Found problems: 85335
2014 Junior Regional Olympiad - FBH, 1
If $a$ and $b$ are digits, how many are there $4$ digit numbers $\overline{3ab4}$ divisible with $9$ . Which numbers are they ($4$ digit numbers)?
2009 Baltic Way, 12
In a quadrilateral $ABCD$ we have $AB||CD$ and $AB=2CD$. A line $\ell$ is perpendicular to $CD$ and contains the point $C$. The circle with centre $D$ and radius $DA$ intersects the line $\ell$ at points $P$ and $Q$. Prove that $AP\perp BQ$.
2017 Greece Junior Math Olympiad, 1
Let $ABCD$ be a square of side $a$. On side $AD$ consider points $E$ and $Z$ such that $DE=a/3$ and $AZ=a/4$. If the lines $BZ$ and $CE$ intersect at point $H$, calculate the area of the triangle $BCH$ in terms of $a$.
2021 USAMTS Problems, 4
Let ABC be a triangle whose vertices are inside a circle $\Omega$. Prove that we can choose two of the vertices of ABC such that there are infinitely many circles $\omega$ that satisfy the following properties:
1. $\omega$ is inside of $\Omega$,
2. $\omega$ passes through the two chosen vertices, and
3. the third vertex is in the interior of $\omega$ .
2017 Iran MO (2nd Round), 2
Let $ABCD$ be an isosceles trapezoid such that $AB \parallel CD$. Suppose that there exists a point $P$ in $ABCD$ such that $\angle APB > \angle ADC$ and $\angle DPC > \angle ABC$. Prove that $$AB+CD>DA+BC.$$
2019 BMT Spring, 12
$2019$ people (all of whom are perfect logicians), labeled from $1$ to $2019$, partake in a paintball duel. First, they decide to stand in a circle, in order, so that Person $1$ has Person $2$ to his left and person $2019$ to his right. Then, starting with Person $1$ and moving to the left, every person who has not been eliminated takes a turn shooting. On their turn, each person can choose to either shoot one non-eliminated person of his or her choice (which eliminates that person from the game), or deliberately miss. The last person standing wins. If, at any point, play goes around the circle once with no one getting eliminated (that is, if all the people playing decide to miss), then automatic paint sprayers will turn on, and end the game with everyone losing. Each person will, on his or her turn, always pick a move that leads to a win if possible, and, if there is still a choice in what move to make, will prefer shooting over missing, and shooting a person closer to his or her left over shooting someone farther from their left. What is the number of the person who wins this game? Put “$0$” if no one wins.
2012 International Zhautykov Olympiad, 1
Do there exist integers $m, n$ and a function $f\colon \mathbb R \to \mathbb R$ satisfying simultaneously the following two conditions?
$\bullet$ i) $f(f(x))=2f(x)-x-2$ for any $x \in \mathbb R$;
$\bullet$ ii) $m \leq n$ and $f(m)=n$.
2016 LMT, 15
For nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Let $g(n)=3^{f(n)}$. Compute
\[\sum_{i=1}^{1000} g(i).\]
[i]Proposed by Nathan Ramesh
2020 Ukrainian Geometry Olympiad - December, 1
The three sides of the quadrilateral are equal, the angles between them are equal, respectively $90^o$ and $150^o$. Find the smallest angle of this quadrilateral in degrees.
Brazil L2 Finals (OBM) - geometry, 2017.5
Let $ABC$ be a triangle with, $AB$ ≠ $AC$, and let $K$ is your incenter. The points $P$ and $Q$ are the points of the intersections of the circumcicle($BCK$) with the line(s) $AB$ and $AC$, respectively. Let $D$ be intersection of $AK$ and $BC$.
Show that $P, Q, D$ are collinears.
2005 Bosnia and Herzegovina Team Selection Test, 1
Let $H$ be an orthocenter of an acute triangle $ABC$. Prove that midpoints of $AB$ and $CH$ and intersection point of angle bisectors of $\angle CAH$ and $\angle CBH$ lie on the same line.
2013 Bosnia And Herzegovina - Regional Olympiad, 3
Find all integers $a$ such that $\sqrt{\frac{9a+4}{a-6}}$ is rational number
2018 China Team Selection Test, 6
Find all pairs of positive integers $(x, y)$ such that $(xy+1)(xy+x+2)$ be a perfect square .
2022 Turkey EGMO TST, 5
We are given three points $A,B,C$ on a semicircle. The tangent lines at $A$ and $B$ to the semicircle meet the extension of the diameter at points $M,N$ respectively. The line passing through $A$ that is perpendicular to the diameter meets $NC$ at $R$, and the line passing through $B$ that is perpendicular to the diameter meets $MC$ at $S$. If the line $RS$ meets the extension of the diameter at $Z$, prove that $ZC$ is tangent to the semicircle.
2019 Purple Comet Problems, 4
Of the students attending a school athletic event, $80\%$ of the boys were dressed in the school colors, $60\%$ of the girls were dressed in the school colors, and $45\% $ of the students were girls. Find the percentage of students attending the event who were wearing the school colors.
2022 Stanford Mathematics Tournament, 1
George is drawing a Christmas tree; he starts with an isosceles triangle $AB_0C_0$ with $AB_0=AC_0=41$ and $B_0C_0=18$. Then, he draws points $B_i$ and $C_i$ on sides $AB_0$ and $AC_0$, respectively, such that $B_iB_{i+1}=1$ and $C_iC_{i+1}=1$ ($B_{41}=C_{41}=A$). Finally, he uses a green crayon to color in triangles $B_iC_iC_{i+1}$ for $i$ from $0$ to $40$. What is the total area that he colors in?
2007 All-Russian Olympiad Regional Round, 11.5
Find all positive integers $ n$ for which there exist integers $ a,b,c$ such that $ a\plus{}b\plus{}c\equal{}0$ and the number $ a^{n}\plus{}b^{n}\plus{}c^{n}$ is prime.
1987 Romania Team Selection Test, 4
Let $ P(X) \equal{} a_{n}X^{n} \plus{} a_{n \minus{} 1}X^{n \minus{} 1} \plus{} \ldots \plus{} a_{1}X \plus{} a_{0}$ be a real polynomial of degree $ n$. Suppose $ n$ is an even number and:
a) $ a_{0} > 0$, $ a_{n} > 0$;
b) $ a_{1}^{2} \plus{} a_{2}^{2} \plus{} \ldots \plus{} a_{n \minus{} 1}^{2}\leq\frac {4\min(a_{0}^{2} , a_{n}^{2})}{n \minus{} 1}$.
Prove that $ P(x)\geq 0$ for all real values $ x$.
[i]Laurentiu Panaitopol[/i]
2010 India Regional Mathematical Olympiad, 2
Let $P_1(x) = ax^2 - bx - c$, $P_2(x) = bx^2 - cx - a$, $P_3(x) = cx^2 - ax - b$ be three quadratic polynomials. Suppose there exists a real number $\alpha$ such that $P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$. Prove that $a = b = c$.
2024-IMOC, A2
Given integer $n \geq 3$ and $x_1$, $x_2$, …, $x_n$ be $n$ real numbers satisfying $|x_1|+|x_2|+…+|x_n|=1$. Find the minimum of
\[|x_1+x_2|+|x_2+x_3|+…+|x_{n-1}+x_n|+|x_n+x_1|.\]
[i]Proposed by snap7822[/i]
2003 Estonia National Olympiad, 2
Find all positive integers $n$ such that $n+ \left[ \frac{n}{6} \right] \ne \left[ \frac{n}{2} \right] + \left[ \frac{2n}{3} \right]$
2008 IMAR Test, 3
Two circles $ \gamma_{1}$ and $ \gamma_{2}$ meet at points $ X$ and $ Y$. Consider the parallel through $ Y$ to the nearest common tangent of the circles. This parallel meets again $ \gamma_{1}$ and $ \gamma_{2}$ at $ A$, and $ B$ respectively. Let $ O$ be the center of the circle tangent to $ \gamma_{1},\gamma_{2}$ and the circle $ AXB$, situated outside $ \gamma_{1}$ and $ \gamma_{2}$ and inside the circle $ AXB.$ Prove that $ XO$ is the bisector line of the angle $ \angle{AXB}.$
[b]Radu Gologan[/b]
2003 All-Russian Olympiad, 2
The diagonals of a cyclic quadrilateral $ABCD$ meet at $O$. Let $S_1, S_2$ be the circumcircles of triangles $ABO$ and $CDO$ respectively, and $O,K$ their intersection points. The lines through $O$ parallel to $AB$ and $CD$ meet $S_1$ and $S_2$ again at $L$ and $M$, respectively. Points $P$ and $Q$ on segments $OL$ and $OM$ respectively are taken such that $OP : PL = MQ : QO$. Prove that $O,K, P,Q$ lie on a circle.
2010 Purple Comet Problems, 10
A baker uses $6\tfrac{2}{3}$ cups of flour when she prepares $\tfrac{5}{3}$ recipes of rolls. She will use $9\tfrac{3}{4}$ cups of flour when she prepares $\tfrac{m}{n}$ recipes of rolls where m and n are relatively prime positive integers. Find $m + n.$
2006 MOP Homework, 1
In how many ways can the set $N ={1, 2, \cdots, n}$ be partitioned in the form $p(N) = A_{1}\cup A_{2}\cup \cdots \cup A_{k},$ where $A_{i}$ consists of elements forming arithmetic progressions, all with the same common positive difference $d_{p}$ and of length at least one? At least two?
[hide="Solution"]
[b]Part 1[/b]
Claim: There are $2^{n}-2$ ways of performing the desired partitioning.
Let $d(k)$ equal the number of ways $N$ can be partitioned as above with common difference $k.$ We are thus trying to evaluate
$\sum_{i=1}^{n-1}d(i)$
[b]Lemma: $d(i) = 2^{n-i}$[/b]
We may divide $N$ into various rows where the first term of each row denotes a residue $\bmod{i}.$ The only exception is the last row, as no row starts with $0$; the last row will start with $i.$ We display the rows as such:
$1, 1+i, 1+2i, 1+3i, \cdots$
$2, 2+i, 2+2i, 2+3i, \cdots$
$\cdots$
$i, 2i, 3i, \cdots$
Since all numbers have one lowest remainder $\bmod{i}$ and we have covered all possible remainders, all elements of $N$ occur exactly once in these rows.
Now, we can take $k$ line segments and partition a given row above; all entries within two segments would belong to the same set. For example, we can have:
$1| 1+i, 1+2i, 1+3i | 1+4i | 1+5i, 1+6i, 1+7i, 1+8i,$
which would result in the various subsets: ${1},{1+i, 1+2i, 1+3i},{1+4i},{1+5i, 1+6i, 1+7i, 1+8i}.$ For any given row with $k$ elements, we can have at most $k-1$ segments. Thus, we can arrange any number of segments where the number lies between $0$ and $k-1$, inclusive, in:
$\binom{k-1}{0}+\binom{k-1}{1}+\cdots+\binom{k-1}{k-1}= 2^{k-1}$
ways. Applying the same principle to the other rows and multiplying, we see that the result always gives us $2^{n-i},$ as desired.
We now proceed to the original proof.
Since $d(i) = 2^{n-i}$ by the above lemma, we have:
$\sum_{i=1}^{n-1}d(i) = \sum_{i=1}^{n-1}2^{n-i}= 2^{n}-2$
Thus, there are $2^{n}-2$ ways of partitioning the set as desired.
[b]Part 2[/b]
Everything is the same as above, except the lemma slightly changes to $d(i) = 2^{n-i}-i.$ Evaluating the earlier sum gives us:
$\sum_{i=1}^{n-1}d(i) = \sum_{i=1}^{n-1}2^{n-i}-i = 2^{n}-\frac{n(n-1)}{2}-2$
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