Found problems: 85335
2012 CHMMC Fall, 3
A particular graph has $6$ vertices, $12$ edges, and has the property that it contains no Eulerian path; a Eulerian path is a route from vertex to vertex along edges that traces each edge exactly once. Determine all the possible degrees of its vertices in no particular order. There are two solutions, and you need to get both to get credit for this problem.
2024 Brazil EGMO TST, 2
Let \( n, k \geq 1 \). In a school, there are \( n \) students and \( k \) clubs. Each student participates in at least one of the clubs. One day, a school uniform was established, which could be either blue or red. Each student chose only one of these colors. Every day, the principal visited one of the clubs, forcing all the students in it to switch the colors of the uniforms they wore.
Assuming that the students are distributed in clubs in such a way that any initial choice of uniforms they make, after a certain number of days, it is possible to have at most one student with one of the colors. Show that
\[
n \geq 2^{n-k-1} - 1.
\]
2019 Korea Junior Math Olympiad., 7
Let $O$ be the circumcenter of an acute triangle $ABC$. Let $D$ be the intersection of the bisector of the angle $A$ and $BC$. Suppose that $\angle ODC = 2 \angle DAO$. The circumcircle of $ABD$ meets the line segment $OA$ and the line $OD$ at $E (\neq A,O)$, and $F(\neq D)$, respectively. Let $X$ be the intersection of the line $DE$ and the line segment $AC$. Let $Y$ be the intersection of the bisector of the angle $BAF$ and the segment $BE$. Prove that $\frac{\overline{AY}}{\overline{BY}}= \frac{\overline{EX}}{\overline{EO}}$.
2011 AMC 8, 13
Two congruent squares, $ABCD$ and $PQRS$, have side length $15$. They overlap to form the $15$ by $25$ rectangle $AQRD$ shown. What percent of the area of rectangle $AQRD$ is shaded?
[asy]
filldraw((0,0)--(25,0)--(25,15)--(0,15)--cycle,white,black);
label("D",(0,0),S);
label("R",(25,0),S);
label("Q",(25,15),N);
label("A",(0,15),N);
filldraw((10,0)--(15,0)--(15,15)--(10,15)--cycle,mediumgrey,black);
label("S",(10,0),S);
label("C",(15,0),S);
label("B",(15,15),N);
label("P",(10,15),N);
[/asy]
$\textbf{(A)}\ 15\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 25$
1993 Turkey MO (2nd round), 4
$a_{n}$ is a sequence of positive integers such that, for every $n\geq 1$, $0<a_{n+1}-a_{n}<\sqrt{a_{n}}$. Prove that for every $x,y\in{R}$ such that $0<x<y<1$ $x< \frac{a_{k}}{a_{m}}<y$ we can find such $k,m\in{Z^{+}}$.
1995 ITAMO, 5
Two non-coplanar circles in space are tangent at a point and have the same tangents at this point. Show that both circles lie on some sphere.
2015 AMC 10, 9
Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders?
$\textbf{(A) }\text{The second height is 10\% less than the first.}$
$\textbf{(B) }\text{The first height is 10\% more than the second.}$
$\textbf{(C) }\text{The second height is 21\% less than the first.}$
$\textbf{(D) }\text{The first height is 21\% more than the second.}$
$\textbf{(E) }\text{The second height is 80\% of the first.}$
1994 China Team Selection Test, 1
Find all sets comprising of 4 natural numbers such that the product of any 3 numbers in the set leaves a remainder of 1 when divided by the remaining number.
1992 Canada National Olympiad, 4
Solve the equation
\[ x^2 \plus{} \frac{x^2}{(x\plus{}1)^2} \equal{} 3\]
2022 Chile National Olympiad, 3
The $19$ numbers $472$ , $473$ , $...$ , $490$ are juxtaposed in some order to form a $57$-digit number. Can any of the numbers thus obtained be prime?
2021 CCA Math Bonanza, I4
Given that nonzero real numbers $x$ and $y$ satisfy $x+\frac{1}{y}=3$ and $y+\frac{1}{x}=4$, what is $xy+\frac{1}{xy}$?
[i]2021 CCA Math Bonanza Individual Round #4[/i]
2023 Chile National Olympiad, 1
Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.
2010 National Olympiad First Round, 16
$11$ different books are on a $3$-shelf bookcase. In how many different ways can the books be arranged such that at most one shelf is empty?
$ \textbf{(A)}\ 75\cdot 11!
\qquad\textbf{(B)}\ 62\cdot 11!
\qquad\textbf{(C)}\ 68\cdot 12!
\qquad\textbf{(D)}\ 12\cdot 13!
\qquad\textbf{(E)}\ 6 \cdot 13!
$
2019 China Western Mathematical Olympiad, 1
Determine all the possible positive integer $n,$ such that $3^n+n^2+2019$ is a perfect square.
2009 Sharygin Geometry Olympiad, 22
Construct a quadrilateral which is inscribed and circumscribed, given the radii of the respective circles and the angle between the diagonals of quadrilateral.
2012 Belarus Team Selection Test, 2
Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$.
[i]Proposed by Japan[/i]
2000 Harvard-MIT Mathematics Tournament, 2
How many positive solutions are there to $x^{10}+7x^9+14x^8+1729x^7-1379x^6=0$? How many positive integer solutions?
2011 Canadian Mathematical Olympiad Qualification Repechage, 5
Each vertex of a regular $11$-gon is colored black or gold. All possible triangles are formed using these vertices. Prove that there are either two congruent triangles with three black vertices or two congruent triangles with three gold vertices.
2025 Serbia Team Selection Test for the IMO 2025, 2
Let $ABC$ be an acute triangle. Let $A'$ be the reflection of point $A$ over the line $BC$. Let $O$ and $H$ be the circumcenter and the orthocenter of triangle $ABC$, respectively, and let $E$ be the midpoint of segment $OH$. Let $D$ and $L$ be the points where the reflection of line $AA'$ with respect to line $OA'$ intersects the circumcircle of triangle $ABC$, where point $D$ lies on the arc $BC$ not containing $A$. If \( M \) is a point on the line \( BC \) such that \( OM \perp AD \), prove that \( \angle MAD = \angle EAL \).
[i]Proposed by Strahinja Gvozdić[/i]
2012 Brazil Team Selection Test, 2
Into each box of a $ 2012 \times 2012 $ square grid, a real number greater than or equal to $ 0 $ and less than or equal to $ 1 $ is inserted. Consider splitting the grid into $2$ non-empty rectangles consisting of boxes of the grid by drawing a line parallel either to the horizontal or the vertical side of the grid. Suppose that for at least one of the resulting rectangles the sum of the numbers in the boxes within the rectangle is less than or equal to $ 1 $, no matter how the grid is split into $2$ such rectangles. Determine the maximum possible value for the sum of all the $ 2012 \times 2012 $ numbers inserted into the boxes.
VI Soros Olympiad 1999 - 2000 (Russia), 11.8
Prove that the plane dividing in equal proportions the surface area and volume of the circumscribed polyhedron passes through the center of the sphere inscribed in this polyhedron.
2006 China National Olympiad, 2
For positive integers $a_1,a_2 ,\ldots,a_{2006}$ such that $\frac{a_1}{a_2},\frac{a_2}{a_3},\ldots,\frac{a_{2005}}{a_{2006}}$ are pairwise distinct, find the minimum possible amount of distinct positive integers in the set$\{a_1,a_2,...,a_{2006}\}$.
2000 Tournament Of Towns, 4
In how many ways can $31$ squares be marked on an $8 \times 8$ chessboard so that no two of the marked squares have a common side?
(R Zhenodarov)
2020 Novosibirsk Oral Olympiad in Geometry, 2
Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?
2016 Ecuador NMO (OMEC), 5
Determine the number of positive integers $N = \overline{abcd}$, with $a, b, c, d$ nonzero digits, which satisfy $(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1$.