A$, and so that two spheres can be constructed through $A$, $B$, and $C$ which are both tangent to $S$. Prove that the sum of their radii is equal to the radius of $S$.
PEN O Problems, 3
Prove that the set of integers of the form $2^{k}-3$ ($k=2,3,\cdots$) contains an infinite subset in which every two members are relatively prime.
2016 CHMMC (Fall), 3
For a positive integer $m$, let $f(m)$ be the number of positive integers $q \le m$ such that $\frac{q^2-4}{m}$ is an integer. How many positive square-free integers $m < 2016$ satisfy $f(m) \ge 16$?
1998 Greece Junior Math Olympiad, 3
Let $k$ be a prime, such as $k\neq 2, 5$, prove that between the first $k$ terms of the sequens $1, 11, 111, 1111,....,1111....1$, where the last term have $k$ ones, is divisible by $k$.
2016 JBMO Shortlist, 3
A $5 \times 5$ table is called regular f each of its cells contains one of four pairwise distinct real numbers,such that each of them occurs exactly one in every $2 \times 2$ subtable.The sum of all numbers of a regular table is called the total sum of the table.With any four numbers,one constructs all possible regular tables,computes their total sums and counts the distinct outcomes.Determine the maximum possible count.
1999 Denmark MO - Mohr Contest, 3
A function $f$ satisfies $$f(x)+xf(1-x)=x$$ for all real numbers $x$. Determine the number $f (2)$. Find $f$ .
2007 ITAMO, 2
We define two polynomials with integer coefficients P,Q to be similar if the coefficients of P are a permutation of the coefficients of Q.
a) if P,Q are similar, then $P(2007)-Q(2007)$ is even
b) does there exist an integer $k > 2$ such that $k \mid P(2007)-Q(2007)$ for all similar polynomials P,Q?
2007 F = Ma, 32
A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$.
The rod is suspended from a distance $kd$ from the center, and undergoes small oscillations with an angular frequency $\beta \sqrt{\frac{g}{d}}$.
Find an expression for $\beta$ in terms of $k$.
$ \textbf{(A)}\ 1+k^2$
$ \textbf{(B)}\ \sqrt{1+k^2}$
$ \textbf{(C)}\ \sqrt{\frac{k}{1+k}}$
$ \textbf{(D)}\ \sqrt{\frac{k^2}{1+k}}$
$ \textbf{(E)}\ \text{none of the above}$
2014 Contests, 2
Paul owes Paula $35$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, and $25$-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$
Kyiv City MO 1984-93 - geometry, 1991.10.5
Diagonal sections of a regular 8-gon pyramid, which are drawn through the smallest and largest diagonals of the base, are equal. At what angle is the plane passing through the vertex, the pyramids and the smallest diagonal of the base inclined to the base?
[hide=original wording]Діагональні перерізи правильної 8-кутної піраміди, які Проведені через найменшу і найбільшу діагоналі основи, рівновеликі. Під яким кутом до основи нахилена площина, що проходить через вершину, піраміди і найменшу діагональ основи?[/hide]
2009 AMC 10, 10
Triangle $ ABC$ has a right angle at $ B$. Point $ D$ is the foot of the altitude from $ B$, $ AD\equal{}3$, and $ DC\equal{}4$. What is the area of $ \triangle{ABC}$?
[asy]unitsize(5mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair B=(0,0), C=(sqrt(28),0), A=(0,sqrt(21));
pair D=foot(B,A,C);
pair[] ps={B,C,A,D};
draw(A--B--C--cycle);
draw(B--D);
draw(rightanglemark(B,D,C));
dot(ps);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,NE);
label("$3$",midpoint(A--D),NE);
label("$4$",midpoint(D--C),NE);[/asy]$ \textbf{(A)}\ 4\sqrt3 \qquad
\textbf{(B)}\ 7\sqrt3 \qquad
\textbf{(C)}\ 21 \qquad
\textbf{(D)}\ 14\sqrt3 \qquad
\textbf{(E)}\ 42$
2013 Korea Junior Math Olympiad, 4
Prove that there exists a prime number $p$ such that the minimum positive integer $n$ such that $p|2^n -1$ is $3^{2013}$.
2010 USAJMO, 3
Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.
2009 Mathcenter Contest, 4
Find the values of the real numbers $x,y,z$ that correspond to the system of equations.
$$8(x+\frac{1}{x}) =15(y+\frac{1}{y}) = 17(z+\frac{1}{z})$$
$$xy + yz + zx=1$$
[i](Heir of Ramanujan)[/i]
1975 Miklós Schweitzer, 5
Let $ \{ f_n \}$ be a sequence of Lebesgue-integrable functions on $ [0,1]$ such that for any Lebesgue-measurable subset $ E$ of $ [0,1]$ the sequence $ \int_E f_n$ is convergent. Assume also that $ \lim_n f_n\equal{}f$ exists almost everywhere. Prove that $ f$ is integrable and $ \int_E f\equal{}\lim_n \int_E f_n$. Is the assertion also true if $ E$ runs only over intervals but we also assume $ f_n \geq 0 ?$ What happens if $ [0,1]$ is replaced by $ [0,\plus{}\infty) ?$
[i]J. Szucs[/i]
2023 Ecuador NMO (OMEC), 2
Let $ABCD$ a cyclic convex quadrilateral. There is a line $l$ parallel to $DC$ containing $A$. Let $P$ a point on $l$ closer to $A$ than to $B$. Let $B'$ the reflection of $B$ over the midpoint of $AD$. Prove that $\angle B'AP = \angle BAC$
2022 IFYM, Sozopol, 7
A graph $ G$ with $ n$ vertices is given. Some $ x$ of its edges are colored red so that each triangle has at most one red edge. The maximum number of vertices in $ G$ that induce a bipartite graph equals $ y.$ Prove that $ n\ge 4x/y.$
1971 IMO Longlists, 3
Let $a, b, c$ be positive real numbers, $0 < a \leq b \leq c$. Prove that for any positive real numbers $x, y, z$ the following inequality holds:
\[(ax+by+cz) \left( \frac xa + \frac yb+\frac zc \right) \leq (x+y+z)^2 \cdot \frac{(a+c)^2}{4ac}.\]
2014 PUMaC Team, 7
Let us consider a function $f:\mathbb{N}\to\mathbb{N}$ for which $f(1)=1$, $f(2n)=f(n)$ and $f(2n+1)=f(2n)+1$. Find the number of values at which the maximum value of $f(n)$ is attained for integer $n$ satisfying $0<n<2014$.
2013 Greece Team Selection Test, 3
Find the largest possible value of $M$ for which $\frac{x}{1+\frac{yz}{x}}+\frac{y}{1+\frac{zx}{y}}+\frac{z}{1+\frac{xy}{z}}\geq M$ for all $x,y,z>0$ with $xy+yz+zx=1$
Russian TST 2017, P1
The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?
2001 Moldova National Olympiad, Problem 5
Show that there are nine distinct nonzero integers such that their sum is a perfect square and the sum of any eight of them is a perfect cube.