Found problems: 85335
2021 Israel TST, 3
What is the smallest value of $k$ for which the inequality
\begin{align*}
ad-bc+yz&-xt+(a+c)(y+t)-(b+d)(x+z)\leq \\
&\leq k\left(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}+\sqrt{x^2+y^2}+\sqrt{z^2+t^2}\right)^2
\end{align*}
holds for any $8$ real numbers $a,b,c,d,x,y,z,t$?
Edit: Fixed a mistake! Thanks @below.
2009 CHKMO, 4
There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$.
2022 AMC 10, 13
Let $\triangle ABC$ be a scalene triangle. Point $P$ lies on $\overline{BC}$ so that $\overline{AP}$ bisects $\angle BAC$. The line through $B$ perpendicular to $\overline{AP}$ intersects the line through $A$ parallel to $\overline{BC}$ at point $D$. Suppose $BP = 2$ and $PC = 3$. What is $AD$ ?
$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }12$
2009 IMC, 3
In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from $1$ to $n$ and let $a_i$ be the number of friends of the $i^{\text{th}}$ resident. Suppose that
\[ \sum_{i=1}^{n}a_i^2=n^2-n \]
Let $k$ be the smallest number of residents (at least three) who can be seated at a round table in such a way that any two neighbors are friends. Determine all possible values of $k.$
2018 Peru IMO TST, 1
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
2013 Romania Team Selection Test, 3
Given a positive integer $n$, consider a triangular array with entries $a_{ij}$ where $i$ ranges from $1$ to $n$ and $j$ ranges from $1$ to $n-i+1$. The entries of the array are all either $0$ or $1$, and, for all $i > 1$ and any associated $j$ , $a_{ij}$ is $0$ if $a_{i-1,j} = a_{i-1,j+1}$, and $a_{ij}$ is $1$ otherwise. Let $S$ denote the set of binary sequences of length $n$, and define a map $f \colon S \to S$ via $f \colon (a_{11}, a_{12},\cdots ,a_{1n}) \to (a_{n1}, a_{n-1,2}, \cdots , a_{1n})$. Determine the number of fixed points of $f$.
2006 May Olympiad, 2
Several prime numbers (some repeated) are written on the board. Mauro added the numbers on the board and Fernando multiplied the numbers on the board. The result obtained by Fernando is equal to $40$ times the result obtained by Mauro. Determine what the numbers on the board can be. Give all chances.
1964 IMO Shortlist, 6
In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centrod if $\triangle ABC$. Line parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_2, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result if point $D_o$ is selected anywhere within $\triangle ABC$?
1998 All-Russian Olympiad, 6
A binary operation $*$ on real numbers has the property that $(a * b) * c = a+b+c$ for all $a$, $b$, $c$. Prove that $a * b = a+b$.
2009 Belarus Team Selection Test, 3
Given trapezoid $ABCD$ ($AD\parallel BC$) with $AD \perp AB$ and $T=AC\cap BD$. A circle centered at point $O$ is inscribed in the trapezoid and touches the side $CD$ at point $Q$. Let $P$ be the intersection point (different from $Q$) of the side $CD$ and the circle passing through $T,Q$ and $O$. Prove that $TP \parallel AD$.
I. Voronovich
2012 239 Open Mathematical Olympiad, 5
Point $M$ is the midpoint of the base $AD$ of trapezoid $ABCD$ inscribed in circle $S$. Rays $AB$ and $DC$ intersect at point $P$, and ray $BM$ intersects $S$ at point $K$. The circumscribed circle of triangle $PBK$ intersects line $BC$ at point $L$. Prove that $\angle{LDP} = 90^{\circ}$.
1996 Estonia National Olympiad, 5
Suppose that $n$ teterahedra are given in space such that any two of them have at least two common vertices, but any three have at most one common vertex. Find the greatest possible $n$.
1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 1
In a class, some pupils learn German, the other learn French. The number of girls learning French and the number of boys learning German total to 16. There are 11 pupils learning French, and there are 10 girls in the class. In addition to the girls learning French, there are 16 pupils. How many pupils are there in the class?
A. 18
B. 21
C. 23
D. 27
E. 31
2019 IOM, 1
Three prime numbers $p,q,r$ and a positive integer $n$ are given such that the numbers
\[ \frac{p+n}{qr}, \frac{q+n}{rp}, \frac{r+n}{pq} \]
are integers. Prove that $p=q=r $.
[i]Nazar Agakhanov[/i]
LMT Guts Rounds, 2020 F12
If the value of the infinite sum
$$\frac{1}{2^2-1^2}+\frac{1}{4^2-2^2}+\frac{1}{8^2-4^2}+\frac{1}{16^2-8^2}+\dots.$$
can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a,b,$ evaluate $a+b.$
[i]Proposed by Alex Li[/i]
2005 Georgia Team Selection Test, 10
Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that
\[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]
2002 Iran MO (3rd Round), 14
A subset $S$ of $\mathbb N$ is [i]eventually linear[/i] iff there are $k,N\in\mathbb N$ that for $n>N,n\in S\Longleftrightarrow k|n$. Let $S$ be a subset of $\mathbb N$ that is closed under addition. Prove that $S$ is eventually linear.
2015 HMNT, 8
Consider an $8\times 8$ grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e. it cannot stay in the same square). What is the expected number of minutes until the rook reaches the upper right corner?
2011 China Girls Math Olympiad, 8
The $A$-excircle $(O)$ of $\triangle ABC$ touches $BC$ at $M$. The points $D,E$ lie on the sides $AB,AC$ respectively such that $DE\parallel BC$. The incircle $(O_1)$ of $\triangle ADE$ touches $DE$ at $N$. If $BO_1\cap DO=F$ and $CO_1\cap EO=G$, prove that the midpoint of $FG$ lies on $MN$.
2012 Polish MO Finals, 3
Triangle $ABC$ with $AB = AC$ is inscribed in circle $o$. Circles $o_1$ and $o_2$ are internally tangent to circle $o$ in points $P$ and $Q$, respectively, and they are tangent to segments $AB$ and $AC$, respectively, and they are disjoint with the interior of triangle $ABC$. Let $m$ be a line tangent to circles $o_1$ and $o_2$, such that points $P$ and $Q$ lie on the opposite side than point $A$. Line $m$ cuts segments $AB$ and $AC$ in points $K$ and $L$, respectively. Prove, that intersection point of lines $PK$ and $QL$ lies on bisector of angle $BAC$.
2013 Ukraine Team Selection Test, 12
$4026$ points were noted on the plane, not three of which lie on a straight line.
The $2013$ points are the vertices of a convex polygon, and the other $2013$ vertices are inside this polygon. It is allowed to paint each point in one of two colors. Coloring will be good if some pairs of dots can be combined segments with the following conditions:
$\bullet$ Each segment connects dots of the same color.
$\bullet$ No two drawn segments intersect at their inner points.
$\bullet$ For an arbitrary pair of dots of the same color, there is a path along the lines from one point to another.
Please note that the sides of the convex $2013$ rectangle are not automatically drawn segments, although some (or all) can be drawn as needed. Prove that the total number of good colors does not depend on the specific locations of the points and find that number.
1975 Putnam, B5
Define $f_{0}(x)=e^x$ and $f_{n+1}(x)=x f_{n}'(x)$. Show that $\sum_{n=0}^{\infty} \frac{f_{n}(1)}{n!}=e^e$.
1999 North Macedonia National Olympiad, 3
Let the two tangents from a point $A$ outside a circle $k$ touch $k$ at $M$ and $N$. A line $p$ through $A$ intersects $k$ at $B$ and $C$, and $D$ is the midpoint of $MN$. Prove that $MN$ bisects the angle $BDC$
2021 Brazil Undergrad MO, Problem 1
Consider the matrices like
$$M=
\left(
\begin{array}{ccc}
a & b & c \\
c & a & b \\
b & c & a
\end{array}
\right)$$
such that $det(M) = 1$.
Show that
a) There are infinitely many matrices like above with $a,b,c \in \mathbb{Q}$
b) There are finitely many matrices like above with $a,b,c \in \mathbb{Z}$
2014 India IMO Training Camp, 1
Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]