Found problems: 85335
2002 Balkan MO, 1
Consider $n$ points $A_1,A_2,A_3,\ldots, A_n$ ($n\geq 4$) in the plane, such that any three are not collinear. Some pairs of distinct points among $A_1,A_2,A_3,\ldots, A_n$ are connected by segments, such that every point is connected with at least three different points. Prove that there exists $k>1$ and the distinct points $X_1,X_2,\ldots, X_{2k}$ in the set $\{A_1,A_2,A_3,\ldots, A_n\}$, such that for every $i\in \overline{1,2k-1}$ the point $X_i$ is connected with $X_{i+1}$, and $X_{2k}$ is connected with $X_1$.
2018 Dutch IMO TST, 1
(a) If $c(a^3+b^3) = a(b^3+c^3) = b(c^3+a^3)$ with $a, b, c$ positive real numbers,
does $a = b = c$ necessarily hold?
(b) If $a(a^3+b^3) = b(b^3+c^3) = c(c^3+a^3)$ with $a, b, c$ positive real numbers,
does $a = b = c$ necessarily hold?
2003 Czech And Slovak Olympiad III A, 5
Show that, for each integer $z \ge 3$, there exist two two-digit numbers $A$ and $B$ in base $z$, one equal to the other one read in reverse order, such that the equation $x^2 -Ax+B$ has one double root. Prove that this pair is unique for a given $z$. For instance, in base $10$ these numbers are $A = 18, B = 81$.
Today's calculation of integrals, 892
Evaluate $\int_0^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1+\cos x}\ dx.$
2020 USA IMO Team Selection Test, 2
Two circles $\Gamma_1$ and $\Gamma_2$ have common external tangents $\ell_1$ and $\ell_2$ meeting at $T$. Suppose $\ell_1$ touches $\Gamma_1$ at $A$ and $\ell_2$ touches $\Gamma_2$ at $B$. A circle $\Omega$ through $A$ and $B$ intersects $\Gamma_1$ again at $C$ and $\Gamma_2$ again at $D$, such that quadrilateral $ABCD$ is convex.
Suppose lines $AC$ and $BD$ meet at point $X$, while lines $AD$ and $BC$ meet at point $Y$. Show that $T$, $X$, $Y$ are collinear.
[i]Merlijn Staps[/i]
2020 Dutch BxMO TST, 5
A set S consisting of $2019$ (different) positive integers has the following property:
[i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i].
What is the maximum number of prime numbers that $S$ can contain?
2023 Silk Road, 2
Let $n$ be a positive integer. Each cell of a $2n\times 2n$ square is painted in one of the $4n^2$ colors (with some colors may be missing). We will call any two-cell rectangle a [i]domino[/I], and a domino is called [i]colorful[/I] if its cells have different colors. Let $k$ be the total number of colorful dominoes in our square; $l$ be the maximum integer such that every partition of the square into dominoes contains at least $l$ colorful dominoes. Determine the maximum possible value of $4l-k$ over all possible colourings of the square.
2022 239 Open Mathematical Olympiad, 2
Point $I{}$ is the center of the circle inscribed in the quadrilateral $ABCD$. Prove that there is a point $K{}$ on the ray $CI$ such that $\angle KBI=\angle KDI=\angle BAI$.
2022 Bolivia Cono Sur TST, P4
Find all right triangles with integer sides and inradius 6.
2018 Baltic Way, 18
Let $n \ge 3$ be an integer such that $4n+1$ is a prime number. Prove that $4n+1$ divides $n^{2n}-1$.
2009 Spain Mathematical Olympiad, 5
Let, $ a,b,c$ real positive numbers with $ abc \equal{} 1$
Prove:
$ (\frac {a}{1 \plus{} ab})^2 \plus{} (\frac {b}{1 \plus{} bc})^2 \plus{} (\frac {c}{1 \plus{} ca})^2\geq \frac {3}{4}$
Thanks!
2009 Math Prize For Girls Problems, 13
The figure below shows a right triangle $ \triangle ABC$.
[asy]unitsize(15);
pair A = (0, 4);
pair B = (0, 0);
pair C = (4, 0);
draw(A -- B -- C -- cycle);
pair D = (2, 0);
real p = 7 - 3sqrt(3);
real q = 4sqrt(3) - 6;
pair E = p + (4 - p)*I;
pair F = q*I;
draw(D -- E -- F -- cycle);
label("$A$", A, N);
label("$B$", B, S);
label("$C$", C, S);
label("$D$", D, S);
label("$E$", E, NE);
label("$F$", F, W);[/asy]
The legs $ \overline{AB}$ and $ \overline{BC}$ each have length $ 4$. An equilateral triangle $ \triangle DEF$ is inscribed in $ \triangle ABC$ as shown. Point $ D$ is the midpoint of $ \overline{BC}$. What is the area of $ \triangle DEF$?
1967 IMO Longlists, 49
Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]
2007-2008 SDML (Middle School), 3
If $n$ is a positive integer such that $1+2+3+\cdots+n=190$, then what is $n$?
1972 Swedish Mathematical Competition, 5
Show that
\[
\int\limits_0^1 \frac{1}{(1+x)^n} dx > 1-\frac{1}{n}
\]
for all positive integers $n$.
1973 Putnam, A6
Prove that it is impossible for seven distinct straight lines to be situated in the euclidean plane so as to have at least six points where exactly three of these lines intersect and at least four points where exactly two of these lines intersect.
2024 LMT Fall, 1
A positive integer $n$ is called "foursic'' if there exists a placement of $0$ in the digits of $n$ such that the resulting number a multiple of $4.$ For example, $14$ is foursic because $104$ is a multiple of $4.$ Find the number of two-digit foursic numbers.
2017 Rioplatense Mathematical Olympiad, Level 3, 2
One have $n$ distinct circles(with the same radius) such that for any $k+1$ circles there are (at least) two circles that intersects in two points. Show that for each line $l$ one can make $k$ lines, each one parallel with $l$, such that each circle has (at least) one point of intersection with some of these lines.
2021 239 Open Mathematical Olympiad, 3
Given is a simple graph with $239$ vertices, such that it is not bipartite and each vertex has degree at least $3$. Find the smallest $k$, such that each odd cycle has length at most $k$.
2013 Saudi Arabia IMO TST, 4
Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.
1994 Moldova Team Selection Test, 2
Prove that every positive rational number can be expressed uniquely as a finite sum of the form $$a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},$$ where $a_n$ are integers such that $0 \leq a_n \leq n-1$ for all $n > 1$.
2009 Today's Calculation Of Integral, 493
In the $ x \minus{} y$ plane, let $ l$ be the tangent line at the point $ A\left(\frac {a}{2},\ \frac {\sqrt {3}}{2}b\right)$ on the ellipse $ \frac {x^2}{a^2} \plus{} \frac {y^2}{b^2}\equal{}1\ (0 < b < 1 < a)$. Let denote $ S$ be the area of the figure bounded by $ l,$ the $ x$ axis and the ellipse.
(1) Find the equation of $ l$.
(2) Express $ S$ in terms of $ a,\ b$.
(3) Find the maximum value of $ S$ with the constraint $ a^2 \plus{} 3b^2 \equal{} 4$.
2012 May Olympiad, 4
Pedro has $111$ blue chips and $88$ white chips. There is a machine that for every $14$ blue chips , it gives $11$ white pieces and for every $7$ white chips, it gives $13$ blue pieces. Decide if Pedro can achieve, through successive operations with the machine, increase the total number of chips by $33$, so that the number of blue chips equals $\frac53$ of the amount of white chips. If possible, indicate how to do it. If not, indicate why.
2008 Thailand Mathematical Olympiad, 6
Let $f : R \to R$ be a function satisfying the inequality $|f(x + y) -f(x) - f(y)| < 1$ for all reals $x, y$.
Show that $\left| f\left( \frac{x}{2008 }\right) - \frac{f(x)}{2008} \right| < 1$ for all real numbers $x$.
2004 Thailand Mathematical Olympiad, 15
Find the largest positive integer $n \le 2004$ such that $3^{3n+3} - 27$ is divisible by $169$.