Found problems: 85335
1976 Euclid, 5
Source: 1976 Euclid Part A Problem 5
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If $\log_8 m+\log_8 \frac{1}{6}=\frac{2}{3}$, then $m$ equals
$\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{2}{3} \qquad \textbf{(C) } \frac{23}{6} \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 24$
2001 India IMO Training Camp, 1
If on $ \triangle ABC$, trinagles $ AEB$ and $ AFC$ are constructed externally such that $ \angle AEB\equal{}2 \alpha$, $ \angle AFB\equal{} 2 \beta$.
$ AE\equal{}EB$, $ AF\equal{}FC$.
COnstructed externally on $ BC$ is triangle $ BDC$ with $ \angle DBC\equal{} \beta$ , $ \angle BCD\equal{} \alpha$.
Prove that 1. $ DA$ is perpendicular to $ EF$.
2. If $ T$ is the projection of $ D$ on $ BC$, then prove that $ \frac{DA}{EF}\equal{} 2 \frac{DT}{BC}$.
2023 VN Math Olympiad For High School Students, Problem 6
a) Given a triangle $ABC$ with $\angle BAC=90^{\circ}$ and the altitude $AH(H$ is on the side $BC).$
Prove that: the [i]Lemoine[/i] point of the triangle $ABC$ is the midpoint of $AH.$
b) If a triangle has its [i]Lemoine[/i] point is the midpoint of $1$ in $3$ symmedian segments, does that triangle need to be a right triangle? Explain why.
2009 IberoAmerican Olympiad For University Students, 7
Let $G$ be a group such that every subgroup of $G$ is subnormal. Suppose that there exists $N$ normal subgroup of $G$ such that $Z(N)$ is nontrivial and $G/N$ is cyclic. Prove that $Z(G)$ is nontrivial. ($Z(G)$ denotes the center of $G$).
[b]Note[/b]: A subgroup $H$ of $G$ is subnormal if there exist subgroups $H_1,H_2,\ldots,H_m=G$ of $G$ such that $H\lhd H_1\lhd H_2 \lhd \ldots \lhd H_m= G$ ($\lhd$ denotes normal subgroup).
2023 Rioplatense Mathematical Olympiad, 2
Let $ABCD$ be a convex quadrilateral with $AB>AD$ and $\angle B=\angle D=90^{\circ}$. Let $P$ be a point in the side $AB$ such that $AP=AD$. The lines $PD$ and $BC$ cut in the point $Q$. The perpendicular line to $AC$ passing by $Q$ cuts $AB$ in the point $R$. Let $S$ be the foot of perpendicular of $D$ to the line $AC$. Prove that $\angle PSQ=\angle RCP$.
2015 Belarus Team Selection Test, 3
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.
[i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]
2011 ISI B.Stat Entrance Exam, 3
Let $\mathbb{R}$ denote the set of real numbers. Suppose a function $f: \mathbb{R} \to \mathbb{R}$ satisfies $f(f(f(x)))=x$ for all $x\in \mathbb{R}$. Show that
[b](i)[/b] $f$ is one-one,
[b](ii)[/b] $f$ cannot be strictly decreasing, and
[b](iii)[/b] if $f$ is strictly increasing, then $f(x)=x$ for all $x \in \mathbb{R}$.
PEN H Problems, 23
Find all $(x,y,z) \in {\mathbb{Z}}^3$ such that $x^{3}+y^{3}+z^{3}=x+y+z=3$.
2020 HK IMO Preliminary Selection Contest, 1
Let $n=(10^{2020}+2020)^2$. Find the sum of all the digits of $n$.
2018 PUMaC Number Theory A, 4
Let $n$ be a positive integer. Let $f(n)$ be the probability that, if divisors $a, b, c$ of $n$ are selected uniformly at random with replacement, then $\gcd(a, \text{lcm}(b, c)) = \text{lcm}(a, \gcd(b, c))$. Let $s(n)$ be the sum of the distinct prime divisors of $n$. If $f(n) < \frac{1}{2018}$, compute the smallest possible value of $s(n)$.
OIFMAT II 2012, 5
Let $ n \in N $. Let's define $ S_n = \{1, ..., n \} $. Let $ x_1 <x_2 <\cdots <x_n $ be any real. Determine the largest possible number of pairs $ (i, j) \in S_n \times S_n $ with $ i \not = j $, for which it is true that $ 1 <| x_i-x_j | <2 $ and justify why said value cannot be higher.
1986 Miklós Schweitzer, 8
Let $a_0=0$, $a_1, \ldots, a_k$ and $b_1, \ldots, b_k$ be arbitrary real numbers.
(i) Show that for all sufficiently large $n$ there exist polynomials $p_n$ of degree at most $n$ for which
$$p_n^{(i)} (-1)=a_i,\,\,\,\,\, p_n^{(i)} (1)=b_i,\,\,\,\,\, i=0, 1, \ldots, k$$
and
$$\max_{|x|\leq 1} |p_n (x)|\leq \frac{c}{n^2}\,\,\,\,\,\,\,\,\,\, (*)$$
where the constant $c$ depends only on the numbers $a_i, b_i$.
(ii) Prove that, in general, (*) cannot be replaced by the relation
$$\lim_{n\to\infty} n^2\cdot \max_{|x|\leq 1} |p_n (x)| = 0$$
[J. Szabados]
1986 Tournament Of Towns, (107) 1
Through vertices $A$ and $B$ of triangle $ABC$ are constructed two lines which divide the triangle into four regions (three triangles and one quadrilateral). It is known that three of them have equal area. Prove that one of these three regions is the quadrilateral .
(G . Galperin , A . Savin, Moscow)
2002 AMC 10, 19
Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach?
$ \text{(A)}\ 2\pi/3 \qquad
\text{(B)}\ 2\pi \qquad
\text{(C)}\ 5\pi/2 \qquad
\text{(D)}\ 8\pi/3 \qquad
\text{(E)}\ 3\pi$
2009 Iran Team Selection Test, 3
Suppose that $ a$,$ b$,$ c$ be three positive real numbers such that $ a\plus{}b\plus{}c\equal{}3$ . Prove that :
$ \frac{1}{2\plus{}a^{2}\plus{}b^{2}}\plus{}\frac{1}{2\plus{}b^{2}\plus{}c^{2}}\plus{}\frac{1}{2\plus{}c^{2}\plus{}a^{2}} \leq \frac{3}{4}$
2018 Auckland Mathematical Olympiad, 1
Find a multiple of $2018$ whose decimal expansion's first four digits are $2017$.
2019 Estonia Team Selection Test, 5
Boeotia is comprised of $3$ islands which are home to $2019$ towns in total. Each flight route connects three towns, each on a different island, providing connections between any two of them in both directions. Any two towns in the country are connected by at most one flight route. Find the maximal number of flight routes in the country
2016 Azerbaijan BMO TST, 4
For all numbers $n\ge 1$ does there exist infinite positive numbers sequence $x_1,x_2,...,x_n$ such that $x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_n}$
2020 Online Math Open Problems, 6
Alexis has $2020$ paintings, the $i$th one of which is a $1\times i$ rectangle for $i = 1, 2, \ldots, 2020$. Compute the smallest integer $n$ for which they can place all of the paintings onto an $n\times n$ mahogany table without overlapping or hanging off the table.
[i]Proposed by Brandon Wang[/i]
1966 IMO Shortlist, 8
We are given a bag of sugar, a two-pan balance, and a weight of $1$ gram. How do we obtain $1$ kilogram of sugar in the smallest possible number of weighings?
2014 Junior Regional Olympiad - FBH, 3
Let $ABCD$ be a trapezoid with base sides $AB$ and $CD$ and let $AB=a$, $BC=b$, $CD=c$, $DA=d$, $AC=m$ and $BD=n$. We know that $m^2+n^2=(a+c)^2$
$a)$ Prove that lines $AC$ and $BD$ are perpendicular
$b)$ Prove that $ac<bd$
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.
2002 Croatia National Olympiad, Problem 2
Let $a,b,c$ be real numbers greater than $1$. Prove the inequality
$$\log_a\left(\frac{b^2}{ac}-b+ac\right)\log_b\left(\frac{c^2}{ab}-c+ab\right)\log_c\left(\frac{a^2}{bc}-a+bc\right)\ge1.$$
2020 SIME, 12
Two sets $S_1$ and $S_2$, which are not necessarily distinct, are each selected randomly and independently from each other among the $512$ subsets of $S = \{1, 2, \ldots ,9\}$. Let $\sigma(X)$ denote the sum of the elements of set $X$. Note that $\sigma(\emptyset) = 0$ where $\emptyset$ denotes the empty set. If $S_1 \cup S_2$ stands for the union of $S_1$ and $S_2$, the probability that $\sigma(S_1 \cup S_2)$ is divisible by $3$ can be expressed as a common fraction of the form $\tfrac{m}{2^n}$ where $m$ is odd and $n$ is a positive integer. Find $m + n$.
1998 Brazil Team Selection Test, Problem 5
Let $p$ be an odd prime integer and $k$ a positive integer not divisible by $p$, $1\le k<2(p+1)$, and let $N=2kp+1$. Prove that the following statements are equivalent:
(i) $N$ is a prime number
(ii) there exists a positive integer $a$, $2\le a<n$, such that $a^{kp}+1$ is divisible by $N$ and $\gcd\left(a^k+1,N\right)=1$.