Found problems: 85335
2021 Canadian Junior Mathematical Olympiad, 2
How many ways are there to permute the first $n$ positive integers such that in the permutation, for each value of $k \le n$, the first $k$ elements of the permutation have distinct remainder mod $k$?
1992 All Soviet Union Mathematical Olympiad, 572
Half the cells of a $2m \times n$ board are colored black and the other half are colored white. The cells at the opposite ends of the main diagonal are different colors. The center of each black cell is connected to the center of every other black cell by a straight line segment, and similarly for the white cells. Show that we can place an arrow on each segment so that it becomes a vector and the vectors sum to zero.
2017 Bosnia Herzegovina Team Selection Test, 5
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that
\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]
1985 Canada National Olympiad, 1
The lengths of the sides of a triangle are 6, 8 and 10 units. Prove that there is exactly one straight line which simultaneously bisects the area and perimeter of the triangle.
2019 Belarus Team Selection Test, 5.2
Let $AA_1$ be the bisector of a triangle $ABC$. Points $D$ and $F$ are chosen on the line $BC$ such that $A_1$ is the midpoint of the segment $DF$. A line $l$, different from $BC$, passes through $A_1$ and intersects the lines $AB$ and $AC$ at points $B_1$ and $C_1$, respectively.
Find the locus of the points of intersection of the lines $B_1D$ and $C_1F$ for all possible positions of $l$.
[i](M. Karpuk)[/i]
2009 Flanders Math Olympiad, 1
In an attempt to beat the Belgian handshake record come on $20/09/2009$ exactly $2009$ Belgians together in a large sports hall. Among them are Nathalie and thomas. During this event, everyone shakes hands with everyone exactly once other attendees. Afterwards, Nathalie says: “I have exactly $5$ times as many Flemish people shaken hands as people from Brussels.” Thomas replies with “I have exactly $3$ times as much Walloons and Brussels people shook hands”. From which region does Nathalie come and from which region comes Thomas?
2023 Abelkonkurransen Finale, 3a
Find all non-negative integers $n$, $a$, and $b$ satisfying
\[2^a + 5^b + 1 = n!.\]
2016 China Team Selection Test, 2
Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.
2014 Saint Petersburg Mathematical Olympiad, 7
$I$ - incenter , $M$- midpoint of arc $BAC$ of circumcircle, $AL$ - angle bisector of triangle $ABC$. $MI$ intersect circumcircle in $K$. Circumcircle of $AKL$ intersect $BC$ at $L$ and $P$.
Prove that $\angle AIP=90$
2023 SG Originals, Q4
On a connected graph $G$, one may perform the following operations:
[list]
[*]choose a vertice $v$, and add a vertice $v'$ such that $v'$ is connected to $v$ and all of its neighbours
[*] choose a vertice $v$ with odd degree and delete it
[/list]
Show that for any connected graph $G$, we may perform a finite number of operations such that the resulting graph is a clique.
Proposed by [i]idonthaveanaopsaccount[/i]
2015 Paraguay Juniors, 5
Camila creates a pattern to write the following numbers:
$2, 4$
$5, 7, 9, 11$
$12, 14, 16, 18, 20, 22$
$23, 25, 27, 29, 31, 33, 35, 37$
$…$
Following the same pattern, what is the sum of the numbers in the tenth row?
IV Soros Olympiad 1997 - 98 (Russia), 10.10
A circle touches the extensions of sides $CA$ and $CB$ of triangle $ABC$, and also touches side $AB$ of this triangle at point $P$. Prove that the radius of the circle tangent to segments $AP$, $CP$ and the circumscribed circle of this triangle is equal to the radius of the inscribed circle in this triangle.
2011 Today's Calculation Of Integral, 751
Find $\lim_{n\to\infty}\left(\frac{1}{n}\int_0^n (\sin ^ 2 \pi x)\ln (x+n)dx-\frac 12\ln n\right).$
2021 China Team Selection Test, 3
Determine the greatest real number $ C $, such that for every positive integer $ n\ge 2 $, there exists $ x_1, x_2,..., x_n \in [-1,1]$, so that
$$\prod_{1\le i<j\le n}(x_i-x_j) \ge C^{\frac{n(n-1)}{2}}$$.
2006 Portugal MO, 4
In the parallelogram $[ABCD], E$ is the midpoint of $[AD]$ and $F$ the orthogonal projection of $B$ on $[CE]$. Prove that the triangle $[ABF]$ is isosceles.
[img]https://1.bp.blogspot.com/-DLmFg8ayEQ4/X4XMohA5TjI/AAAAAAAAMnk/thlIKnNUiCkuu9cg1Aq7Zltz8SenmFWuwCLcBGAsYHQ/s0/2006%2Bportugal%2Bp4.png[/img]
2024 Indonesia Regional, 4
Find the number of positive integer pairs $1\leqslant a,b \leqslant 2027$ that satisfy
\[ 2027 \mid a^6+b^5+b^2.\]
(Note: For integers $a$ and $b$, the notation $a \mid b$ means that there is an integer $c$ such that $ac=b$.)
[i]Proposed by Valentio Iverson, Indonesia[/i]
2005 Olympic Revenge, 2
Let $\Gamma$ be a circumference, and $A,B,C,D$ points of $\Gamma$ (in this order).
$r$ is the tangent to $\Gamma$ at point A.
$s$ is the tangent to $\Gamma$ at point D.
Let $E=r \cap BC,F=s \cap BC$.
Let $X=r \cap s,Y=AF \cap DE,Z=AB \cap CD$
Show that the points $X,Y,Z$ are collinear.
Note: assume the existence of all above points.
VMEO II 2005, 12
a) Find all real numbers $k$ such that there exists a positive constant $c_k$ satisfying $$(x^2 + 1)(y^2 + 1)(z^2 + 1) \ge c_k(x + y + z)^k$$ for all positive real numbers.
b) Given the numbers $k$ found, determine the largest number $c_k$.
2004 Postal Coaching, 10
A convex quadrilateral $ABCD$ has an incircle. In each corner a circle is inscribed that also externally touches the two circles inscribed in the adjacent corners. Show that at least two circles have the same size.
2015 Geolympiad Summer, 4.
Let $ABC$ be a triangle and $I$ be its incenter. Let $D$ be the intersection of the exterior bisectors of $\angle BAC$ and $\angle BIC$, $E$ be the intersection of the exterior bisectors of $\angle ABC$ and $\angle AIC$, and $F$ be the intersection of the exterior bisectors of $\angle ACB$ and $\angle AIB$. Prove that $D$, $E$, $F$ are collinear
2020 Ukrainian Geometry Olympiad - April, 3
Let $H$ be the orthocenter of the acute-angled triangle $ABC$. Inside the segment $BC$ arbitrary point $D$ is selected. Let $P$ be such that $ADPH$ is a parallelogram. Prove that $\angle BCP< \angle BHP$.
1999 AMC 12/AHSME, 21
A circle is circumscribed about a triangle with sides $ 20$, $ 21$, and $ 29$, thus dividing the interior of the circle into four regions. Let $ A$, $ B$, and $ C$ be the areas of the non-triangular regions, with $ C$ being the largest. Then
$ \textbf{(A)}\ A \plus{} B \equal{} C\qquad
\textbf{(B)}\ A \plus{} B \plus{} 210 \equal{} C\qquad
\textbf{(C)}\ A^2 \plus{} B^2 \equal{} C^2\qquad \\
\textbf{(D)}\ 20A \plus{} 21B \equal{} 29C\qquad
\textbf{(E)}\ \frac{1}{A^2} \plus{} \frac{1}{B^2} \equal{} \frac{1}{C^2}$
2002 Baltic Way, 19
Let $n$ be a positive integer. Prove that the equation
\[x+y+\frac{1}{x}+\frac{1}{y}=3n\]
does not have solutions in positive rational numbers.
1990 Romania Team Selection Test, 11
In a group of $n$ persons,
(i) each person is acquainted to exactly $k$ others,
(ii) any two acquainted persons have exactly $l$ common acquaintances,
(iii) any two non-acquainted persons have exactly $m$ common acquaintances.
Prove that $m(n-k -1) = k(k -l -1)$.
2000 Swedish Mathematical Competition, 4
The vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$.