Found problems: 85335
1963 IMO, 5
Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$
1967 IMO Longlists, 21
Without using tables, find the exact value of the product:
\[P = \prod^7_{k=1} \cos \left(\frac{k \pi}{15} \right).\]
2012 Kosovo National Mathematical Olympiad, 3
Solve the recurrence $R_0=1, R_n=nR_{n-1}+2^n\cdot n!$.
2005 Gheorghe Vranceanu, 2
Let be a twice-differentiable function $ f:(0,\infty )\longrightarrow\mathbb{R} $ that admits a polynomial function of degree $ 1 $ or $ 2, $ namely, $ \alpha :(0,\infty )\longrightarrow\mathbb{R} $ as its asymptote. Prove the following propositions:
[b]a)[/b] $ f''>0\implies f-\alpha >0 $
[b]b)[/b] $ \text{supp} f''=(0,\infty )\wedge f-\alpha >0\implies f''=0 $
2014 CHMMC (Fall), 6
Suppose the transformation $T$ acts on points in the plane like this:
$$T(x, y) = \left( \frac{x}{x^2 + y^2}, \frac{-y}{x^2 + y^2}\right).$$
Determine the area enclosed by the set of points of the form $T(x, y)$, where $(x, y)$ is a point on the edge of a length-$2$ square centered at the origin with sides parallel to the axes.
1996 Singapore Team Selection Test, 3
Let $S$ be a sequence $n_1, n_2,..., n_{1995}$ of positive integers such that $n_1 +...+ n_{1995 }=m < 3990$. Prove that for each integer $q$ with $1 \le q \le m$, there is a sequence $n_{i_1} , n_{i_2} , ... , n_{i_k}$ , where $1 \le i_1 < i_2 < ...< i_k \le 1995$, $n_{i_1} + ...+ n_{i_k} = q$ and $k$ depends on $q$.
1999 All-Russian Olympiad Regional Round, 8.1
A father and two sons went to visit their grandmother, who Raya lives $33$ km from the city. My father has a motor roller, the speed of which $25$ km/h, and with a passenger - $20$ km/h (with two passengers on a scooter It’s impossible to move). Each of the brothers walks along the road at a speed of $5$ km/h. Prove that all three can get to grandma's in $3$ hours
2000 Estonia National Olympiad, 4
On the side $AC$ of the triangle $ABC$, choose any point $D$ different from the vertices $A$ and C. Let $O_1$ and $O_2$ be circumcenters the triangles $ABD$ and $CBD$, respectively. Prove that the triangles $O_1DO_2$ and $ABC$ are similar.
1995 All-Russian Olympiad Regional Round, 11.2
A planar section of a parallelepiped is a regular hexagon. Show that this parallelepiped is a cube.
1996 AMC 12/AHSME, 23
The sum of the lengths of the twelve edges of a rectangular box is $140$, and the distance from one corner of the box to the farthest corner is $21$. The total surface area of the box is
$\text{(A)}\ 776 \qquad \text{(B)}\ 784 \qquad \text{(C)}\ 798 \qquad \text{(D)}\ 800 \qquad \text{(E)}\ 812$
2007 Belarusian National Olympiad, 5
Let $O$ be the intersection point of the diagonals of the convex quadrilateral $ABCD$, $AO = CO$. Points $P$ and $Q$ are marked on the segments $AO$ and $CO$, respectively, such that $PO = OQ$. Let $N$ and $K$ be the intersection points of the sides $AB$, $CD$, and the lines $DP$ and $BQ$ respectively.
Prove that the points $N$, $O$, and $K$ are colinear.
2009 Princeton University Math Competition, 7
We have a $6 \times 6$ square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw?
2010 Indonesia TST, 1
Sequence ${u_n}$ is defined with $u_0=0,u_1=\frac{1}{3}$ and
$$\frac{2}{3}u_n=\frac{1}{2}(u_{n+1}+u_{n-1})$$ $\forall n=1,2,...$
Show that $|u_n|\leq1$ $\forall n\in\mathbb{N}.$
2002 Kazakhstan National Olympiad, 8
$ N $ grasshoppers are lined up in a row. At any time, one grasshopper is allowed to jump over exactly two grasshoppers standing to the right or left of him. At what $ n $ can grasshoppers rearrange themselves in reverse order?
1984 Vietnam National Olympiad, 1
$(a)$ Find a polynomial with integer coefficients of the smallest degree having $\sqrt{2} + \sqrt[3]{3}$ as a root.
$(b)$ Solve $1 +\sqrt{1 + x^2}(\sqrt{(1 + x)^3}-\sqrt{(1- x)^3}) = 2\sqrt{1 - x^2}$.
2002 AMC 12/AHSME, 11
Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages $ 40$ miles per hour, he arrives at his workplace three minutes late. When he averages $ 60$ miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time?
$ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 58$
2013 Junior Balkan Team Selection Tests - Romania, 2
Weights of $1$ g, $2$ g,$ ...$ , $200$ g are placed on the two pans of a balance such that on each pan there are $100$ weights and the balance is in equilibrium. Prove that one can swap $50$ weights from one pan with $50$ weights from the other pan such that the balance remains in equilibrium.
Kvant Magazine
2019 Yasinsky Geometry Olympiad, p6
In the triangle $ABC$ it is known that $BC = 5, AC - AB = 3$. Prove that $r <2$ .
(here $r$ is the radius of the circle inscribed in the triangle $ABC$).
(Mykola Moroz)
2012-2013 SDML (Middle School), 2
If $\frac{a}{3}=b$ and $\frac{b}{4}=c$, what is the value of $\frac{ab}{c^2}$?
$\text{(A) }12\qquad\text{(B) }36\qquad\text{(C) }48\qquad\text{(D) }60\qquad\text{(E) }144$
2008 JBMO Shortlist, 5
Find all triples $(x, y, z)$ of real positive numbers, which satisfy the system $\begin{cases} \frac{1}{x}+\frac{4}{y}+\frac{9}{z}=3 \\ x + y + z \le 12 \end{cases}$
2010 Regional Olympiad of Mexico Northeast, 1
Sofia has $5$ pieces of paper on a table. He takes some of the pieces, cuts each one into $5$ little pieces, and puts them back on the table. She repeats this procedure several times until she gets tired. Could Sofia end up with $2010$ pieces on the table?
2016 Romania Team Selection Tests, 2
Given a positive integer $k$ and an integer $a\equiv 3 \pmod{8}$, show that $a^m+a+2$ is divisible by $2^k$ for some positive integer $m$.
1976 Putnam, 5
In the $(x,y)-$plane, if $R$ is the set of points inside and on a convex polygon, let $D(x,y)$ be the distance from $(x,y)$ to the nearest point of $R.$
(a) Show that there exists constants $a,b,c,$ independent of $R$, such that $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-D(x,y)} dxdy =a+bL+cA,$$ where $L$ is the perimeter of $R$ and $A$ is the area of $R.$
(b) Find the values of $a,b$ and $c.$
2018 Junior Balkan Team Selection Tests - Romania, 1
Determine the positive integers $n \ge 3$ such that, for every integer $m \ge 0$, there exist integers $a_1, a_2,..., a_n$ such that $a_1 + a_2 +...+ a_n = 0$ and $a_1a_2 + a_2a_3 + ...+a_{n-1}a_n + a_na_1 = -m$
Alexandru Mihalcu
1982 IMO Longlists, 56
Let $f(x) = ax^2 + bx+ c$ and $g(x) = cx^2 + bx + a$. If $|f(0)| \leq 1, |f(1)| \leq 1, |f(-1)| \leq 1$, prove that for $|x| \leq 1$,
[b](a)[/b] $|f(x)| \leq 5/4$,
[b](b)[/b] $|g(x)| \leq 2$.