Found problems: 85335
2000 Turkey Junior National Olympiad, 3
$f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies the equation
\[f(x)f(y)-af(xy)=x+y\]
, for every real numbers $x,y$. Find all possible real values of $a$.
2009 Purple Comet Problems, 25
The polynomial $P(x)=a_0+a_1x+a_2x^2+...+a_8x^8+2009x^9$ has the property that $P(\tfrac{1}{k})=\tfrac{1}{k}$ for $k=1,2,3,4,5,6,7,8,9$. There are relatively prime positive integers $m$ and $n$ such that $P(\tfrac{1}{10})=\tfrac{m}{n}$. Find $n-10m$.
2003 AMC 12-AHSME, 4
It takes Mary $ 30$ minutes to walk uphill $ 1$ km from her home to school, but it takes her only $ 10$ minutes to walk from school to home along the same route. What is her average speed, in km/hr, for the round trip?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 3.125 \qquad
\textbf{(C)}\ 3.5 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 4.5$
1965 Bulgaria National Olympiad, Problem 1
The numbers $2,3,7$ have the property that the product of any two of them increased by $1$ is divisible by the third number. Prove that this triple of integer numbers greater than $1$ is the only triple with the given property.
2010 Balkan MO, 3
A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$.
Prove that $S$ can be covered by a strip of width $2$.
1997 Slovenia Team Selection Test, 5
A square $ (n \minus{} 1) \times (n \minus{} 1)$ is divided into $ (n \minus{} 1)^2$ unit squares in the usual manner. Each of the $ n^2$ vertices of these squares is to be coloured red or blue. Find the number of different colourings such that each unit square has exactly two red vertices. (Two colouring schemse are regarded as different if at least one vertex is coloured differently in the two schemes.)
2011 Iran MO (3rd Round), 1
Suppose that $S\subseteq \mathbb Z$ has the following property: if $a,b\in S$, then $a+b\in S$. Further, we know that $S$ has at least one negative element and one positive element. Is the following statement true?
There exists an integer $d$ such that for every $x\in \mathbb Z$, $x\in S$ if and only if $d|x$.
[i]proposed by Mahyar Sefidgaran[/i]
2004 Nicolae Coculescu, 3
Solve in $ \mathcal{M}_2(\mathbb{R}) $ the equation $ X^3+X+2I=0. $
[i]Florian Dumitrel[/i]
2007 Indonesia TST, 3
For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.
2016 CCA Math Bonanza, L3.2
Let $a_0 = 1$ and define the sequence $\{a_n\}$ by \[a_{n+1} = \frac{\sqrt{3}a_n - 1}{a_n + \sqrt{3}}.\] If $a_{2017}$ can be expressed in the form $a+b\sqrt{c}$ in simplest radical form, compute $a+b+c$.
[i]2016 CCA Math Bonanza Lightning #3.2[/i]
2019 Junior Balkan Team Selection Tests - Romania, 3
Let $d$ be the tangent at $B$ to the circumcircle of the acute scalene triangle $ABC$. Let $K$ be the orthogonal projection of the orthocenter, $H$, of triangle $ABC$ to the line $d$ and $L$ the midpoint of the side $AC$. Prove that the triangle $BKL$ is isosceles.
2006 AMC 12/AHSME, 18
An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?
$ \textbf{(A) } 120 \qquad \textbf{(B) } 121 \qquad \textbf{(C) } 221 \qquad \textbf{(D) } 230 \qquad \textbf{(E) } 231$
2006 Stanford Mathematics Tournament, 2
A customer enters a supermarket. The probability that the customer buys bread is .60, the probability that the customer buys milk is .50, and the probability that the customer buys both bread and milk is .30. What is the probability that the customer would buy either bread or milk or both?
2012 Romania Team Selection Test, 1
Let $\Delta ABC$ be a triangle. The internal bisectors of angles $\angle CAB$ and $\angle ABC$ intersect segments $BC$, respectively $AC$ in $D$, respectively $E$. Prove that \[DE\leq (3-2\sqrt{2})(AB+BC+CA).\]
2023 Princeton University Math Competition, 14
14. Kelvin the frog is hopping on the coordinate plane $\mathbb{R}^{2}$. He starts at the origin, and every second, he hops one unit to the right, left, up, or down, such that he always remains in the first quadrant $\{(x, y): x \geq 0, y \geq 0\}$. In how many ways can Kelvin make his first 14 jumps such that his 14 th jump lands at the origin?
1999 Gauss, 8
The average of 10, 4, 8, 7, and 6 is
$\textbf{(A)}\ 33 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 7$
2020-2021 Fall SDPC, 4
Let $ABC$ be an acute scalene triangle, let $D$ be a point on the $A$-altitude, and let the circle with diameter $AD$ meet $AC$, $AB$, and the circumcircle of $ABC$ at $E$, $F$, $G$, respectively. Let $O$ be the circumcenter of $ABC$, let $AO$ meet $EF$ at $T$, and suppose the circumcircles of $ABC$ and $GTO$ meet at $X \neq G$. Then, prove that $AX$, $DG$, and $EF$ concur.
V Soros Olympiad 1998 - 99 (Russia), 10.3
Without using a calculator, find out which number is greater:
$$29^{200}\cdot 2^{151} \,\,\, or \,\,\, 5^{279} \cdot 3^{300}$$
2001 Tuymaada Olympiad, 6
On the side $AB$ of an isosceles triangle $AB$ ($AC=BC$) lie points $P$ and $Q$ such that $\angle PCQ \le \frac{1}{2} \angle ACB$. Prove that $PQ \le \frac{1}{2} AB$.
2013 Danube Mathematical Competition, 3
Determine the natural numbers $m,n$ such as $85^m-n^4=4$
2019 IberoAmerican, 5
Don Miguel places a token in one of the $(n+1)^2$ vertices determined by an $n \times n$ board. A [i]move[/i] consists of moving the token from the vertex on which it is placed to an adjacent vertex which is at most $\sqrt2$ away, as long as it stays on the board. A [i]path[/i] is a sequence of moves such that the token was in each one of the $(n+1)^2$ vertices exactly once. What is the maximum number of diagonal moves (those of length $\sqrt2$) that a path can have in total?
2021 IMO Shortlist, A1
Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$.
[i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]
1990 AMC 8, 4
Which of the following could not be the unit's digit [one's digit] of the square of a whole number?
$ \text{(A)}\ 1\qquad\text{(B)}\ 4\qquad\text{(C)}\ 5\qquad\text{(D)}\ 6\qquad\text{(E)}\ 8 $
2000 Mexico National Olympiad, 4
Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$. (For example, if $a = 2$ and $b = 4$, the first three terms are $5,14,32$.) What is the maximum possible number of primes that can occur before encoutering the first composite term?
2008 India Regional Mathematical Olympiad, 6
Let $BCDK$ be a convex quadrilateral such that $BC=BK$ and $DC=DK$. $A$ and $E$ are points such that $ABCDE$ is a convex pentagon such that $AB=BC$ and $DE=DC$ and $K$ lies in the interior of the pentagon $ABCDE$. If $\angle ABC=120^{\circ}$ and $\angle CDE=60^{\circ}$ and $BD=2$ then determine area of the pentagon $ABCDE$.