Found problems: 85335
2022 Putnam, A2
Let $n$ be an integer with $n\geq 2.$ Over all real polynomials $p(x)$ of degree $n,$ what is the largest possible number of negative coefficients of $p(x)^2?$
2018 Hanoi Open Mathematics Competitions, 13
A competition room of HOMC has $m \times n$ students where $m, n$ are integers larger than $2$. Their seats are arranged in $m$ rows and $n$ columns. Before starting the test, every student takes a handshake with each of his/her adjacent students (in the same row or in the same column). It is known that there are totally $27$ handshakes. Find the number of students in the room.
2021 Saudi Arabia JBMO TST, 3
We have $n > 2$ nonzero integers such that everyone of them is divisible by the sum of the other $n - 1$ numbers, Show that the sum of the $n$ numbers is precisely $0$.
2019 Auckland Mathematical Olympiad, 3
There is a finite number of polygons in a plane and each two of them have a point in common. Prove that there exists a line which crosses every polygon.
1997 Putnam, 1
A rectangle, $HOMF$, has sides $HO=11$ and $OM=5$. A triangle $\Delta ABC$ has $H$ as orthocentre, $O$ as circumcentre, $M$ be the midpoint of $BC$, $F$ is the feet of altitude from $A$. What is the length of $BC$ ?
[asy]
unitsize(0.3 cm);
pair F, H, M, O;
F = (0,0);
H = (0,5);
O = (11,5);
M = (11,0);
draw(H--O--M--F--cycle);
label("$F$", F, SW);
label("$H$", H, NW);
label("$M$", M, SE);
label("$O$", O, NE);
[/asy]
2001 AIME Problems, 1
Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$?
1985 Traian Lălescu, 1.3
Let $ G $ be a finite group of odd order having, at least, three elements. For $ a\in G $ denote $ n(a) $ as the number of ways $ a $ can be written as a product of two distinct elements of $ G. $
Prove that $ \sum_{\substack{a\in G\\a\neq\text{id}}} n(a) $ is a perfect square.
1969 Canada National Olympiad, 10
Let $ABC$ be the right-angled isosceles triangle whose equal sides have length 1. $P$ is a point on the hypotenuse, and the feet of the perpendiculars from $P$ to the other sides are $Q$ and $R$. Consider the areas of the triangles $APQ$ and $PBR$, and the area of the rectangle $QCRP$. Prove that regardless of how $P$ is chosen, the largest of these three areas is at least $2/9$.
2012 Sharygin Geometry Olympiad, 19
Two circles with radii 1 meet in points $X, Y$, and the distance between these points also is equal to $1$. Point $C$ lies on the first circle, and lines $CA, CB$ are tangents to the second one. These tangents meet the first circle for the second time in points $B', A'$. Lines $AA'$ and $BB'$ meet in point $Z$. Find angle $XZY$.
2007 ITest, 60
Let $T=\text{TNFTPP}$. Triangle $ABC$ has $AB=6T-3$ and $AC=7T+1$. Point $D$ is on $BC$ so that $AD$ bisects angle $BAC$. The circle through $A$, $B$, and $D$ has center $O_1$ and intersects line $AC$ again at $B'$, and likewise the circle through $A$, $C$, and $D$ has center $O_2$ and intersects line $AB$ again at $C'$. If the four points $B'$, $C'$, $O_1$, and $O_2$ lie on a circle, find the length of $BC$.
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]