Found problems: 85335
2024 JHMT HS, 4
Let $x$ be a real number satisfying
\[ \sqrt[3]{125-x^3}-\sqrt[3]{27-x^3}=7. \]
Compute $|\sqrt[3]{125-x^3}+\sqrt[3]{27-x^3}|$.
2021 Belarusian National Olympiad, 11.2
Points $A_1,B_1,C_1$ lie on sides $BC, CA, AB$ of an acute-angled triangle, respectively. Denote by $P, Q, R$ the intersections of $BB_1$ and $CC_1$, $CC_1$ and $AA_1$, $AA_1$ and $BB_1$. If triangle $PQR$ is similar to $ABC$ and $\angle AB_1C_1 = \angle BC_1A_1 = \angle CA_1B_1$, prove that $ABC$ is equilateral.
2011 Czech and Slovak Olympiad III A, 4
Consider a quadratic polynomial $ax^2+bx+c$ with real coefficients satisfying $a\ge 2$, $b\ge 2$, $c\ge 2$. Adam and Boris play the following game. They alternately take turns with Adam first. On Adam’s turn, he can choose one of the polynomial’s coefficients and replace it with the sum of the other two coefficients. On Boris’s turn, he can choose one of the polynomial’s coefficients and replace it with the product of the other two coefficients. The winner is the player who first produces a polynomial with two distinct real roots. Depending on the values of $a$, $b$ and $c$, determine who has a winning strategy.
2006 Purple Comet Problems, 6
The positive integers $v, w, x, y$, and $z$ satisfy the equation \[v + \frac{1}{w + \frac{1}{x + \frac{1}{y + \frac{1}{z}}}} = \frac{222}{155}.\] Compute $10^4 v + 10^3 w + 10^2 x + 10 y + z$.
2016 Iran Team Selection Test, 5
Let $AD,BF,CE$ be altitudes of triangle $ABC$.$Q$ is a point on $EF$ such that $QF=DE$ and $F$ is between $E,Q$.$P$ is a point on $EF$ such that $EP=DF$ and $E$ is between $P,F$.Perpendicular bisector of $DQ$ intersect with $AB$ at $X$ and perpendicular bisector of $DP$ intersect with $AC$ at $Y$.Prove that midpoint of $BC$ lies on $XY$.
2018 PUMaC Live Round, Calculus 1
Freddy the king of flavortext has an infinite chest of coins. For each number \(p\) in the interval \([0, 1]\), Freddy has a coin that has probability \(p\) of coming up heads. Jenny the Joyous pulls out a random coin from the chest and flips it 10 times, and it comes up heads every time. She then flips the coin again. If the probability that the coin comes up heads on this 11th flip is \(\frac{p}{q}\) for two integers \(p, q\), find \(p + q\).
Note: flavortext is made up
2004 Singapore Team Selection Test, 1
Let $x_0, x_1, x_2, \ldots$ be the sequence defined by
$x_i= 2^i$ if $0 \leq i \leq 2003$
$x_i=\sum_{j=1}^{2004} x_{i-j}$ if $i \geq 2004$
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by 2004.
1972 All Soviet Union Mathematical Olympiad, 165
Let $O$ be the intersection point of the diagonals of the convex quadrangle $ABCD$ . Prove that the line drawn through the points of intersection of the medians of triangles $AOB$ and $COD$ is orthogonal to the line drawn through the points of intersection of the heights of triangles $BOC$ and $AOD$ .
2012 Bosnia And Herzegovina - Regional Olympiad, 2
Let $a$, $b$, $c$ and $d$ be integers such that $ac$, $bd$ and $bc+ad$ are divisible with positive integer $m$. Show that numbers $bc$ and $ad$ are divisible with $m$
2003 Mexico National Olympiad, 1
Find all positive integers with two or more digits such that if we insert a $0$ between the units and tens digits we get a multiple of the original number.
2007 Tournament Of Towns, 2
Initially, the number $1$ and two positive numbers $x$ and $y$ are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write on the blackboard, in a finite number of moves, the number
[list][b]a)[/b] $x^2$;
[b]b)[/b] $xy$?[/list]
2017 VJIMC, 4
A positive integer $t$ is called a Jane's integer if $t = x^3+y^2$ for some positive integers $x$ and $y$. Prove
that for every integer $n \ge 2$ there exist infinitely many positive integers $m$ such that the set of $n^2$ consecutive
integers $\{m+1,m+2,\dotsc,m+n^2\}$ contains exactly $n + 1$ Jane's integers.
2018 Turkey Team Selection Test, 3
A Retired Linguist (R.L.) writes in the first move a word consisting of $n$ letters, which are all different. In each move, he determines the maximum $i$, such that the word obtained by reversing the first $i$ letters of the last word hasn't been written before, and writes this new word. Prove that R.L. can make $n!$ moves.
2017 Saudi Arabia Pre-TST + Training Tests, 9
Let $ABC$ be a triangle inscribed in circle $(O)$, with its altitudes $BH_b, CH_c$ intersect at orthocenter $H$ ($H_b \in AC$, $H_c \in AB$). $H_bH_c$ meets $BC$ at $P$. Let $N$ be the midpoint of $AH, L$ be the orthogonal projection of $O$ on the symmedian with respect to angle $A$ of triangle $ABC$. Prove that $\angle NLP = 90^o$.
2017 Junior Balkan Team Selection Tests - Moldova, Problem 6
Let $a,b$ and $c$ be real numbers such that $|a+b|+|b+c|+|c+a|=8.$
Find the maximum and minimum value of the expression $P=a^2+b^2+c^2.$
2025 6th Memorial "Aleksandar Blazhevski-Cane", P4
Let $ABCDE$ be a pentagon such that $\angle DCB < 90^{\circ} < \angle EDC$. The circle with diameter $BD$ intersects the line $BC$ again at $F$, and the circle with diameter $DE$ intersects the line $CE$ again at $G$. Prove that the second intersection ($\neq D$) of the circumcircle of $\triangle DFG$ and the circle with diameter $AD$ lies on $AC$.
Proposed by [i]Petar Filipovski[/i]
2022 Romania Team Selection Test, 3
Let $n\geq 2$ be an integer. Let $a_{ij}, \ i,j=1,2,\ldots,n$ be $n^2$ positive real numbers satisfying the following conditions:
[list=1]
[*]For all $i=1,\ldots,n$ we have $a_{ii}=1$ and,
[*]For all $j=2,\ldots,n$ the numbers $a_{ij}, \ i=1,\ldots, j-1$ form a permutation of $1/a_{ji}, \ i=1,\ldots, j-1.$
[/list]
Given that $S_i=a_{i1}+\cdots+a_{in}$, determine the maximum value of the sum $1/S_1+\cdots+1/S_n.$
2015 Czech-Polish-Slovak Junior Match, 4
Let $ABC$ ne a right triangle with $\angle ACB=90^o$. Let $E, F$ be respecitvely the midpoints of the $BC, AC$ and $CD$ be it's altitude. Next, let $P$ be the intersection of the internal angle bisector from $A$ and the line $EF$. Prove that $P$ is the center of the circle inscribed in the triangle $CDE$ .
1997 Romania Team Selection Test, 3
The vertices of a regular dodecagon are coloured either blue or red. Find the number of all possible colourings which do not contain monochromatic sub-polygons.
[i]Vasile Pop[/i]
2001 AMC 12/AHSME, 12
How many positive integers not exceeding 2001 are multiple of 3 or 4 but not 5?
$ \textbf{(A)} \ 768 \qquad \textbf{(B)} \ 801 \qquad \textbf{(C)} \ 934 \qquad \textbf{(D)} \ 1067 \qquad \textbf{(E)} \ 1167$
2009 Princeton University Math Competition, 2
It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has $k$ weights of integral values. What is the minimum $k$ for which there exist weights that satisfy this condition?
1943 Eotvos Mathematical Competition, 3
Let $a < b < c < d$ be real numbers and $(x,y, z,t)$ be any permutation of $a$,$b$, $c$ and $d$. What are the maximum and minimum values of the expression $$(x - y)^2 + (y- z)^2 + (z - t)^2 + (t - x)^2?$$
2015 May Olympiad, 4
We say that a number is [i]superstitious [/i] when it is equal to $13$ times the sum of its digits . Find all superstitious numbers.
May Olympiad L2 - geometry, 2008.2
Let $ABCD$ be a rectangle and $P$ be a point on the side$ AD$ such that $\angle BPC = 90^o$. The perpendicular from $A$ on $BP$ cuts $BP$ at $M$ and the perpendicular from $D$ on $CP$ cuts $CP$ in $N$. Show that the center of the rectangle lies in the $MN$ segment.
1996 Polish MO Finals, 1
Find all pairs $(n,r)$ with $n$ a positive integer and $r$ a real such that $2x^2+2x+1$ divides $(x+1)^n - r$.