Found problems: 85335
2005 Lithuania Team Selection Test, 1
Find the smallest integer $n$ such that an $n\times n$ square can be partitioned into $40\times 40$ and $49\times 49$ squares, with both types of squares present in the partition, if
a) $40|n$; b) $49|n$; c) $n\in \mathbb N$.
1984 AMC 12/AHSME, 11
A calculator has a key which replaces the displayed entry with its square, and another key which replaces the displayed entry with its reciprocal. Let $y$ be the final result if one starts with an entry $x \neq 0$ and alternately squares and reciprocates $n$ times each. Assuming the calculator is completely accurate (e.g., no roundoff or overflow), then $y$ equals
A. $x^{((-2)^n)}$
B. $x^{2n}$
C. $x^{-2n}$
D. $x^{-(2^n)}$
E. $x^{((-1)^n 2n)}$
2017 Regional Olympiad of Mexico Northeast, 3
Prove that there is no pair of relatively prime positive integers $(a, b)$ that satisfy the equation
$$a^3 + 2017a = b^3 -2017b.$$
1952 Putnam, A7
Directed lines are drawn from the center of a circle, making angles of $0, \pm 1, \pm 2, \pm 3, \ldots$ (measured in radians from a prime direction). If these lines meet the circle in points $P_0, P_1, P_{-1}, P_2, P_{-2}, \ldots,$ show that there is no interval on the circumference of the circle which does not contain some $P_{\pm i}.$ (You may assume that $\pi$ is irrational.)
2015 JBMO Shortlist, NT1
What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?
2004 Kazakhstan National Olympiad, 8
Let $ ABCD$ be a convex quadrilateral. The perpendicular bisectors of its sides $ AB$ and $ CD$ meet at $ Y$. Denote by $ X$ a point inside the quadrilateral $ ABCD$ such that $ \measuredangle ADX \equal{} \measuredangle BCX < 90^{\circ}$ and $ \measuredangle DAX \equal{} \measuredangle CBX < 90^{\circ}$. Show that $ \measuredangle AYB \equal{} 2\cdot\measuredangle ADX$.
VI Soros Olympiad 1999 - 2000 (Russia), 9.3
On the coordinate plane, the parabola $y = x^2$ and the points $A(x_1, x_1^2)$, $B(x_2, x_2^2)$ are set such that $x_1=-998$, $x_2 =1999$ The segments $BX_1$, $AX_2$, $BX_3$, $AX_4$,..., $BX_{1997}$, $AX_{1998}$ and $X_k$ are constructed succesively with $(x_k,0)$, $1 \le k \le 1998$ and $x_3$, $x_4$,..., $x_{1998}$ are abscissas of the points of intersection of the parabola with segments $BX_1$, $AX_2$, $BX_3$, $AX_4$,..., $BX_{1997}$, $AX_{1998}$. Find the value $\frac{1}{x_{1999}}+\frac{1}{x_{2000}}$
2000 Moldova National Olympiad, Problem 2
Solve the system
\begin{align*}
36x^2y-27y^3&~=~8,\\
4x^3-27xy^2&~=~4.\end{align*}
2006 France Team Selection Test, 1
Let $ABCD$ be a square and let $\Gamma$ be the circumcircle of $ABCD$. $M$ is a point of $\Gamma$ belonging to the arc $CD$ which doesn't contain $A$. $P$ and $R$ are respectively the intersection points of $(AM)$ with $[BD]$ and $[CD]$, $Q$ and $S$ are respectively the intersection points of $(BM)$ with $[AC]$ and $[DC]$.
Prove that $(PS)$ and $(QR)$ are perpendicular.
1987 AIME Problems, 3
By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?
2019 Greece Team Selection Test, 4
Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.
2024 LMT Fall, 24
Let $ABC$ be a triangle with $AB=13, BC=15, AC=14$. Let $P$ be the point such that $AP$ $=$ $CP$ $=$ $\tfrac12 BP$. Find $AP^2$.
2001 Junior Balkan Team Selection Tests - Romania, 4
Three students write on the blackboard next to each other three two-digit squares. In the end, they observe that the 6-digit number thus obtained is also a square. Find this number!
2019-IMOC, N4
Given a sequence of prime numbers $p_1, p_2,\cdots$ , with the following property:
$p_{n+2}$ is the largest prime divisor of $p_n+p_{n+1}+2018$
Show that the set $\{p_i\}_{i\in \mathbb{N}}$ is finite.
2015 NIMO Problems, 4
Find the sum of all positive integers $1\leq k\leq 99$ such that there exist positive integers $a$ and $b$ with the property that \[x^{100}-ax^k+b=(x^2-2x+1)P(x)\] for some polynomial $P$ with integer coefficients.
[i]Proposed by David Altizio[/i]
VMEO IV 2015, 10.1
Where $n$ is a positive integer, the sequence $a_n$ is determined by the formula $$a_{n+1}=\frac{1}{a_1 + a_2 +... + a_n} -\sqrt2, \,a_1 = 1.$$ Find the limit of the sequence $S_n$ defined by $S_n=a_1 + a_2 +... + a_n$.
2013 National Chemistry Olympiad, 59
All of the following atoms comprise part of a peptide functional group except:
$ \textbf{(A)}\ \text{Hydrogen} \qquad\textbf{(B)}\ \text{Nitrogen}\qquad$
${\textbf{(C)}\ \text{Oxygen} \qquad\textbf{(D)}}\ \text{Phosphorous} \qquad$
1991 Polish MO Finals, 3
If $x, y, z$ are real numbers satisfying $x^2 +y^2 +z^2 = 2$, prove the inequality
\[ x + y + z \leq 2 + xyz \]
When does equality occur?
2023 Kyiv City MO Round 1, Problem 4
Find all pairs $(m, n)$ of positive integers, for which number $2^n - 13^m$ is a cube of a positive integer.
[i]Proposed by Oleksiy Masalitin[/i]
2007 Today's Calculation Of Integral, 232
For $ f(x)\equal{}1\minus{}\sin x$, let $ g(x)\equal{}\int_0^x (x\minus{}t)f(t)\ dt.$
Show that $ g(x\plus{}y)\plus{}g(x\minus{}y)\geq 2g(x)$ for any real numbers $ x,\ y.$
2015 Israel National Olympiad, 3
Prove that the number $\left(\frac{76}{\frac{1}{\sqrt[3]{77}-\sqrt[3]{75}}-\sqrt[3]{5775}}+\frac{1}{\frac{76}{\sqrt[3]{77}+\sqrt[3]{75}}+\sqrt[3]{5775}}\right)^3$ is an integer.
2020 MIG, 23
There exists a positive integer $b$ such that the base-$10$ fraction $\tfrac{59}{48}$ can be expressed as $1.\overline{14}_b$ (or $1.141414\ldots_b$), a value in base $b$. Find $b$.
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2009 Tournament Of Towns, 3
In each square of a $101\times 101$ board, except the central one, is placed either a sign " turn" or a sign " straight". The chess piece " car" can enter any square on the boundary of the board from outside (perpendicularly to the boundary). If the car enters a square with the sign " straight" then it moves to the next square in the same direction, otherwise (in case it enters a square with the sign " turn") it turns either to the right or to the left ( its choice). Can one place the signs in such a way that the car never enter the central square?
2023 MOAA, 9
Let $\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$. Let $I$ be the center of the inscribed circle of $\triangle{ABC}$. If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$, then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.
[color=#00f]Note that this problem is null because a diagram is impossible.[/color]
[i]Proposed by Andy Xu[/i]
1973 USAMO, 1
Two points $ P$ and $ Q$ lie in the interior of a regular tetrahedron $ ABCD$. Prove that angle $ PAQ < 60^\circ$.