Found problems: 85335
2018 AMC 8, 4
The twelve-sided figure shown has been drawn on $1 \text{ cm}\times 1 \text{ cm}$ graph paper. What is the area of the figure in $\text{cm}^2$?
[asy]
unitsize(8mm);
for (int i=0; i<7; ++i) {
draw((i,0)--(i,7),gray);
draw((0,i+1)--(7,i+1),gray);
}
draw((1,3)--(2,4)--(2,5)--(3,6)--(4,5)--(5,5)--(6,4)--(5,3)--(5,2)--(4,1)--(3,2)--(2,2)--cycle,black+2bp);
[/asy]
$\textbf{(A) } 12 \qquad \textbf{(B) } 12.5 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 13.5 \qquad \textbf{(E) } 14$
2014 Harvard-MIT Mathematics Tournament, 5
Let $\mathcal{C}$ be a circle in the $xy$ plane with radius $1$ and center $(0, 0, 0)$, and let $P$ be a point in space with coordinates $(3, 4, 8)$. Find the largest possible radius of a sphere that is contained entirely in the slanted cone with base $\mathcal{C}$ and vertex $P$.
PEN H Problems, 62
Solve the equation $7^x -3^y =4$ in positive integers.
1997 Iran MO (3rd Round), 4
Let $x, y, z$ be real numbers greater than $1$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$. Prove that
\[\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}\leq \sqrt{x+y+z}.\]
1994 Mexico National Olympiad, 6
Show that we cannot tile a $10 x 10$ board with $25$ pieces of type $A$, or with $25$ pieces of type $B$, or with $25$ pieces of type $C$.
2005 Italy TST, 3
The function $\psi : \mathbb{N}\rightarrow\mathbb{N}$ is defined by $\psi (n)=\sum_{k=1}^n\gcd (k,n)$.
$(a)$ Prove that $\psi (mn)=\psi (m)\psi (n)$ for every two coprime $m,n \in \mathbb{N}$.
$(b)$ Prove that for each $a\in\mathbb{N}$ the equation $\psi (x)=ax$ has a solution.
2009 Indonesia TST, 3
Find integer $ n$ with $ 8001 < n < 8200$ such that $ 2^n \minus{} 1$ divides $ 2^{k(n \minus{} 1)! \plus{} k^n} \minus{} 1$ for all integers $ k > n$.
2014 Sharygin Geometry Olympiad, 1
The incircle of a right-angled triangle $ABC$ touches its catheti $AC$ and $BC$ at points $B_1$ and $A_1$, the hypotenuse touches the incircle at point $C_1$. Lines $C_1A_1$ and $C_1B_1$ meet $CA$ and $CB$ respectively at points $B_0$ and $A_0$. Prove that $AB_0 = BA_0$.
(J. Zajtseva, D. Shvetsov )
2009 Indonesia TST, 3
Find all function $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that
\[ f(x \plus{} y)(f(x) \minus{} y) \equal{} xf(x) \minus{} yf(y)
\]
for all $ x,y \in \mathbb{R}$.
2004 Junior Balkan MO, 1
Prove that the inequality \[ \frac{ x+y}{x^2-xy+y^2 } \leq \frac{ 2\sqrt 2 }{\sqrt{ x^2 +y^2 } } \] holds for all real numbers $x$ and $y$, not both equal to 0.
2012 Purple Comet Problems, 9
Find the value of $x$ that satisfies $\log_{3}(\log_9x)=\log_9(\log_3x)$
1986 Czech And Slovak Olympiad IIIA, 6
Assume that $M \subset N$ has the property that every two numbers $m,n$ of $M$ satisfy $|m-n| \ge mn/25$.
Prove that the set $M$ contains no more than $9$ elements.
Decide whether there exists such set M.
2003 Junior Balkan Team Selection Tests - Romania, 2
Consider the prime numbers $n_1< n_2 <...< n_{31}$. Prove that if $30$ divides $n_1^4 + n_2^4+...+n_{31}^4$, then among these numbers one can find three consecutive primes.
1982 IMO Shortlist, 8
A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex figure are perpendicular). Prove that the center of the circle is a center of symmetry of the figure.
2023 Balkan MO Shortlist, A5
Are there polynomials $P, Q$ with real coefficients, such that $P(P(x))\cdot Q(Q(x))$ has exactly $2023$ distinct real roots and $P(Q(x)) \cdot Q(P(x))$ has exactly $2024$ distinct real roots?
2016 Mathematical Talent Reward Programme, SAQ: P 4
For any given $k$ points in a plane, we define the diameter of the points as the maximum distance between any two points among the given points. Suppose $n$ points are there in a plane with diameter $d$. Show that we can always find a circle with radius $\frac{\sqrt{3}}{2}d$ such that all points lie inside the circle.
2000 USAMO, 3
A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
2015 AIME Problems, 4
In an isosceles trapezoid, the parallel bases have lengths $\log3$ and $\log192$, and the altitude to these bases has length $\log16$. The perimeter of the trapezoid can be written in the form $\log2^p3^q$, where $p$ and $q$ are positive integers. Find $p+q$.
1989 Czech And Slovak Olympiad IIIA, 3
For given coprime numbers $p > q > 0$, find all pairs of real numbers $c,d$ such that for the sets
$$A = \left\{ \left[n\frac{p}{q}\right] , n \in N \right\} \ \ and \ \ B = \{[cn + d], n \in N\}$$
where $A \cap B = \emptyset$, $A \cup B = N$, where $N = \{1, 2, 3, ...\}$ is the set of all natural numbers.
PEN O Problems, 25
Let $A$ be a non-empty set of positive integers. Suppose that there are positive integers $b_{1}$, $\cdots$, $b_{n}$ and $c_{1}$, $\cdots$, $c_{n}$ such that [list] [*] for each $i$ the set $b_{i}A+c_{i}=\{b_{i}a+c_{i}\vert a \in A \}$ is a subset of $A$, [*] the sets $b_{i}A+c_{i}$ and $b_{j}A+c_{j}$ are disjoint whenever $i \neq j$.[/list] Prove that \[\frac{1}{b_{1}}+\cdots+\frac{1}{b_{n}}\le 1.\]
2021-IMOC qualification, C1
There are $3n$ $A$s and $2n$ $B$s in a string, where $n$ is a positive integer, prove that you can find a substring in this string that contains $3$ $A$s and $2$ $B$s.
2004 All-Russian Olympiad Regional Round, 9.6
Positive numbers $x, y, z$ are such that the absolute value of the difference of any two of them are less than $2$. Prove that $$ \sqrt{xy +1}+\sqrt{yz + 1}+\sqrt{zx+ 1} > x+ y + z.$$
2009 Regional Olympiad of Mexico Center Zone, 2
Let $p \ge 2$ be a prime number and $a \ge 1$ a positive integer with $p \neq a$. Find all pairs $(a,p)$ such that: $a+p \mid a^2+p^2$
2022 Moldova Team Selection Test, 1
Show that for every integer $n \geq 2$ the number
$$a=n^{5n-1}+n^{5n-2}+n^{5n-3}+n+1$$
is a composite number.
1996 IMO Shortlist, 7
let $ V$ be a finitive set and $ g$ and $ f$ be two injective surjective functions from $ V$to$ V$.let $ T$ and $ S$ be two sets such that they are defined as following"
$ S \equal{} \{w \in V: f(f(w)) \equal{} g(g(w))\}$
$ T \equal{} \{w \in V: f(g(w)) \equal{} g(f(w))\}$
we know that $ S \cup T \equal{} V$, prove:
for each $ w \in V : f(w) \in S$ if and only if $ g(w) \in S$