Found problems: 85335
1994 Iran MO (2nd round), 1
Let $\overline{a_1a_2a_3\ldots a_n}$ be the representation of a $n-$digits number in base $10.$ Prove that there exists a one-to-one function like $f : \{0, 1, 2, 3, \ldots, 9\} \to \{0, 1, 2, 3, \ldots, 9\}$ such that $f(a_1) \neq 0$ and the number $\overline{ f(a_1)f(a_2)f(a_3) \ldots f(a_n) }$ is divisible by $3.$
1999 Baltic Way, 17
Does there exist a finite sequence of integers $c_1,c_2,\ldots ,c_n$ such that all the numbers $a+c_1,a+c_2,\ldots ,a+c_n$ are primes for more than one but not infinitely many different integers $a$?
2007 Hanoi Open Mathematics Competitions, 14
How many ordered pairs of integers (x; y) satisfy
the equation: 2$x^2$ + $y^2$ + xy = 2(x + y)?
2012 Greece Team Selection Test, 4
Let $n=3k$ be a positive integer (with $k\geq 2$). An equilateral triangle is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two). We colour the points of the grid with three colours (red, blue and green) such that each two neighboring points have different colour.
Finally, the colour of a "trapezoid" will be the colour of the midpoint of its big base.
Find the number of all "trapezoids" in the grid (not necessarily disjoint) and determine the number of red, blue and green "trapezoids".
2025 Belarusian National Olympiad, 10.3
Given two angles $ACT$ and $TCB$, where $A$, $C$ and $B$ lie on a line in that order. A circle $\alpha$ is inscribed in the first angle, and $\beta$ in the second. $\alpha$ is tangent to $AB$ and $CT$ at points $A$ and $E$, and $\beta$ is tangent to $AE$ and $BF$ at $B$ and $F \neq E$. Lines $AE$ and $BF$ intersect at $P$. Circumcircle $\omega$ of triangle $PEF$ intersects $\alpha$ and $\beta$ at $X$ and $Y$ respectively.
Prove that $AX$ and $BY$ intersect on $\omega$.
[i]Matsvei Zorka[/i]
2014 Saint Petersburg Mathematical Olympiad, 2
There are $40$ points on the two parallel lines. We divide it to pairs, such that line segments, that connects point in pair, do not intersect each other ( endpoint from one segment cannot lies on another segment). Prove, that number of ways to do it is less than $3^{39}$
2007 China Team Selection Test, 2
After multiplying out and simplifying polynomial $ (x \minus{} 1)(x^2 \minus{} 1)(x^3 \minus{} 1)\cdots(x^{2007} \minus{} 1),$ getting rid of all terms whose powers are greater than $ 2007,$ we acquire a new polynomial $ f(x).$ Find its degree and the coefficient of the term having the highest power. Find the degree of $ f(x) \equal{} (1 \minus{} x)(1 \minus{} x^{2})...(1 \minus{} x^{2007})$ $ (mod$ $ x^{2008}).$
2015 Auckland Mathematical Olympiad, 3
In the calculation $HE \times EH = WHEW$, where different letters stand for different nonzero digits. Find the values of all the letters.
2011 AIME Problems, 6
Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \le a<b<c<d \le 10$, and $a+d>b+c$. How many ordered quadruples are there?
2020 Simon Marais Mathematics Competition, A3
Determine the set of real numbers $\alpha$ that can be expressed in the form \[\alpha=\sum_{n=0}^{\infty}\frac{x_{n+1}}{x_n^3}\]
where $x_0,x_1,x_2,\dots$ is an increasing sequence of real numbers with $x_0=1$.
1992 IMTS, 3
In a mathematical version of baseball, the umpire chooses a positive integer $m$, $m \leq n$, and you guess positive integers to obtain information about $m$. If your guess is smaller than the umpire's $m$, he calls it a "ball"; if it is greater than or equal to $m$, he calls it a "strike." To "hit" it you must state the the correct value of $m$ after the 3rd strike or the 6th guess, whichever comes first. What is the largest $n$ so that there exists a strategy that will allow you to bat 1.000, i.e. always state $m$ correctly? Describe your strategy in detail.
2012 AMC 10, 5
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is $10\%$. She leaves a $15\%$ tip on the prices of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of $27.50$ for dinner. What is the cost of here dinner without tax or tip?
$ \textbf{(A)}\ \$18\qquad\textbf{(B)}\ \$20\qquad\textbf{(C)}\ \$21\qquad\textbf{(D)}\ \$22\qquad\textbf{(E)}\ \$24$
2022/2023 Tournament of Towns, P5
Given an integer $h > 1$. Let's call a positive common fraction (not necessarily irreducible) [i]good[/i] if the sum of its numerator and denominator is equal to $h$. Let's say that a number $h$ is [i]remarkable[/i] if every positive common fraction whose denominator is less than $h$ can be expressed in terms of good fractions (not necessarily various) using the operations of addition and subtraction.
Prove that $h$ is remarkable if and only if it is prime.
(Recall that an common fraction has an integer numerator and a natural denominator.)
2010 Contests, 2
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that we have $f(x + y) = f(x) + f(y) + f(xy)$ for all $ x,y\in \mathbb{R}$
2005 Taiwan TST Round 2, 1
Positive integers $a,b,c,d$ satisfy $a+c=10$ and \[\displaystyle S=\frac{a}{b} + \frac{c}{d} <1.\] Find the maximum value of $S$.
2012 CHKMO, 4
In $\triangle ABC$, $AB>AC$. In the circumcircle $(O)$ of $\triangle ABC$, $M$ is the midpoint of arc $BAC$. The incircle $(I)$ of $\triangle ABC$ touches $BC$ at $D$, the line through $D$ parallel to $AI$ intersects $(I)$ again at $P$. Prove that $AP$ and $IM$ intersect at a point on $(O)$.
1997 Turkey Junior National Olympiad, 3
$1$ or $-1$ is written in $50$ letters. These letters are put into $50$ envelopes. If you ask, you can learn the product of numbers written into any three letters. At least, how many questions are required to find the product of all of the $50$ numbers?
2023 China Second Round, 4
Let $a=1+10^{-4}$. Consider some $2023\times 2023$ matrix with each entry a real in $[1,a]$. Let $x_i$ be the sum of the elements of the $i$-th row and $y_i$ be the sum of the elements of the $i$-th column for each integer $i\in [1,n]$. Find the maximum possible value of $\dfrac{y_1y_2\cdots y_{2023}}{x_1x_2\cdots x_{2023}}$ (the answer may be expressed in terms of $a$).
2007 India IMO Training Camp, 1
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.
[i]Proposed by Harmel Nestra, Estionia[/i]
2010 South East Mathematical Olympiad, 2
For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.
2015 Denmark MO - Mohr Contest, 3
Triangle $ABC$ is equilateral. The point $D$ lies on the extension of $AB$ beyond $B$, the point $E$ lies on the extension of $CB$ beyond $B$, and $|CD| = |DE|$. Prove that $|AD| = |BE|$.
[img]https://1.bp.blogspot.com/-QnAXFw3ijn0/XzR0YjqBQ3I/AAAAAAAAMU0/0TvhMQtBNjolYHtgXsQo2OPGJzEYSfCwACLcBGAsYHQ/s0/2015%2BMohr%2Bp3.png[/img]
2016 Peru IMO TST, 2
Determine how many $100$-positive integer sequences satisfy the two conditions following:
- At least one term of the sequence is equal to $4$ or $5$.
- Any two adjacent terms differ as a maximum in $2$.
2001 Paraguay Mathematical Olympiad, 3
Find a $10$-digit number, in which no digit is zero, that is divisible by the sum of their digits.
2014 Hanoi Open Mathematics Competitions, 15
Let $a_1,a_2,...,a_9 \ge - 1$ and $a^3_1+a^3_2+...+a^3_9= 0$.
Determine the maximal value of $M = a_1 + a_2 + ... + a_9$.
2021 Denmark MO - Mohr Contest, 1
Georg has a set of sticks. From these sticks he must create a closed figure with the property that each stick makes right angles with its neighbouring sticks. All the sticks must be used. If the sticks have the lengths $1, 1, 2, 2, 2, 3, 3$ and $4$, the figure might for example look like this: [img]https://cdn.artofproblemsolving.com/attachments/9/7/c16a3143a52ec6f442208c63b41f2df1ae735c.png[/img]
(a) Prove that he can create such a figure if the sticks have the lengths $1, 1, 1, 2, 2, 3, 4$ and $4$.
(b) Prove that it cannot be done if the sticks have the lengths $1, 2, 2, 3, 3, 3, 4, 4$ and $4$.
(c) Determine whether it is doable if the sticks have the lengths $1, 2, 2, 2, 3, 3, 3, 4, 4$ and $5$.