Found problems: 85335
2017 Princeton University Math Competition, A5/B7
Let $p(n) = n^4-6n^2-160$. If $a_n$ is the least odd prime dividing $q(n) = |p(n-30) \cdot p(n+30)|$, find $\sum_{n=1}^{2017} a_n$. ($a_n = 3$ if $q(n) = 0$.)
1956 Moscow Mathematical Olympiad, 342
Given three numbers $x, y, z$ denote the absolute values of the differences of each pair by $x_1,y_1, z_1$. From $x_1, y_1, z_1$ form in the same fashion the numbers $x_2, y_2, z_2$, etc. It is known that $x_n = x,y_n = y, z_n = z$ for some $n$. Find $y$ and $z$ if $x = 1$.
2021 Portugal MO, 1
Joana divided $365$ by all integers from $1$ to $365$ and added all the remainders. Then she divided $366$ by all the integers from $1$ to $366$ and also added all the remainders. Which of the two sums is greater and what is the difference between them?
Brazil L2 Finals (OBM) - geometry, 2015.3
Let $ABC$ be a triangle and $n$ a positive integer. Consider on the side $BC$ the points $A_1, A_2, ..., A_{2^n-1}$ that divide the side into $2^n$ equal parts, that is, $BA_1=A_1A_2=...=A_{2^n-2}A_{2^n-1}=A_{2^n-1}C$. Set the points $B_1, B_2, ..., B_{2^n-1}$ and $C_1, C_2, ..., C_{2^n-1}$ on the sides $CA$ and $AB$, respectively, analogously. Draw the line segments $AA_1, AA_2, ..., AA_{2^n-1}$, $BB_1, BB_2, ..., BB_{2^n-1}$ and $CC_1, CC_2, ..., CC_{2^n-1}$. Find, in terms of $n$, the number of regions into which the triangle is divided.
2018 CMIMC Individual Finals, 2
How many integer values of $k$, with $1 \leq k \leq 70$, are such that $x^{k}-1 \equiv 0 \pmod{71}$ has at least $\sqrt{k}$ solutions?
2023 Iran MO (3rd Round), 1
In triangle $\triangle ABC$ , $M, N$ are midpoints of $AC,AB$ respectively. Assume that $BM,CN$ cuts $(ABC)$ at $M',N'$ respectively. Let $X$ be on the extention of $BC$ from $B$ st $\angle N'XB=\angle ACN$. And define $Y$ similarly on the extention of $BC$ from $C$. Prove that $AX=AY$.
2014 IMO Shortlist, N6
Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$. On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$.
[i]Proposed by Serbia[/i]
2022 Cyprus JBMO TST, 1
Determine all real numbers $x\in\mathbb{R}$ for which
\[
\left\lfloor \frac{x}{2} \right\rfloor + \left\lfloor \frac{x}{3} \right\rfloor=x-2022.
\]
The notation $\lfloor z \rfloor$, for $z\in\mathbb{R}$, denotes the largest integer which is less than or equal to $z$. For example:
\[\lfloor 3.98 \rfloor =3 \quad \text{and} \quad \lfloor 0.14 \rfloor =0.\]
1987 IMO Longlists, 54
Let $n$ be a natural number. Solve in integers the equation
\[x^n + y^n = (x - y)^{n+1}.\]
1972 IMO Longlists, 33
A rectangle $ABCD$ is given whose sides have lengths $3$ and $2n$, where $n$ is a natural number. Denote by $U(n)$ the number of ways in which one can cut the rectangle into rectangles of side lengths $1$ and $2$.
$(a)$ Prove that
\[U(n + 1)+U(n -1) = 4U(n);\]
$(b)$ Prove that
\[U(n) =\frac{1}{2\sqrt{3}}[(\sqrt{3} + 1)(2 +\sqrt{3})^n + (\sqrt{3} - 1)(2 -\sqrt{3})^n].\]
2002 China Team Selection Test, 1
Given triangle $ ABC$ and $ AB\equal{}c$, $ AC\equal{}b$ and $ BC\equal{}a$ satisfying $ a \geq b \geq c$, $ BE$ and $ CF$ are two interior angle bisectors. $ P$ is a point inside triangle $ AEF$. $ R$ and $ Q$ are the projections of $ P$ on sides $ AB$ and $ AC$.
Prove that $ PR \plus{} PQ \plus{} RQ < b$.
2019 AMC 8, 23
After Euclid High School's last basketball game, it was determined that $\frac{1}{4}$ of the team's points were scored by Alexa and $\frac{2}{7}$ were scored by Brittany. Chelsea scored 15 points. None of the other 7 team members scored more than 2 points. What was the total number of points scored by the other 7 team members?
$\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14$
2010 Today's Calculation Of Integral, 636
Let $a>1$ be a constant. In the $xy$-plane, let $A(a,\ 0),\ B(a,\ \ln a)$ and $C$ be the intersection point of the curve $y=\ln x$ and the $x$-axis. Denote by $S_1$ the area of the part bounded by the $x$-axis, the segment $BA$ and the curve $y=\ln x$
(1) For $1\leq b\leq a$, let $D(b,\ \ln b)$. Find the value of $b$ such that the area of quadrilateral $ABDC$ is the closest to $S_1$ and find the area $S_2$.
(2) Find $\lim_{a\rightarrow \infty} \frac{S_2}{S_1}$.
[i]1992 Tokyo University entrance exam/Science[/i]
2015 Belarus Team Selection Test, 1
Find all positive integers $n$ such that $n=q(q^2-q-1)=r(2r+1)$ for some primes $q$ and $r$.
B.Gilevich
2012 AIME Problems, 15
Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac mn$, where m and n are relatively prime positive integers. Find $m + n$.
2022 Nigerian MO round 3, Problem 1
Integer sequence $(x_{n})$ is defined as follows;
$x_{1} = 1$, and for each integer $n \geq 1$, $x_{n+1}$ is equal to the largest number that can be obtained by permutation of the digits of $x_{n}+2$. Find the smallest $n$ for which the decimal representation of $x_{n}$ contains exactly $2022$ digits
2000 All-Russian Olympiad Regional Round, 8.1
Non-zero numbers $a$ and $b$ satisfy the equality $$a^2b^2(a^2b^2 + 4) = 2(a^6 + b^6).$$ Prove that at least one of them is irrational.
2000 Belarus Team Selection Test, 2.4
In a triangle $ABC$ with $AC = b \ne BC = a$, points $E,F$ are taken on the sides $AC,BC$ respectively such that $AE = BF =\frac{ab}{a+b}$. Let $M$ and $N$ be the midpoints of $AB$ and $EF$ respectively, and $P$ be the intersection point of the segment $EF$ with the bisector of $\angle ACB$. Find the ratio of the area of $CPMN$ to that of $ABC$.
2023 Harvard-MIT Mathematics Tournament, 24
Let $AXBY$ be a cyclic quadrilateral, and let line $AB$ and line $XY$ intersect at $C.$ Suppose $AX \cdot AY = 6, BX \cdot BY=5,$ and $CX \cdot CY=4.$ Compute $AB^2.$
2003 Czech And Slovak Olympiad III A, 4
Let be given an obtuse angle $AKS$ in the plane. Construct a triangle $ABC$ such that $S$ is the midpoint of $BC$ and $K$ is the intersection point of $BC$ with the bisector of $\angle BAC$.
Taiwan TST 2015 Round 1, 3
Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle.
[i]Proposed by Jack Edward Smith, UK[/i]
2014 Polish MO Finals, 3
In an acute triangle $ABC$ point $D$ is the point of intersection of altitude $h_a$ and side $BC$, and points $M, N$ are orthogonal projections of point $D$ on sides $AB$ and $AC$. Lines $MN$ and $AD$ cross the circumcircle of triangle $ABC$ at points $P, Q$ and $A, R$. Prove that point $D$ is the center of the incircle of $PQR$.
2011 Tuymaada Olympiad, 1
Red, blue, and green children are arranged in a circle. When a teacher asked the red children that have a green neighbor to raise their hands, $20$ children raised their hands. When she asked the blue children that have a green neighbor to raise their hands, $25$ children raised their hands. Prove that some child that raised her hand had two green neighbors.
2006 QEDMO 2nd, 3
Prove the inequality
$\frac{b^2+c^2-a^2}{a\left(b+c\right)}+\frac{c^2+a^2-b^2}{b\left(c+a\right)}+\frac{a^2+b^2-c^2}{c\left(a+b\right)}\geq\frac32$
for any three positive reals $a$, $b$, $c$.
[i]Comment.[/i] This was an attempt of creating a contrast to the (rather hard) inequality at the QEDMO before. However, it turned out to be more difficult than I expected (a wrong solution was presented during the competition).
Darij
2014 IMO, 4
Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$. Let $M$ and $N$ be the points on $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$ and $Q$ is the midpoint of $AN$. Prove that the intersection of $BM$ and $CN$ is on the circumference of triangle $ABC$.
[i]Proposed by Giorgi Arabidze, Georgia.[/i]