Found problems: 85335
2004 Belarusian National Olympiad, 6
At a mathematical olympiad, eight problems were given to 30 contestants. In order to take the difficulty of each problem into account, the jury decided to assign weights to the problems as follows: a problem is worth $n$ points if it was not solved by exactly $n$ contestants. For example, if a problem was solved by all contestants, then it is worth no points. (It is assumed that there are no partial marks for a problem.) Ivan got less points than any other contestant. Find the greatest score he can have.
2019 Jozsef Wildt International Math Competition, W. 66
If $0 < a \leq b$ then$$\frac{2}{\sqrt{3}}\tan^{-1}\left(\frac{2(b^2 - a^2)}{(a^2+2)(b^2+2)}\right)\leq \int \limits_a^b \frac{(x^2+1)(x^2+x+1)}{(x^3 + x^2 + 1) (x^3 + x + 1)}dx\leq \frac{4}{\sqrt{3}}\tan^{-1}\left(\frac{(b-a)\sqrt{3}}{a+b+2(1+ab)}\right)$$
2017 MMATHS, 4
In a triangle $ABC$, let $A_0$ be the point where the interior angle bisector of angle $A$ meets with side $BC$. Similarly define $B_0$ and $C_0$. Prove that $\angle B_0A_0C_0 = 90^o$ if and only if $\angle BAC = 120^o$.
2009 Bosnia And Herzegovina - Regional Olympiad, 1
Prove that for every positive integer $m$ there exists positive integer $n$ such that $m+n+1$ is perfect square and $mn+1$ is perfect cube of some positive integers
1999 All-Russian Olympiad Regional Round, 10.2
Given a circle $\omega$, a point $A$ lying inside $\omega$, and point $B$ ($B \ne A$). All possible triangles $BXY$ are considered, such that the points $X$ and $Y$ lie on $\omega$ and the chord $XY$ passes through the point $A$. Prove that the centers of the circumcircles of the triangles $BXY$ lie on the same straight line.
Kvant 2023, M2766
Let $n{}$ be a natural number. The playing field for a "Master Sudoku" is composed of the $n(n+1)/2$ cells located on or below the main diagonal of an $n\times n$ square. A teacher secretly selects $n{}$ cells of the playing field and tells his student
[list]
[*]the number of selected cells on each row, and
[*]that there is one selected cell on each column.
[/list]The teacher's selected cells form a Master Sudoku if his student can determine them with the given information. How many Master Sudokus are there?
[i]Proposed by T. Amdeberkhan, M. Ruby and F. Petrov[/i]
2014 HMNT, 10
Let $ABCDEF$ be a convex hexagon with the following properties.
(a) $\overline{AC}$ and $\overline{AE}$ trisect $\angle BAF$.
(b) $\overline{BE} \parallel \overline{CD}$ and $\overline{CF} \parallel \overline{DE}$.
(c) $AB = 2AC = 4AE = 8AF$.
Suppose that quadrilaterals $ACDE$ and $ADEF$ have area $2014$ and $1400$, respectively. Find the area of quadrilateral $ABCD$.
2019 Vietnam TST, P6
In the real axis, there is bug standing at coordinate $x=1$. Each step, from the position $x=a$, the bug can jump to either $x=a+2$ or $x=\frac{a}{2}$. Show that there are precisely $F_{n+4}-(n+4)$ positions (including the initial position) that the bug can jump to by at most $n$ steps.
Recall that $F_n$ is the $n^{th}$ element of the Fibonacci sequence, defined by $F_0=F_1=1$, $F_{n+1}=F_n+F_{n-1}$ for all $n\geq 1$.
2007 Germany Team Selection Test, 3
Points $ A_{1}$, $ B_{1}$, $ C_{1}$ are chosen on the sides $ BC$, $ CA$, $ AB$ of a triangle $ ABC$ respectively. The circumcircles of triangles $ AB_{1}C_{1}$, $ BC_{1}A_{1}$, $ CA_{1}B_{1}$ intersect the circumcircle of triangle $ ABC$ again at points $ A_{2}$, $ B_{2}$, $ C_{2}$ respectively ($ A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$). Points $ A_{3}$, $ B_{3}$, $ C_{3}$ are symmetric to $ A_{1}$, $ B_{1}$, $ C_{1}$ with respect to the midpoints of the sides $ BC$, $ CA$, $ AB$ respectively. Prove that the triangles $ A_{2}B_{2}C_{2}$ and $ A_{3}B_{3}C_{3}$ are similar.
2024 New Zealand MO, 3
Let $A,B,C,D,E$ be five different points on the circumference of a circle in that (cyclic) order. Let $F$ be the intersection of chords $BD$ and $CE$. Show that if $AB=AE=AF$ then lines $AF$ and $CD$ are perpendicular.
Swiss NMO - geometry, 2006.2
Let $ABC$ be an equilateral triangle and let $D$ be an inner point of the side $BC$. A circle is tangent to $BC$ at $D$ and intersects the sides $AB$ and $AC$ in the inner points $M, N$ and $P, Q$ respectively. Prove that $|BD| + |AM| + |AN| = |CD| + |AP| + |AQ|$.
2023 Romania National Olympiad, 1
We consider real positive numbers $a,b,c$ such that $a + b + c = 3.$
Prove that $a^2 + b^2 + c^2 + a^2b + b^2 c + c^2 a \ge 6.$
1961 AMC 12/AHSME, 28
If $2137^{753}$ is multiplied out, the units' digit in the final product in the final product is:
${{ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 7}\qquad\textbf{(E)}\ 9} $
2016 Tuymaada Olympiad, 6
The numbers $a$, $b$, $c$, $d$ satisfy $0<a \leq b \leq d \leq c$ and
${a+c=b+d}$. Prove that for every internal point $P$ of a segment with
length $a$ this segment is a side of a circumscribed quadrilateral with consecutive sides
$a$, $b$, $c$, $d$, such that its incircle contains~$P$.
2016 Peru Cono Sur TST, P6
Two circles $\omega_1$ and $\omega_2$, which have centers $O_1$ and $O_2$, respectively, intersect at $A$ and $B$. A line $\ell$ that passes through $B$ cuts to $\omega_1$ again at $C$ and cuts to $\omega_2$ again in $D$, so that points $C, B, D$ appear in that order. The tangents of $\omega_1$ and $\omega_2$ in $C$ and $D$, respectively, intersect in $E$. Line $AE$ intersects again to the circumscribed circumference of the triangle $AO_1O_2$ in $F$. Try that the length of the $EF$ segment is constant, that is, it does not depend on the choice of $\ell$.
2005 Pan African, 2
Let $S$ be a set of integers with the property that any integer root of any non-zero polynomial with coefficients in $S$ also belongs to $S$. If $0$ and $1000$ are elements of $S$, prove that $-2$ is also an element of $S$.
2004 AMC 12/AHSME, 9
The point $ (\minus{}3, 2)$ is rotated $ 90^\circ$ clockwise around the origin to point $ B$. Point $ B$ is then reflected over the line $ y \equal{} x$ to point $ C$. What are the coordinates of $ C$?
$ \textbf{(A)}\ ( \minus{} 3, \minus{} 2)\qquad \textbf{(B)}\ ( \minus{} 2, \minus{} 3)\qquad \textbf{(C)}\ (2, \minus{} 3)\qquad \textbf{(D)}\ (2,3)\qquad \textbf{(E)}\ (3,2)$
1985 Tournament Of Towns, (087) 3
A certain class of $32$ pupils is organised into $33$ clubs , so that each club contains $3$ pupils and no two clubs have the same composition. Prove that there are two clubs which have exactly one common member.
LMT Team Rounds 2021+, 1
Derek and Jacob have a cake in the shape a rectangle with dimensions $14$ inches by $9$ inches. They make a deal to split it: Derek takes home the portion of the cake that is less than one inch from the border, while Jacob takes home the remainder of the cake. Let $D : J$ be the ratio of the amount of cake Derek took to the amount of cake Jacob took, where $D$ and $J$ are relatively prime positive integers. Find $D + J$.
2018 All-Russian Olympiad, 7
Given a sequence of positive integers $a_1,a_2,a_3,...$ defined by $a_n=\lfloor n^{\frac{2018}{2017}}\rfloor$. Show that there exists a positive integer $N$ such that among any $N$ consecutive terms in the sequence, there exists a term whose decimal representation contain digit $5$.
2010 IMC, 5
Suppose that for a function $f: \mathbb{R}\to \mathbb{R}$ and real numbers $a<b$ one has $f(x)=0$ for all $x\in (a,b).$ Prove that $f(x)=0$ for all $x\in \mathbb{R}$ if
\[\sum^{p-1}_{k=0}f\left(y+\frac{k}{p}\right)=0\]
for every prime number $p$ and every real number $y.$
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]
2021 IMO Shortlist, C3
Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$.
Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$.
2025 Poland - Second Round, 6
Let $1\le k\le n$. Suppose that the sequence $a_1, a_2, \ldots, a_n$ satisfies $0\le a_1 \le a_2 \le \ldots \le a_k$ and $0 \le a_n \le a_{n-1} \le \ldots \le a_k$. The sequence $b_1, b_2, \ldots, b_n$ is the nondecreasing permutation of $a_1, a_2, \ldots, a_n$. Prove that
\[\sum_{i=1}^n \sum_{j=1}^n (j-i)^2a_ia_j \le \sum_{i=1}^n \sum_{j=1}^n (j-i)^2b_ib_j \]
2020 Bulgaria EGMO TST, 2
Let $ABC$ be an acute triangle with orthocenter $H$ and altitudes $AA_1$, $BB_1$, $CC_1$. The lines $AB$ and $A_1B_1$ intersect at $C_2$ and $\ell_C$ is the line through the midpoint of $CH$, perpendicular to $CC_2$. The lines $\ell_A$ and $\ell_B$ are defined analogously. Prove that the lines $\ell_A$, $\ell_B$ and $\ell_C$ are concurrent.