Found problems: 85335
2009 Indonesia TST, 4
Sixteen people for groups of four people such that each two groups have at most two members in common. Prove that there exists a set of six people in which every group is not properly contained in it.
1991 Romania Team Selection Test, 3
Prove the following identity for every $ n\in N$:
$ \sum_{j\plus{}h\equal{}n,j\geq h}\frac{(\minus{}1)^h2^{j\minus{}h}\binom{j}{h}}{j}\equal{}\frac{2}{n}$
1987 India National Olympiad, 9
Prove that any triangle having two equal internal angle bisectors (each measured from a vertex to the opposite side) is isosceles.
2007 Nicolae Coculescu, 4
Prove that $ p $ divides $ \varphi (1+a^p) , $ where $ a\ge 2 $ is a natural number, $ p $ is a prime, and $ \varphi $ is Euler's totient.
[i]Cristinel Mortici[/i]
1982 Tournament Of Towns, (029) 3
$60$ symbols, each of which is either $X$ or $O$, are written consecutively on a strip of paper. This strip must then be cut into pieces with each piece containing symbols symmetric about their centre, e.g. $O, XX, OXXXXX, XOX$, etc.
(a) Prove that there is a way of cutting the strip so that there are no more than $24$ such pieces.
(b) Give an example of such an arrangement of the signs for which the number of pieces cannot be less than $15$.
(c) Try to improve the result of (b).
1973 Bundeswettbewerb Mathematik, 2
In a planar lake, every point can be reached by a straight line from the point $A$. The same holds for the point $B$. Show that this holds for every point on the segment $[AB]$, too.
2009 All-Russian Olympiad Regional Round, 10.4
Circles $\omega_1$ and $\omega_2$ touch externally at the point $O$. Points $A$ and $B$ on the circle $\omega_1$ and points $C$ and $D$ on the circle $\omega_2$ are such that $AC$ and $BD$ are common external tangents to circles. Line $AO$ intersects segment $CD$ at point $M$ and straight line $CO$ intersexts $\omega_1$ again at point $N$. Prove that the points $B$, $M$ and $N$ lie on the same straight line.
2020 CCA Math Bonanza, I2
Circles $\omega$ and $\gamma$ are drawn such that $\omega$ is internally tangent to $\gamma$, the distance between their centers are $5$, and the area inside of $\gamma$ but outside of $\omega$ is $100\pi$. What is the sum of the radii of the circles?
[asy]
size(3cm);
real lw=0.4, dr=0.3;
real r1=14, r2=9;
pair A=(0,0), B=(r1-r2,0);
draw(A--B,dashed);
draw(circle(A,r1),linewidth(lw)); draw(circle(B,r2),linewidth(lw));
filldraw(circle(A,dr)); filldraw(circle(B,dr));
label("$5$",(A+B)/2,dir(-90));
label("$\gamma$",A+r1*dir(135),dir(135)); label("$\omega$",B+r2*dir(135),dir(135));
[/asy]
[i]2020 CCA Math Bonanza Individual Round #2[/i]
2017 Iran MO (3rd round), 3
Find all functions $f:\mathbb{R^+}\rightarrow\mathbb{R^+}$ such that
$$\frac{x+f(y)}{xf(y)}=f(\frac{1}{y}+f(\frac{1}{x}))$$
for all positive real numbers $x$ and $y$.
1984 IMO Longlists, 31
Let $f_1(x) = x^3+a_1x^2+b_1x+c_1 = 0$ be an equation with three positive roots $\alpha>\beta>\gamma > 0$. From the equation $f_1(x) = 0$, one constructs the equation $f_2(x) = x^3 +a_2x^2 +b_2x+c_2 = x(x+b_1)^2 -(a_1x+c_1)^2 = 0$. Continuing this process, we get equations $f_3,\cdots, f_n$. Prove that
\[\lim_{n\to\infty}\sqrt[2^{n-1}]{-a_n} = \alpha\]
Kharkiv City MO Seniors - geometry, 2018.11.4
The line $\ell$ parallel to the side $BC$ of the triangle $ABC$, intersects its sides $AB,AC$ at the points $D,E$, respectively. The circumscribed circle of triangle $ABC$ intersects line $\ell$ at points $F$ and $G$, such that points $F,D,E,G$ lie on line $\ell$ in this order. The circumscribed circles of the triangles $FEB$ and $DGC$ intersect at points $P$ and $Q$. Prove that points $A, P$ and $Q$ are collinear.
2007 IMC, 4
Let $ n > 1$ be an odd positive integer and $ A = (a_{ij})_{i, j = 1..n}$ be the $ n \times n$ matrix with
\[ a_{ij}= \begin{cases}2 & \text{if }i = j \\ 1 & \text{if }i-j \equiv \pm 2 \pmod n \\ 0 & \text{otherwise}\end{cases}.\]
Find $ \det A$.
1937 Moscow Mathematical Olympiad, 035
Given three points that are not on the same straight line. Three circles pass through each pair of the points so that the tangents to the circles at their intersection points are perpendicular to each other. Construct the circles.
2013 District Olympiad, 1
Prove that the equation
$$\frac{1}{\sqrt{x} +\sqrt{1006}}+\frac{1}{\sqrt{2012 -x} +\sqrt{1006}}=\frac{2}{\sqrt{x} +\sqrt{2012 -x}}$$
has $2013$ integer solutions.
2025 Ukraine National Mathematical Olympiad, 8.1
There are \(n\) numbers arranged in a circle, and each number equals the absolute value of the difference between its two neighbors. Is it necessarily true that all numbers are equal to zero if:
a) \(n=2025\);
b) \(n=2024\)?
[i]Proposed by Anton Trygub[/i]
2017 Bosnia And Herzegovina - Regional Olympiad, 4
It is given isosceles triangle $ABC$ ($AB=AC$) such that $\angle BAC=108^{\circ}$. Angle bisector of angle $\angle ABC$ intersects side $AC$ in point $D$, and point $E$ is on side $BC$ such that $BE=AE$. If $AE=m$, find $ED$
2024 Indonesia TST, 2
Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.
2007 Putnam, 2
Suppose that $ f: [0,1]\to\mathbb{R}$ has a continuous derivative and that $ \int_0^1f(x)\,dx\equal{}0.$
Prove that for every $ \alpha\in(0,1),$
\[ \left|\int_0^{\alpha}f(x)\,dx\right|\le\frac18\max_{0\le x\le 1}|f'(x)|\]
1995 All-Russian Olympiad, 1
Can the numbers $1,2,3,\ldots,100$ be covered with $12$ geometric progressions?
[i]A. Golovanov[/i]
2010 Today's Calculation Of Integral, 534
Find the indefinite integral $ \int \frac{x^3}{(x\minus{}1)^3(x\minus{}2)}\ dx$.
2005 Romania Team Selection Test, 4
We consider a polyhedra which has exactly two vertices adjacent with an odd number of edges, and these two vertices are lying on the same edge.
Prove that for all integers $n\geq 3$ there exists a face of the polyhedra with a number of sides not divisible by $n$.
2016 Latvia National Olympiad, 2
Triangle $ABC$ has incircle $\omega$ and incenter $I$. On its sides $AB$ and $BC$ we pick points $P$ and $Q$ respectively, so that $PI = QI$ and $PB > QB$. Line segment $QI$ intersects $\omega$ in $T$. Draw a tangent line to $\omega$ passing through $T$; it intersects the sides $AB$ and $BC$ in $U$ and $V$ respectively. Prove that $PU = UV + VQ$!
2004 Indonesia MO, 4
There exists 4 circles, $ a,b,c,d$, such that $ a$ is tangent to both $ b$ and $ d$, $ b$ is tangent to both $ a$ and $ c$, $ c$ is both tangent to $ b$ and $ d$, and $ d$ is both tangent to $ a$ and $ c$. Show that all these tangent points are located on a circle.
1995 Argentina National Olympiad, 6
The $27$ points $(a,b,c)$ of the space are marked such that $a$, $b$ and $c$ take the values $0$, $1$ or $2$. We will call these points "junctures". Using $54$ rods of length $1$, all the joints that are at a distance of $1$ are joined together. A cubic structure of $2\times 2\times 2$ is thus formed. An ant starts from a juncture $A$ and moves along the rods; When it reaches a juncture it turns $90^\circ$ and changes rod. If the ant returns to $A$ and has not visited any juncture more than once except $A$, which it visited $2$ times, at the beginning of the walk and at the end of it, what is the greatest length that the path of the ant can have?
2007 Thailand Mathematical Olympiad, 1
In a circle $\odot O$, radius $OA$ is perpendicular to radius $OB$. Chord $AC$ intersects $OB$ at $E$ so that the length of arc $AC$ is one-third the circumference of $\odot O$. Point $D$ is chosen on $OB$ so that $CD \perp AB$. Suppose that segment $AC$ is $2$ units longer than segment $OD$. What is the length of segment $AC$?