Found problems: 85335
2000 Stanford Mathematics Tournament, 9
Edward's formula for the stock market predicts correctly that the price of HMMT is directly proportional to a secret quantity $ x$ and inversely proportional to $ y$, the number of hours he slept the night before. If the price of HMMT is $ \$12$ when $ x\equal{}8$ and $ y\equal{}4$, how many dollars does it cost when $ x\equal{}4$ and $ y\equal{}8$?
1999 Junior Balkan Team Selection Tests - Romania, 3
Let be a subset of the interval $ (0,1) $ that contains $ 1/2 $ and has the property that if a number is in this subset, then, both its half and its successor's inverse are in the same subset. Prove that this subset contains all the rational numbers of the interval $ (0,1). $
2012 European Mathematical Cup, 2
Let $S$ be the set of positive integers. For any $a$ and $b$ in the set we have $GCD(a, b)>1$. For any $a$, $b$ and $c$ in the set we have $GCD(a, b, c)=1$. Is it possible that $S$ has $2012$ elements?
[i]Proposed by Ognjen Stipetić.[/i]
2022 IFYM, Sozopol, 5
Find the number of subsets of $\{1, 2,... , 2100\}$ such that each has sum of the elements giving a remainder of $3$ when divided by $7$.
2011 Argentina National Olympiad Level 2, 1
On the board were written the numbers from $1$ to $k$ (where $k$ is an unknown positive integer). One of the numbers was erased. The average of the remaining numbers is $25.25$. Which number was erased?
2002 Mediterranean Mathematics Olympiad, 4
If $a, b, c$ are non-negative real numbers with $ a^2 \plus{} b^2 \plus{} c^2 \equal{} 1$, prove that:
\[ \frac {a}{b^2 \plus{} 1} \plus{} \frac {b}{c^2 \plus{} 1} \plus{} \frac {c}{a^2 \plus{} 1} \geq \frac {3}{4}(a\sqrt {a} \plus{} b\sqrt {b} \plus{} c\sqrt {c})^2\]
2011 IMO Shortlist, 3
Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$.
[i]Proposed by Japan[/i]
2005 Alexandru Myller, 2
Let $f:[0,1]\to\mathbb R$ be an increasing function. Prove that if $\int_0^1f(x)dx=\int_0^1\left(\int_0^xf(t)dt\right)dx=0$ then $f(x)=0,\forall x\in(0,1)$.
[i]Mihai Piticari[/i]
1999 All-Russian Olympiad, 6
Prove that for all natural numbers $n$, \[ \sum_{k=1}^{n^2} \left\{ \sqrt{k} \right\} \le \frac{n^2-1}{2}. \] Here, $\{x\}$ denotes the fractional part of $x$.
TNO 2008 Junior, 7
A $5 \times 5$ grid is given, called $f_1$:
\[
\begin{array}{ccccc}
1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 \\
-1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 \\
\end{array}
\]
A new grid $f_{n+1}$ is constructed where each cell is equal to the product of its neighboring cells in grid $f_n$.
(a) Find the grids $f_6$ and $f_7$.
(b) Find the grids $f_{2008}$ and $f_{2009}$.
(c) Find $f_{2n}$ and $f_{2n+1}$ for any $n \in \mathbb{N}$.
*Note: Neighboring cells are those that share an edge, not just a vertex.*
2024 Middle European Mathematical Olympiad, 4
Determine all polynomials $P(x)$ with integer coefficients such that $P(n)$ is divisible by $\sigma(n)$ for all positive integers $n$. (As usual, $\sigma(n)$ denotes the sum of all positive divisors of $n$.)
1971 Spain Mathematical Olympiad, 3
If $0 < p$, $0 < q$ and $p +q < 1$ prove $$(px + qy)^2 \le px^2 + qy^2$$
Kvant 2021, M2640
In convex pentagon $ABCDE$ points $A_1$, $B_1$, $C_1$, $D_1$, $E_1$ are intersections of pairs of diagonals $(BD, CE)$, $(CE, DA)$, $(DA, EB)$, $(EB, AC)$ and $(AC, BD)$ respectively. Prove that if four of quadrilaterals $AB_{1}A_{1}B$, $BC_{1}B_{1}C$, $CD_{1}C_{1}D$, $DE_{1}D_{1}E$ and $EA_{1}E_{1}A$ are cyclic then the fifth one is also cyclic.
PEN S Problems, 34
Let $S_{n}$ be the sum of the digits of $2^n$. Prove or disprove that $S_{n+1}=S_{n}$ for some positive integer $n$.
PEN H Problems, 61
Solve the equation $2^x -5 =11^{y}$ in positive integers.
2022 Moldova Team Selection Test, 6
Let $A$ be a point outside of the circle $\Omega$. Tangents from $A$ touch $\Omega$ in points $B$ and $C$. Point $C$, collinear with $A$ and $P$, is between $A$ and $P$, such that the circumcircle of triangle $ABP$ intersects $\Omega$ again in point $E$. Point $Q$ is on the segment $BP$, such that $\angle PEQ=2 \cdot \angle APB$. Prove that the lines $BP$ and $CQ$ are perpendicular.
2011 All-Russian Olympiad, 1
In every cell of a table with $n$ rows and ten columns, a digit is written. It is known that for every row $A$ and any two columns, you can always find a row that has different digits from $A$ only when it intersects with two columns. Prove that $n\geq512$.
2022 Serbia Team Selection Test, P3
Let $n$ be an odd positive integer. Given are $n$ balls - black and white, placed on a circle. For a integer $1\leq k \leq n-1$, call $f(k)$ the number of balls, such that after shifting them with $k$ positions clockwise, their color doesn't change.
a) Prove that for all $n$, there is a $k$ with $f(k) \geq \frac{n-1}{2}$.
b) Prove that there are infinitely many $n$ (and corresponding colorings for them) such that $f(k)\leq \frac{n-1}{2}$ for all $k$.
2015 Junior Balkan Team Selection Test, 4
The diagonals $AD$, $BE$, $CF$ of cyclic hexagon $ABCDEF$ intersect in $S$ and $AB$ is parallel to $CF$ and lines $DE$ and $CF$ intersect each other in $M$. Let $N$ be a point such that $M$ is the midpoint of $SN$. Prove that circumcircle of $\triangle ADN$ is passing through midpoint of segment $CF$.
2005 National Olympiad First Round, 13
Let $ABCD$ be an isosceles trapezoid such that its diagonal is $\sqrt 3$ and its base angle is $60^\circ$, where $AD \parallel BC$. Let $P$ be a point on the plane of the trapezoid such that $|PA|=1$ and $|PD|=3$. Which of the following can be the length of $[PC]$?
$
\textbf{(A)}\ \sqrt 6
\qquad\textbf{(B)}\ 2\sqrt 2
\qquad\textbf{(C)}\ 2 \sqrt 3
\qquad\textbf{(D)}\ 3\sqrt 3
\qquad\textbf{(E)}\ \sqrt 7
$
1981 Yugoslav Team Selection Test, Problem 3
Let $a,b$ be nonnegative integers. Prove that $5a>7b$ if and only if there exist nonnegative integers $x,y,z,t$ such that
\begin{align*}
x+2y+3z+7t&=a,\\
y+2z+5t&=b.
\end{align*}
2014 ELMO Shortlist, 13
Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$.
[i]Proposed by David Stoner[/i]
1964 Dutch Mathematical Olympiad, 5
Consider a sequence of non-negative integers g$_1,g_2,g_3,...$ each consisting of three digits (numbers smaller than $100$ are also written with three digits; the number $27$, for example, is written as $027$). Each number consists of the preceding by taking the product of the three digits that make up the preceding. The resulting sequence is of course dependent on the choice of $g_1$ (e.g. $g_1 = 359$ leads to $g_2= 135$, $g_3= 015$, $g_4 = 000$).Prove that independent of the choice of $g_1$:
(a) $g_{n+1}\le g_n$
(b) $g_{10}= 000$.
LMT Speed Rounds, 2011.15
Given that $20N^2$ is a divisor of $11!,$ what is the greatest possible integer value of $N?$
2018 India Regional Mathematical Olympiad, 6
Define a sequence $\{a_n\}_{n\geq 1}$ of real numbers by \[a_1=2,\qquad a_{n+1} = \frac{a_n^2+1}{2}, \text{ for } n\geq 1.\] Prove that \[\sum_{j=1}^{N} \frac{1}{a_j + 1} < 1\] for every natural number $N$.