Found problems: 85335
2006 China Girls Math Olympiad, 1
Let $a>0$, the function $f: (0,+\infty) \to R$ satisfies $f(a)=1$, if for any positive reals $x$ and $y$, there is \[f(x)f(y)+f \left( \frac{a}{x}\right)f \left( \frac{a}{y}\right) =2f(xy)\] then prove that $f(x)$ is a constant.
PEN A Problems, 99
Let $n \ge 2$ be a positive integer, with divisors \[1=d_{1}< d_{2}< \cdots < d_{k}=n \;.\] Prove that \[d_{1}d_{2}+d_{2}d_{3}+\cdots+d_{k-1}d_{k}\] is always less than $n^{2}$, and determine when it divides $n^{2}$.
2014 Dutch Mathematical Olympiad, 4
A quadruple $(p, a, b, c)$ of positive integers is called a Leiden quadruple if
- $p$ is an odd prime number,
- $a, b$, and $c$ are distinct and
- $ab + 1, bc + 1$ and $ca + 1$ are divisible by $p$.
a) Prove that for every Leiden quadruple $(p, a, b, c)$ we have $p + 2 \le \frac{a+b+c}{3}$ .
b) Determine all numbers $p$ for which a Leiden quadruple $(p, a, b, c)$ exists with $p + 2 = \frac{a+b+c}{3} $
1988 AIME Problems, 3
Find $(\log_2 x)^2$ if $\log_2 (\log_8 x) = \log_8 (\log_2 x)$.
2011 HMNT, 5
For any finite sequence of positive integers $\pi$, let $S(\pi)$ be the number of strictly increasing sub sequences in $\pi$ with length $2$ or more. For example, in the sequence $\pi = \{3, 1, 2, 4\}$, there are five increasing sub-sequences: $\{3, 4\}$, $\{1, 2\}$, $\{1, 4\}$, $\{2, 4\}$, and \${1, 2, 4\}, so $S(\pi) = 5$. In an eight-player game of Fish, Joy is dealt six cards of distinct values, which she puts in a random order $\pi$ from left to right in her hand. Determine
$$\sum_{\pi} S(\pi),$$
where the sum is taken over all possible orders $\pi$ of the card values.
1985 Austrian-Polish Competition, 2
Suppose that $n\ge 8$ persons $P_1,P_2,\dots,P_n$ meet at a party. Assume that $P_k$ knows $k+3$ persons for $k=1,2,\dots,n-6$. Further assume that each of $P_{n-5},P_{n-4},P_{n-3}$ knows $n-2$ persons, and each of $P_{n-2},P_{n-1},P_n$ knows $n-1$ persons. Find all integers $n\ge 8$ for which this is possible.
(It is understood that "to know" is a symmetric nonreflexive relation: if $P_i$ knows $P_j$ then $P_j$ knows $P_i$; to say that $P_i$ knows $p$ persons means: knows $p$ persons other than herself/himself.)
2018 Junior Regional Olympiad - FBH, 1
Four buddies bought a ball. First one paid half of the ball price. Second one gave one third of money that other three gave. Third one paid a quarter of sum paid by other three. Fourth paid $5\$$. How much did the ball cost?
2014 Singapore Senior Math Olympiad, 4
Find the smallest number among the following numbers:
$ \textbf{(A) }\sqrt{55}-\sqrt{52}\qquad\textbf{(B) }\sqrt{56}-\sqrt{53}\qquad\textbf{(C) }\sqrt{77}-\sqrt{74}\qquad\textbf{(D) }\sqrt{88}-\sqrt{85}\qquad\textbf{(E) }\sqrt{70}-\sqrt{67} $
2024 Kyiv City MO Round 2, Problem 4
Let $BD$ be an altitude of $\triangle ABC$ with $AB < BC$ and $\angle B > 90^\circ$. Let $M$ be the midpoint of $AC$, and point $K$ be symmetric to point $D$ with respect to point $M$. A perpendicular drawn from point $M$ to the line $BC$ intersects line $AB$ at point $L$. Prove that $\angle MBL = \angle MKL$.
[i]Proposed by Oleksandra Yakovenko[/i]
Maryland University HSMC part II, 2007
[b]p1.[/b] One hundred hobbits sit in a circle. The hobbits realize that whenever a hobbit and his two neighbors add up their total rubles, the sum is always $2007$. Prove that each hobbit has $669$ rubles.
[b]p2.[/b] There was a young lady named Chris,
Who, when asked her age, answered this:
"Two thirds of its square
Is a cube, I declare."
Now what was the age of the miss?
(a) Find the smallest possible age for Chris. You must justify your answer.
(Note: ages are positive integers; "cube" means the cube of a positive integer.)
(b) Find the second smallest possible age for Chris. You must justify your answer.
(Ignore the word "young.")
[b]p3.[/b] Show that $$\sum_{n=1}^{2007}\frac{1}{n^3+3n^2+2n}<\frac14$$
[b]p4.[/b] (a) Show that a triangle $ABC$ is isosceles if and only if there are two distinct points $P_1$ and $P_2$ on side $BC$ such that the sum of the distances from $P_1$ to the sides $AB$ and $AC$ equals the sum of the distances from $P_2$ to the sides $AB$ and $AC$.
(b) A convex quadrilateral is such that the sum of the distances of any interior point to its four sides is constant. Prove that the quadrilateral is a parallelogram.
(Note: "distance to a side" means the shortest distance to the line obtained by extending the side.)
[b]p5.[/b] Each point in the plane is colored either red or green. Let $ABC$ be a fixed triangle. Prove that there is a triangle $DEF$ in the plane such that $DEF$ is similar to $ABC$ and the vertices of $DEF$ all have the same color.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1996 AMC 12/AHSME, 24
The sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, \ldots$ consists of 1’s separated by blocks of 2’s with n 2’s in the nth block. The sum of the first $1234$ terms of this sequence is
$\text{(A)}\ 1996 \qquad \text{(B)}\ 2419 \qquad \text{(C)}\ 2429 \qquad \text{(D)}\ 2439 \qquad \text{(E)}\ 2449$
1998 Taiwan National Olympiad, 3
Let $ m,n$ be positive integers, and let $ F$ be a family of $ m$-element subsets of $ \{1,2,...,n\}$ satisfying $ A\cap B \not \equal{} \emptyset$ for all $ A,B\in F$. Determine the maximum possible number of elements in $ F$.
2008 National Olympiad First Round, 35
What is the least real value of the expression $\sqrt{x^2-6x+13} + \sqrt{x^2-14x+58}$ where $x$ is a real number?
$
\textbf{(A)}\ \sqrt {39}
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ \frac {43}6
\qquad\textbf{(D)}\ 2\sqrt 2 + \sqrt {13}
\qquad\textbf{(E)}\ \text{None of the above}
$
2001 239 Open Mathematical Olympiad, 3
The circles $ S_1 $ and $ S_2 $ intersect at points $ A $ and $ B $. Circle $ S_3 $ externally touches $ S_1 $ and $ S_2 $ at points $ C $ and $ D $ respectively. Let $ PQ $ be a chord cut by the line $ AB $ on circle $ S_3 $, and $ K $ be the midpoint of $ CD $. Prove that $ \angle PKC = \angle QKC $.
1952 Putnam, B2
Find the surface generated by the solutions of \[ \frac {dx}{yz} = \frac {dy}{zx} = \frac{dz}{xy}, \] which intersects the circle $y^2+ z^2 = 1, x = 0.$
2013 Harvard-MIT Mathematics Tournament, 1
Let $a$ and $b$ be real numbers such that $ \frac {ab}{a^2 + b^2} = \frac {1}{4} $. Find all possible values of $ \frac {|a^2-b^2|}{a^2+b^2} $.
2024 All-Russian Olympiad, 6
The altitudes of an acute triangle $ABC$ with $AB<AC$ intersect at a point $H$, and $O$ is the center of the circumcircle $\Omega$. The segment $OH$ intersects the circumcircle of $BHC$ at a point $X$, different from $O$ and $H$. The circumcircle of $AOX$ intersects the smaller arc $AB$ of $\Omega$ at point $Y$. Prove that the line $XY$ bisects the segment $BC$.
[i]Proposed by A. Tereshin[/i]
2011 Greece Junior Math Olympiad, 1
Let $ABC$ be a triangle with $\angle BAC=120^o$, which the median $AD$ is perpendicular to side $AB$ and intersects the circumscribed circle of triangle $ABC$ at point $E$. Lines $BA$ and $EC$ intersect at $Z$. Prove that
a) $ZD \perp BE$
b) $ZD=BC$
2010 AMC 10, 10
Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?
$ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 30$
2015 Bosnia Herzegovina Team Selection Test, 4
Let $X$ be a set which consists from $8$ consecutive positive integers. Set $X$ is divided on two disjoint subsets $A$ and $B$ with equal number of elements. If sum of squares of elements from set $A$ is equal to sum of squares of elements from set $B$, prove that sum of elements of set $A$ is equal to sum of elements of set $B$.
2015 May Olympiad, 2
$6$ indistinguishable coins are given, $4$ are authentic, all of the same weight, and $2$ are false, one is more light than the real ones and the other one, heavier than the real ones. The two false ones together weigh same as two authentic coins. Find two authentic coins using a balance scale twice only by two plates, no weights.
Clarification: A two-pan scale only reports if the left pan weighs more, equal or less that right.
2023 VIASM Summer Challenge, Problem 1
Find all relatively distinct integers $m, n, p\in \mathbb{Z}_{\ne 0}$ such that the polynomial
$$F(x) = x(x - m)(x - n)(x - p) + 1$$is reducible in $\mathbb{Z}[x].$
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 $\left\lfloor \frac{10}{p} \right\rfloor$.
2007 IMO Shortlist, 7
For a prime $ p$ and a given integer $ n$ let $ \nu_p(n)$ denote the exponent of $ p$ in the prime factorisation of $ n!$. Given $ d \in \mathbb{N}$ and $ \{p_1,p_2,\ldots,p_k\}$ a set of $ k$ primes, show that there are infinitely many positive integers $ n$ such that $ d\mid \nu_{p_i}(n)$ for all $ 1 \leq i \leq k$.
[i]Author: Tejaswi Navilarekkallu, India[/i]
1998 South africa National Olympiad, 2
Find the maximum value of \[ \sin{2\alpha} + \sin{2\beta} + \sin{2\gamma} \] where $\alpha,\beta$ and $\gamma$ are positive and $\alpha + \beta + \gamma = 180^{\circ}$.