Found problems: 85335
Kyiv City MO 1984-93 - geometry, 1987.9.4
Inscribe a triangle in a given circle, if its smallest side is known, as well as the point of intersection of altitudes lying outside the circle.
2017 IMO Shortlist, C1
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]
2010 China Team Selection Test, 1
Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings:
(1) $\sum_{v\in V} f(v)=|E|$;
(2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the number of uncolored vertices adjacent to $v$.
Let $m(G)$ be the number of good maps. Prove that if every vertex in $G$ is adjacent to at least one another vertex, then $n\leq m(G)\leq n!$.
2017 QEDMO 15th, 8
For a function $f: R\to R $ , $ f (2017)> 0$ as well as $f (x^2 + yf (z)) = xf (x) + zf (y)$ for all $x,y,z \in R$ is known. What is the value of $f (-42)$?
2009 Tournament Of Towns, 4
We only know that the password of a safe consists of $7$ different digits. The safe will open if we enter $7$ different digits, and one of them matches the corresponding digit of the password. Can we open this safe in less than $7$ attempts?
[i](5 points for Juniors and 4 points for Seniors)[/i]
2020 Ukrainian Geometry Olympiad - December, 3
Given convex $1000$-gon. Inside this polygon, $1020$ points are chosen so that no $3$ of the $2020$ points do not lie on one line. Polygon is cut into triangles so that these triangles have vertices only those specified $2020$ points and each of these points is the vertex of at least one of cutting triangles. How many such triangles were formed?
2010 Contests, 1
Solve in the integers the diophantine equation
$$x^4-6x^2+1 = 7 \cdot 2^y.$$
PEN N Problems, 14
One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.
1999 AMC 8, 14
In trapezoid $ABCD$ , the sides $AB$ and $CD$ are equal. The perimeter of $ABCD$ is
[asy]
draw((0,0)--(4,3)--(12,3)--(16,0)--cycle);
draw((4,3)--(4,0),dashed);
draw((3.2,0)--(3.2,.8)--(4,.8));
label("$A$",(0,0),SW);
label("$B$",(4,3),NW);
label("$C$",(12,3),NE);
label("$D$",(16,0),SE);
label("$8$",(8,3),N);
label("$16$",(8,0),S);
label("$3$",(4,1.5),E);[/asy]
$ \text{(A)}\ 27\qquad\text{(B)}\ 30\qquad\text{(C)}\ 32\qquad\text{(D)}\ 34\qquad\text{(E)}\ 48 $
2007 iTest Tournament of Champions, 1
Find the remainder when $3^{2007}$ is divided by $2007$.