Found problems: 85335
1999 Baltic Way, 5
The point $(a,b)$ lies on the circle $x^2+y^2=1$. The tangent to the circle at this point meets the parabola $y=x^2+1$ at exactly one point. Find all such points $(a,b)$.
2003 Tournament Of Towns, 6
Let $O$ be the center of insphere of a tetrahedron $ABCD$. The sum of areas of faces $ABC$ and $ABD$ equals the sum of areas of faces $CDA$ and $CDB$. Prove that $O$ and midpoints of $BC, AD, AC$ and $BD$ belong to the same plane.
2010 Tournament Of Towns, 5
A circle is divided by $2N$ points into $2N$ arcs of length $1$. These points are joined in pairs to form $N$ chords. Each chord divides the circle into two arcs, the length of each being an even integer. Prove that $N$ is even.
2021 Alibaba Global Math Competition, 11
Let $M$ be a compact orientable $2n$-manifold with boundary, where $n \ge 2$. Suppose that $H_0(M;\mathbb{Q}) \cong \mathbb{Q}$ and $H_i(M;\mathbb{Q})=0$ for $i>0$. Prove that the order of $H_{n-1}(\partial M; \mathbb{Z})$ is a square number.
1990 Greece National Olympiad, 3
In a triangle $ABC$ with medians $AD$ and $BE$ , holds that $\angle CAD= \angle CBE=30^o$. Prove that triangle $ABC$ is equilateral.
2021 BMT, 6
Compute the sum of all positive integers $n$ such that $n^n$ has 325 positive integer divisors. (For example, $4^4=256$ has 9 positive integer divisors: 1, 2, 4, 8, 16, 32, 64, 128, 256.)
2014 Sharygin Geometry Olympiad, 8
A convex polygon $P$ lies on a flat wooden table. You are allowed to drive some nails into the table. The nails must not go through $P$, but they may touch its boundary. We say that a set of nails blocks $P$ if the nails make it impossible to move $P$ without lifting it off the table. What is the minimum number of nails that suffices to block any convex polygon $P$?
(N. Beluhov, S. Gerdgikov)
2017 ASDAN Math Tournament, 3
Triangle $ABC$ has $AB=4,BC=6,CA=5$. Let $M$ be the midpoint of $\overline{BC}$ and $P$ the point on the circumcircle of $\triangle ABC$ such that $\angle MPA=90^\circ$. Let points $D$ and $E$ lie on $\overline{AC}$ and $\overline{AB}$ respectively such that $\overline{BD}\perp\overline{AC}$ and $\overline{CE}\perp\overline{AB}$. Find $\tfrac{PD}{PE}$.
2016 Saudi Arabia BMO TST, 3
Let $d$ be a positive integer. Show that for every integer $S$, there exist a positive integer $n$ and a sequence $a_1, ..., a_n \in \{-1, 1\}$ such that $S = a_1(1 + d)^2 + a_2(1 + 2d)^2 + ... + a_n(1 + nd)^2$.
2014 Purple Comet Problems, 1
In the diagram below $ABCD$ is a square and both $\triangle CFD$ and $\triangle CBE$ are equilateral. Find the degree measure of $\angle CEF$.
[asy]
size(4cm);
pen dps = fontsize(10);
defaultpen(dps);
pair temp = (1,0);
pair B = (0,0);
pair C = rotate(45,B)*temp;
pair D = rotate(270,C)*B;
pair A = rotate(270,D)*C;
pair F = rotate(60 ,D)*C;
pair E = rotate(60 ,C)*B;
label("$B$",B,SW*.5);
label("$C$",C,W*2);
label("$D$",D,NW*.5);
label("$A$",A,W);
label("$F$",F,N*.5);
label("$E$",E,S*.5);
draw(A--B--C--D--cycle^^D--F--C--E--B^^F--E);
[/asy]
1951 Moscow Mathematical Olympiad, 201
To prepare for an Olympiad $20$ students went to a coach. The coach gave them $20$ problems and it turned out that
(a) each of the students solved two problems and
(b) each problem was solved by twostudents.
Prove that it is possible to organize the coaching so that each student would discuss one of the problems that (s)he had solved, and so that all problems would be discussed.
2015 Switzerland - Final Round, 10
Find the largest natural number $n$ such that for all real numbers $a, b, c, d$ the following holds:
$$(n + 2)\sqrt{a^2 + b^2} + (n + 1)\sqrt{a^2 + c^2} + (n + 1)\sqrt{a^2 + d^2} \ge n(a + b + c + d)$$
2017 BMT Spring, 7
A light has been placed on every lattice point (point with integer coordinates) on the (infi nite) 2$D$ plane. De ne the Chebyshev distance between points $(x_1,y_1)$ and $(x_2, y_2)$ to be $\ max (|x_1 - x_2|, |y_1 -y_2|)$. Each light is turned on with probability $\frac{1}{2^{d/2}}$ , where $d$ is the Chebyshev distance from that point to the origin. What is expected number of lights that have all their directly adjacent lights turned on? (Adjacent points being points such that $|x_1-x_2|+|y_1- y_2| =1$.)
2021 Saudi Arabia IMO TST, 5
Let $ABC$ be a non isosceles triangle with incenter $I$ . The circumcircle of the triangle $ABC$ has radius $R$. Let $AL$ be the external angle bisector of $\angle BAC $with $L \in BC$. Let $K$ be the point on perpendicular bisector of $BC$ such that $IL \perp IK$.Prove that $OK=3R$.
2023 Singapore Senior Math Olympiad, 4
Find all positive integers $m, n$ satisfying $n!+2^{n-1}=2^m$.
1978 IMO Longlists, 10
Show that for any natural number $n$ there exist two prime numbers $p$ and $q, p \neq q$, such that $n$ divides their difference.
2008 Greece Junior Math Olympiad, 3
Find the greatest value of positive integer $ x$ , such that the number
$ A\equal{} 2^{182} \plus{} 4^x \plus{} 8^{700}$
is a perfect square .
1999 Tournament Of Towns, 6
On a large chessboard $2n$ of its $1 \times 1$ squares have been marked such thar the rook (which moves only horizontally or vertically) can visit all the marked squares without jumpin over any unmarked ones. Prove that the figure consisting of all the marked squares can be cut into rectangles.
(A Shapovalov)
2009 Stanford Mathematics Tournament, 9
Find the shortest distance between the point $(6,12)$ and the parabola given by the equation $x=\frac{y^2}{2}$
2017 CMIMC Number Theory, 3
For how many triples of positive integers $(a,b,c)$ with $1\leq a,b,c\leq 5$ is the quantity \[(a+b)(a+c)(b+c)\] not divisible by $4$?
2004 Oral Moscow Geometry Olympiad, 2
Construct a triangle $ABC$ given angle $A$ and the medians drawn from vertices $B$ and $C$.
2013 JBMO TST - Macedonia, 1
Let $ x $ be a real number such that $ x^3 $ and $ x^2+x $ are rational numbers. Prove that $ x $ is rational.
2018 Dutch IMO TST, 2
Suppose a triangle $\vartriangle ABC$ with $\angle C = 90^o$ is given. Let $D$ be the midpoint of $AC$, and let $E$ be the foot of the altitude through $C$ on $BD$. Show that the tangent in $C$ of the circumcircle of $\vartriangle AEC$ is perpendicular to $AB$.
2016 Kazakhstan National Olympiad, 3
Circles $\omega_1 , \omega_2$ intersect at points $X,Y$ and they are internally tangent to circle $\Omega$ at points $A,B$,respectively.$AB$ intersect with $\omega_1 , \omega_2$ at points $A_1,B_1$ ,respectively.Another circle is internally tangent to $\omega_1 , \omega_2$ and $A_1B_1$ at $Z$.Prove that $\angle AXZ =\angle BXZ$.(C.Ilyasov)
2013 Flanders Math Olympiad, 2
$2013$ smurfs are sitting at a large round table. Each of them has two tickets. on each card represents a number from $\{1, 2, . . ., 2013\}$ such that each of the numbers from this set occurs exactly twice. Every smurf takes the card every minute with the smaller of the two numbers, it smurfs on to its left neighbor and receives a card from his right neighbor. Show that there will come a time when a smurf has two cards with the same number.