Found problems: 85335
2012 Vietnam National Olympiad, 3
Find all $f:\mathbb{R} \to \mathbb{R}$ such that:
(a) For every real number $a$ there exist real number $b$:$f(b)=a$
(b) If $x>y$ then $f(x)>f(y)$
(c) $f(f(x))=f(x)+12x.$
Fractal Edition 2, P4
Show that:
$$
1+\frac{1}{4}+\frac{1}{9}+\dots+\frac{1}{2023^2}+\frac{1}{2024^2} < 2.
$$
2014 PUMaC Team, 6
Find the sum of positive integer solutions of $x$ for $\dfrac{x^2}{1716-x}=p$, where $p$ is a prime. (If there are no solutions, answer $0$.)
2017 China Girls Math Olympiad, 3
Given $a_i\ge 0,x_i\in\mathbb{R},(i=1,2,\ldots,n)$. Prove that
$$((1-\sum_{i=1}^n a_i\cos x_i)^2+(1-\sum_{i=1}^n a_i\sin x_i)^2)^2\ge 4(1-\sum_{i=1}^n a_i)^3$$
2006 Stanford Mathematics Tournament, 2
In a given sequence $\{S_1,S_2,...,S_k\}$, for terms $n\ge3$, $S_n=\sum_{i=1}^{n-1} i\cdot S_{n-i}$. For example, if the first two elements are 2 and 3, respectively, the third entry would be $1\cdot3+2\cdot2=7$, and the fourth would be $1\cdot7+2\cdot3+3\cdot2=19$, and so on. Given that a sequence of integers having this form starts with 2, and the 7th element is 68, what is the second element?
2021 Baltic Way, 8
We are given a collection of $2^{2^k}$ coins, where $k$ is a non-negative integer. Exactly one coin is fake.
We have an unlimited number of service dogs. One dog is sick but we do not know which one.
A test consists of three steps: select some coins from the collection of all coins; choose a service dog; the dog smells all of the selected coins at once.
A healthy dog will bark if and only if the fake coin is amongst them. Whether the sick dog will bark or not is random. \\
Devise a strategy to find the fake coin, using at most $2^k+k+2$ tests, and prove that it works.
2010 HMNT, 5
Circle $O$ has chord $AB$. A circle is tangent to $O$ at $T$ and tangent to$ AB$ at $X$ such that $AX = 2XB$. What is $\frac{AT}{BT}$ ?
2023 MIG, 13
Five cards numbered $1,2,3,4,$ and $5$ are given to Paige, Quincy, Ronald, Selena, and Terrence. Paige, Quincy, and Ronald have the following conversation:
[list=disc]
[*]Paige: My number is between is between Selena's number and Quincy's number.
[*]Quincy: My number is between Ronald's number and Terrence's number.
[*]Ronald: My number is between Paige's number and Quincy's number.
[/list]
Who received the card numbered $3$?
$\textbf{(A) } \text{Paige}\qquad\textbf{(B) } \text{Quincy}\qquad\textbf{(C) } \text{Ronald}\qquad\textbf{(D) } \text{Selena}\qquad\textbf{(E) } \text{Terrence}$
2010 Korea - Final Round, 3
There are $ n$ websites $ 1,2,\ldots,n$ ($ n \geq 2$). If there is a link from website $ i$ to $ j$, we can use this link so we can move website $ i$ to $ j$.
For all $ i \in \left\{1,2,\ldots,n - 1 \right\}$, there is a link from website $ i$ to $ i+1$.
Prove that we can add less or equal than $ 3(n - 1)\log_{2}(\log_{2} n)$ links so that for all integers $ 1 \leq i < j \leq n$, starting with website $ i$, and using at most three links to website $ j$. (If we use a link, website's number should increase. For example, No.7 to 4 is impossible).
Sorry for my bad English.
2023 Estonia Team Selection Test, 2
Let $n$ be a positive integer. Find all polynomials $P$ with real coefficients such that $$P(x^2+x-n^2)=P(x)^2+P(x)$$ for all real numbers $x$.
1961 IMO, 5
Construct a triangle $ABC$ if $AC=b$, $AB=c$ and $\angle AMB=w$, where $M$ is the midpoint of the segment $BC$ and $w<90$. Prove that a solution exists if and only if \[ b \tan{\dfrac{w}{2}} \leq c <b \] In what case does the equality hold?
2016 Irish Math Olympiad, 9
Show that the number $a^3$ where $a=\frac{251}{ \frac{1}{\sqrt[3]{252}-5\sqrt[3]{2}}-10\sqrt[3]{63}}+\frac{1}{\frac{251}{\sqrt[3]{252}+5\sqrt[3]{2}}+10\sqrt[3]{63}}$
is an integer and find its value
1985 Swedish Mathematical Competition, 6
X-wich has a vibrant club-life. For every pair of inhabitants there is exactly one club to which they both belong. For every pair of clubs there is exactly one person who is a member of both. No club has fewer than $3$ members, and at least one club has $17$ members. How many people live in X-wich?
2014 Federal Competition For Advanced Students, P2, 3
(i) For which triangles with side lengths $a, b$ and $c$ apply besides the triangle inequalities $a + b> c, b + c> a$ and $c + a> b$ also the inequalities $a^2 + b^2> c^2, b^2 + c^2> a^2$ and $a^2 + c^2> b^2$ ?
(ii) For which triangles with side lengths $a, b$ and $c$ apply besides the triangle inequalities $a + b> c, b + c> a$ and $c + a> b$ also for all positive natural $n$ the inequalities $a^n + b^n> c^n, b^n + c^n> a^n$ and $a^n + c^n> b^n$ ?
OMMC POTM, 2024 5
Every integer $> 2024$ is given a color, white or black. The product of any two white integers is a black integer. Prove that there are two black integers that have a difference of one.
2017 Argentina National Math Olympiad Level 2, 5
Let $ABCD$ be a convex quadrilateral with $AB = BD = 8$ and $CD = DA = 6$. Let $P$ be a point on side $AB$ such that $DP$ is bisector of angle $\angle ADB$ and let $Q$ be a point on side $BC$ such that $DQ$ is bisector of angle $\angle CDB$. Calculate the radius of the circumcircle of triangle $DPQ$.
Note: The circumcircle of a triangle is the circle that passes through its three vertices.
1984 AMC 12/AHSME, 1
$\frac{1000^2}{252^2 - 248^2}$ equals
$\textbf{(A) }62,500\qquad \textbf{(B) }1000\qquad\textbf{(C) }500\qquad\textbf{(D) }250\qquad\textbf{(E) } \frac{1}{2}$
1984 AMC 12/AHSME, 18
A point $(x,y)$ is to be chosen in the coordinate plane so that it is equally distant from the x-axis, the y-axis, and the line $x+y = 2$. Then $x$ is
A. $\sqrt{2} - 1$
B. $\frac{1}{2}$
C. $2 - \sqrt{2}$
D. 1
E. Not uniquely determined
2020 Balkan MO, 4
Let $a_1=2$ and, for every positive integer $n$, let $a_{n+1}$ be the smallest integer strictly greater than $a_n$ that has more positive divisors than $a_n$. Prove that $2a_{n+1}=3a_n$ only for finitely many indicies $n$.
[i] Proposed by Ilija Jovčevski, North Macedonia[/i]
2017-IMOC, N5
Find all functions $f:\mathbb N\to\mathbb N$ such that
$$f(x)+f(y)\mid x^2-y^2$$holds for all $x,y\in\mathbb N$.
1994 Vietnam National Olympiad, 2
$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.
2011 AMC 12/AHSME, 23
Let $f(z)=\frac{z+a}{z+b}$ and $g(z)=f(f(z))$, where $a$ and $b$ are complex numbers. Suppose that $|a|=1$ and $g(g(z))=z$ for all $z$ for which $g(g(z))$ is defined. What is the difference between the largest and smallest possible values of $|b|$?
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ \sqrt{2}-1 \qquad
\textbf{(C)}\ \sqrt{3}-1 \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ 2$
2011 District Olympiad, 4
Let be a ring $ A. $ Denote with $ N(A) $ the subset of all nilpotent elements of $ A, $ with $ Z(A) $ the center of $ A, $ and with $ U(A) $ the units of $ A. $ Prove:
[b]a)[/b] $ Z(A)=A\implies N(A)+U(A)=U(A) . $
[b]b)[/b] $ \text{card} (A)\in\mathbb{N}\wedge a+U(A)\subset U(A)\implies a\in N(A) . $
2001 Croatia Team Selection Test, 1
Consider $A = \{1, 2, ..., 16\}$. A partition of $A$ into nonempty sets $A_1, A_2,..., A_n$ is said to be good if none of the Ai contains elements $a, b, c$ (not necessarily distinct) such that $a = b + c$.
(a) Find a good partition $\{A_1, A_2, A_3, A_4\}$ of $A$.
(b) Prove that no partition $\{A_1, A_2, A_3\}$ of $A$ is good
2016 Singapore Senior Math Olympiad, 3
For any integer $n \ge 1$, show that
$$\sum_{k=1}^{n} \frac{2^k}{\sqrt{k+0.5}} \le 2^{n+1}\sqrt{n+1}-\frac{4n^{3/2}}{3}$$