Found problems: 85335
2015 Germany Team Selection Test, 1
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]
[i]Proposed by Titu Andreescu, USA[/i]
1998 French Mathematical Olympiad, Problem 5
Let $A$ be a set of $n\ge3$ points in the plane, no three of which are collinear. Show that there is a set $S$ of $2n-5$ points in the plane such that, for each triangle with vertices in $A$, there exists a point in $S$ which is strictly inside that triangle.
2023 Benelux, 1
Find all functions $f\colon\mathbb{R}\to\mathbb{R}$ such that
$(x-y)\bigl(f(x)+f(y)\bigr)\leqslant f\bigl(x^2-y^2\bigr)$ for all $x,y\in\mathbb{R}$.
2025 Korea Winter Program Practice Test, P7
There are $2025$ positive integers $a_1, a_2, \cdots, a_{2025}$ are placed around a circle. For any $k = 1, 2, \cdots, 2025$, $a_k \mid a_{k-1} + a_{k+1}$ where indices are considered modulo $n$. Prove that there exists a positive integer $N$ such that satisfies the following condition.
[list]
[*] [b](Condition)[/b] For any positive integer $n > N$, when $a_1 = n^n$, $a_1, a_2, \cdots, a_{2025}$ are all multiples of $n$.
[/list]
1974 IMO Shortlist, 5
Let $A_r,B_r, C_r$ be points on the circumference of a given circle $S$. From the triangle $A_rB_rC_r$, called $\Delta_r$, the triangle $\Delta_{r+1}$ is obtained by constructing the points $A_{r+1},B_{r+1}, C_{r+1} $on $S$ such that $A_{r+1}A_r$ is parallel to $B_rC_r$, $B_{r+1}B_r$ is parallel to $C_rA_r$, and $C_{r+1}C_r$ is parallel to $A_rB_r$. Each angle of $\Delta_1$ is an integer number of degrees and those integers are not multiples of $45$. Prove that at least two of the triangles $\Delta_1,\Delta_2, \ldots ,\Delta_{15}$ are congruent.
2016 CMIMC, 2
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of (positive) integers such that $k$ divides $\gcd(a_{k-1},a_k)$ for all $k\geq 2$. Compute the smallest possible value of $a_1+a_2+\cdots+a_{10}$.
2021 Peru IMO TST, P1
For any positive integer $n$, we define $S(n)$ to be the sum of its digits in the decimal representation. Prove that for any positive integer $m$, there exists a positive integer $n$ such that $S(n)-S(n^2)>m$.
1975 Chisinau City MO, 93
Prove that $(a^2 + b^2 + c^2)^ 2 = 2 (a^4 + b^4 + c^4)$ if $a + b + c = 0$.
2013 HMNT, 5
In triangle $ABC$, $\angle BAC=60^o$/ Let $\omega$ be a circle tangent to segment $AB$ at point $D$ and segment $AC$ at point $E$. Suppose $\omega$ intersects segment $BC$ at points $F$ and $G$ such that$ F$ lies in between $B$ and $G$. Given that $AD = FG = 4$ and $BF = \frac12$ , find the length of $CG$.
2023 Tuymaada Olympiad, 3
Prove that for every positive integer $n \geq 2$, $$\frac{\sum_{1\leq i \leq n} \sqrt[3]{\frac{i}{n+1}}}{n} \leq \frac{\sum_{1\leq i \leq n-1} \sqrt[3]{\frac{i}{n}}}{n-1}.$$
2016 CCA Math Bonanza, L4.3
Let $ABC$ be a non-degenerate triangle with perimeter $4$ such that $a=bc\sin^2A$. If $M$ is the maximum possible area of $ABC$ and $m$ is the minimum possible area of $ABC$, then $M^2+m^2$ can be expressed in the form $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $a+b$.
[i]2016 CCA Math Bonanza Lightning #4.3[/i]
2023 Balkan MO Shortlist, G4
Let $O$ and $H$ be the circumcenter and orthocenter of a scalene triangle $ABC$, respectively. Let $D$ be the intersection point of the lines $AH$ and $BC$. Suppose the line $OH$ meets the side $BC$ at $X$. Let $P$ and $Q$ be the second intersection points of the circumcircles of $\triangle BDH$ and $\triangle CDH$ with the circumcircle of $\triangle ABC$, respectively. Show that the four points $P, D, Q$ and $X$ lie on a circle.
2018 Serbia National Math Olympiad, 1
Let $\triangle ABC$ be a triangle with incenter $I$. Points $P$ and $Q$ are chosen on segmets $BI$ and $CI$ such that $2\angle PAQ=\angle BAC$. If $D$ is the touch point of incircle and side $BC$ prove that $\angle PDQ=90$.
2000 Harvard-MIT Mathematics Tournament, 25
Find the next number in the sequence $131, 111311, 311321, 1321131211,\cdots$
1962 AMC 12/AHSME, 23
In triangle $ ABC$, $ CD$ is the altitude to $ AB$ and $ AE$ is the altitude to $ BC.$ If the lengths of $ AB, CD,$ and $ AE$ are known, the length of $ DB$ is:
$ \textbf{(A)}\ \text{not determined by the information given} \qquad$
$ \textbf{(B)}\ \text{determined only if A is an acute angle} \qquad$
$ \textbf{(C)}\ \text{determined only if B is an acute angle} \qquad$
$ \textbf{(D)}\ \text{determined only in ABC is an acute triangle} \qquad$
$ \textbf{(E)}\ \text{none of these is correct}$
2017 Harvard-MIT Mathematics Tournament, 10
Let $\mathbb{N}$ denote the natural numbers. Compute the number of functions $f:\mathbb{N}\rightarrow \{0, 1, \dots, 16\}$ such that $$f(x+17)=f(x)\qquad \text{and} \qquad f(x^2)\equiv f(x)^2+15 \pmod {17}$$ for all integers $x\ge 1$.
2010 CHMMC Fall, 2
Alfonso teaches Francis how to draw a spiral in the plane: First draw half of a unit circle. Starting at one of the ends, draw half a circle with radius $1/2$. Repeat this process at the endpoint of each half circle, where each time the radius is half of the previous half-circle. Assuming you can’t stop Francis from drawing the entire spiral, compute the total length of the spiral.
1956 Moscow Mathematical Olympiad, 325
On sides $AB$ and $CB$ of $\vartriangle ABC$ there are drawn equal segments, $AD$ and $CE$, respectively, of arbitrary length (but shorter than min($AB,BC$)). Find the locus of midpoints of all possible segments $DE$.
2008 Poland - Second Round, 2
We are given a triangle $ ABC$ such that $ AC \equal{} BC$. There is a point $ D$ lying on the segment $ AB$, and $ AD < DB$. The point $ E$ is symmetrical to $ A$ with respect to $ CD$. Prove that:
\[\frac {AC}{CD} \equal{} \frac {BE}{BD \minus{} AD}\]
2017 HMIC, 3
Let $v_1, v_2, \ldots, v_m$ be vectors in $\mathbb{R}^n$, such that each has a strictly positive first coordinate. Consider the following process. Start with the zero vector $w = (0, 0, \ldots, 0) \in \mathbb{R}^n$. Every round, choose an $i$ such that $1 \le i \le m$ and $w \cdot v_i \le 0$, and then replace $w$ with $w + v_i$.
Show that there exists a constant $C$ such that regardless of your choice of $i$ at each step, the process is guaranteed to terminate in (at most) $C$ rounds. The constant $C$ may depend on the vectors $v_1, \ldots, v_m$.
2019 Online Math Open Problems, 25
The sequence $f_0, f_1, \dots$ of polynomials in $\mathbb{F}_{11}[x]$ is defined by $f_0(x) = x$ and $f_{n+1}(x) = f_n(x)^{11} - f_n(x)$ for all $n \ge 0$. Compute the remainder when the number of nonconstant monic irreducible divisors of $f_{1000}(x)$ is divided by $1000$.
[i]Proposed by Ankan Bhattacharya[/i]
2019 Regional Olympiad of Mexico West, 4
Let $ABC$ be a triangle. $M$ the midpoint of $AB$ and $L$ the midpoint of $BC$. We denote by $G$ the intersection of $AL$ with $CM$ and we take $E$ a point such that $G$ is the midpoint of the segment $AE$. Prove that the quadrilateral $MCEB$ is cyclic if and only if $MB = BG$.
2005 Iran MO (3rd Round), 2
Suppose $O$ is circumcenter of triangle $ABC$. Suppose $\frac{S(OAB)+S(OAC)}2=S(OBC)$. Prove that the distance of $O$ (circumcenter) from the radical axis of the circumcircle and the 9-point circle is \[\frac {a^2}{\sqrt{9R^2-(a^2+b^2+c^2)}}\]
2016 Math Prize for Girls Olympiad, 3
Let $n$ be a positive integer. Let $x_1$, $x_2$, $\ldots\,$, $x_n$ be a sequence of $n$ real numbers. Say that a sequence $a_1$, $a_2$, $\ldots\,$, $a_n$ is [i]unimodular[/i] if each $a_i$ is $\pm 1$. Prove that
\[
\sum a_1 a_2 \ldots a_n (a_1x_1 + a_2x_2 + \cdots + a_nx_n)^n = 2^{n} n!\, x_1 x_2 \ldots x_n ,
\]
where the sum is over all $2^{n}$ unimodular sequences $a_1$, $a_2$, $\ldots\,$, $a_n$.
1997 Irish Math Olympiad, 3
Let $ A$ be a subset of $ \{ 0,1,2,...,1997 \}$ containing more than $ 1000$ elements. Prove that either $ A$ contains a power of $ 2$ (that is, a number of the form $ 2^k$ with $ k\equal{}0,1,2,...)$ or there exist two distinct elements $ a,b \in A$ such that $ a\plus{}b$ is a power of $ 2$.