Found problems: 85335
1967 Miklós Schweitzer, 3
Prove that if an infinite, noncommutative group $ G$ contains a proper normal subgroup with a commutative factor group, then $ G$ also contains an infinite proper normal subgroup.
[i]B. Csakany[/i]
2014 AMC 10, 3
Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip?
$ \textbf {(A) } 30 \qquad \textbf {(B) } \frac{400}{11} \qquad \textbf {(C) } \frac{75}{2} \qquad \textbf {(D) } 40 \qquad \textbf {(E) } \frac{300}{7} $
2015 German National Olympiad, 6
Prove that for all $x,y,z>0$, the inequality
\[\frac{x+y+z}{3}+\frac{3}{\frac{1}{x}+\frac{1}{y}+\frac{1}{z}} \ge 5 \sqrt[3]{\frac{xyz}{16}}\]
holds. Determine if equality can hold and if so, in which cases it occurs.
2008 Germany Team Selection Test, 3
Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$
[i]Author: Gerhard Wöginger, Netherlands[/i]
2006 Macedonia National Olympiad, 5
All segments joining $n$ points (no three of which are collinear) are coloured in one of $k$ colours. What is the smallest $k$ for which there always exists a closed polygonal line with the vertices at some of the $n$ points, whose sides are all of the same colour?
2012 Romania Team Selection Test, 3
Determine all finite sets $S$ of points in the plane with the following property: if $x,y,x',y'\in S$ and the closed segments $xy$ and $x'y'$ intersect in only one point, namely $z$, then $z\in S$.
1987 Traian Lălescu, 1.1
Consider the parabola $ P:x-y^2-(p+3)y-p=0,p\in\mathbb{R}^*. $ Show that $ P $ intersects the coordonate axis at three points, and that the circle formed by these three points passes through a fixed point.
2001 National Olympiad First Round, 3
How many primes $p$ are there such that $2p^4-7p^2+1$ is equal to square of an integer?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ \text{Infinitely many}
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2023 CMIMC Algebra/NT, 7
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Compute $\displaystyle \sum_{i=1}^{\phi(2023)} \dfrac{\gcd(i,\phi(2023))}{\phi(2023)}$.
[i]Proposed by Giacomo Rizzo[/i]
2019 Korea National Olympiad, 4
Let $(x_1, y_1, z_1), (x_2, y_2, z_2), \cdots, (x_{19}, y_{19}, z_{19})$ be integers. Prove that there exist pairwise distinct subscripts $i, j, k$ such that $x_i+x_j+x_k$, $y_i+y_j+y_k$, $z_i+z_j+z_k$ are all multiples of $3$.
2016 Switzerland - Final Round, 1
Let $ABC$ be a triangle with $\angle BAC = 60^o$. Let $E$ be the point on the side $BC$ , such that $2 \angle BAE = \angle ACB$ . Let $D$ be the second intersection of $AB$ and the circumcircle of the triangle $AEC$ and $P$ be the second intersection of $CD$ and the circumcircle of the triangle $DBE$. Calculate the angle $\angle BAP$.
2014 District Olympiad, 3
Let $(A,+,\cdot)$ be an unit ring with the property: for all $x\in A$,
\[ x+x^{2}+x^{3}=x^{4}+x^{5}+x^{6} \]
[list=a]
[*]Let $x\in A$ and let $n\geq2$ be an integer such that $x^{n}=0$. Prove that $x=0$.
[*]Prove that $x^{4}=x$, for all $x\in A$.[/list]
2017 Princeton University Math Competition, A4/B6
The four faces of a tetrahedral die are labelled $0, 1, 2,$ and $3,$ and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the face labelled $0,$ it disappears. If it lands on the face labelled $1,$ nothing happens. If it lands on the face labelled $2$ or $3,$ there will then be $2$ or $3$ copies of the die, respectively (including the original). Suppose the die and all its copies are continually rolled, and let $p$ be the probability that they will all eventually disappear. Find $\left\lfloor \frac{10}{p} \right\rfloor$.
1997 National High School Mathematics League, 10
Bottom surface of triangular pyramid $S-ABC$ is an isosceles right triangle (hypotenuse is $AB$). $SA=SB=SC=AB=2$, and $S,A,B,C$ are on a sphere with center of $O$. The distance of $O$ to plane $ABC$ is________.
1998 Hong kong National Olympiad, 1
In a concyclic quadrilateral $PQRS$,$\angle PSR=\frac{\pi}{2}$ , $H,K$ are perpendicular foot from $Q$ to sides $PR,RS$ , prove that $HK$ bisect segment$SQ$.
2007 France Team Selection Test, 2
Let $a,b,c,d$ be positive reals such taht $a+b+c+d=1$.
Prove that: \[6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}.\]
2020 Purple Comet Problems, 29
Find the number of distinguishable $2\times 2\times 2$ cubes that can be formed by gluing together two blue, two green, two red, and two yellow $1\times 1\times 1$ cubes. Two cubes are indistinguishable if one can be rotated so that the two cubes have identical coloring patterns.
2013 Macedonian Team Selection Test, Problem 2
a) Denote by $S(n)$ the sum of digits of a positive integer $n$. After the decimal point, we write one after the other the numbers $S(1),S(2),...$. Show that the number obtained is irrational.
b) Denote by $P(n)$ the product of digits of a positive integer $n$. After the decimal point, we write one after the other the numbers $P(1),P(2),...$. Show that the number obtained is irrational.
2014 Thailand TSTST, 2
Let $a, b, c$ be positive real numbers. Prove that $$\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}\geq\sqrt{\frac{2ab}{3a+b+2c}}+\sqrt{\frac{2bc}{3b+c+2a}}+\sqrt{\frac{2ca}{3c+a+2b}}.$$
2023-24 IOQM India, 8
Given a $2 \times 2$ tile and seven dominoes ( $2 \times 1$ tile), find the number of ways of tiling (that is, cover without leaving gaps and without overlapping of any two tiles) a $2 \times 7$ rectangle using some of these tiles.
2013 Mexico National Olympiad, 6
Let $A_1A_2 ... A_8$ be a convex octagon such that all of its sides are equal and its opposite sides are parallel. For each $i = 1, ... , 8$, define $B_i$ as the intersection between segments $A_iA_{i+4}$ and $A_{i-1}A_{i+1}$, where $A_{j+8} = A_j$ and $B_{j+8} = B_j$ for all $j$. Show some number $i$, amongst 1, 2, 3, and 4 satisfies
\[\frac{A_iA_{i+4}}{B_iB_{i+4}} \leq \frac{3}{2}\]
2024 Romania EGMO TST, P3
Given acute angle triangle $ ABC$. Let $ CD$be the altitude , $ H$ be the orthocenter and $ O$ be the circumcenter of $ \triangle ABC$ The line through point $ D$ and perpendicular with $ OD$ , is intersect $ BC$ at $ E$. Prove that $ \angle DHE \equal{} \angle ABC$.
1971 Kurschak Competition, 2
Given any $22$ points in the plane, no three collinear. Show that the points can be divided into $11$ pairs, so that the $11$ line segments defined by the pairs have at least five different intersections
1960 Miklós Schweitzer, 3
[b]3.[/b] Let $f(z)$ with $f(0)=1$ be regular in the unit disk and let
$\left [\frac{\partial^2 \mid f(z)\mid}{\partial x\partial y} \right ] _{z=0} =1$.
Show thatthe area of the image of the unit disk by $w= f(z)$ (taken with multiplicity) is not less than $\frac {1} {2}$ .[b](f. 6)[/b]
2023 Czech-Polish-Slovak Match, 4
Let $p, q$ and $r$ be positive real numbers such that the equation $$\lfloor pn \rfloor + \lfloor qn \rfloor + \lfloor rn \rfloor = n$$ is satisfied for infinitely many positive integers $n{}$.
(a) Prove that $p, q$ and $r$ are rational.
(b) Determine the number of positive integers $c$ such that there exist positive integers $a$ and $b$, for which the equation $$\left \lfloor \frac{n}{a} \right \rfloor+\left \lfloor \frac{n}{b} \right \rfloor+\left \lfloor \frac{cn}{202} \right \rfloor=n$$ is satisfied for infinitely many positive integers $n{}$.