Found problems: 85335
2025 JBMO TST - Turkey, 3
Find all positive real solutions $(a, b, c)$ to the following system:
$$
\begin{aligned}
a^2 + \frac{b}{a} &= 8, \\
ab + c^2 &= 18, \\
3a + b + c &= 9\sqrt{3}.
\end{aligned}
$$
2012 Princeton University Math Competition, A6
Consider a pool table with the shape of an equilateral triangle. A ball of negligible size is initially placed at the center of the table. After it has been hit, it will keep moving in the direction it was hit towards and bounce off any edges with perfect symmetry. If it eventually reaches the midpoint of any edge, we mark the midpoint of the entire route that the ball has travelled through. Repeating this experiment, how many points can we mark at most?
[img]https://cdn.artofproblemsolving.com/attachments/5/d/3ae7aad4271b9a417826f3bc8dd7b43aeca5d4.png[/img]
2006 Switzerland Team Selection Test, 1
Let $n$ be natural number and $1=d_1<d_2<\ldots <d_k=n$ be the positive divisors of $n$.
Find all $n$ such that $2n = d_5^2+ d_6^2 -1$.
2015 ELMO Problems, 4
Let $a > 1$ be a positive integer. Prove that for some nonnegative integer $n$, the number $2^{2^n}+a$ is not prime.
[i]Proposed by Jack Gurev[/i]
2007 China Team Selection Test, 3
Show that there exists a positive integer $ k$ such that $ k \cdot 2^{n} \plus{} 1$ is composite for all $ n \in \mathbb{N}_{0}$.
2020 CMIMC Team, 13
Given $10$ points arranged in a equilateral triangular grid of side length $4$, how many ways are there to choose two distinct line segments, with endpoints on the grid, that intersect in exactly one point (not necessarily on the grid)?
MMPC Part II 1996 - 2019, 2017
[b]p1.[/b] Consider a normal $8 \times 8$ chessboard, where each square is labelled with either $1$ or $-1$. Let $a_k$ be the product of the numbers in the $k$th row, and let $b_k$ be the product of the numbers in the $k$th column. Find, with proof, all possible values of $\sum^8_{k=1}(a_kb_k)$.
[b]p2.[/b] Let $\overline{AB}$ be a line segment with $AB = 1$, and $P$ be a point on $\overline{AB}$ with $AP = x$, for some $0 < x < 1$. Draw circles $C_1$ and $C_2$ with $\overline{AP}$, $\overline{PB}$ as diameters, respectively. Let $\overline{AB_1}$, $\overline{AB_2}$ be tangent to $C_2$ at $B_1$ and $B_2$, and let $\overline{BA_1}$;$\overline{BA_2}$ be tangent to $C_1$ at $A_1$ and $A_2$. Now $C_3$ is a circle tangent to $C_2$, $\overline{AB_1}$, and $\overline{AB_2}$; $C_4$ is a circle tangent to $C_1$, $\overline{BA_1}$, and $\overline{BA_2}$.
(a) Express the radius of $C_3$ as a function of $x$.
(b) Prove that $C_3$ and $C_4$ are congruent.
[img]https://cdn.artofproblemsolving.com/attachments/c/a/fd28ad91ed0a4893608b92f5ccbd01088ae424.png[/img]
[b]p3.[/b] Suppose that the graphs of $y = (x + a)^2$ and $x = (y + a)^2$ are tangent to one another at a point on the line $y = x$. Find all possible values of $a$.
[b]p4.[/b] You may assume without proof or justification that the infinite radical expressions $\sqrt{a-\sqrt{a-\sqrt{a-\sqrt{a-...}}}}$ and $\sqrt{a-\sqrt{a+\sqrt{a-\sqrt{a+...}}}}$ represent unique values for $a > 2$.
(a) Find a real number $a$ such that $$\sqrt{a-\sqrt{a-\sqrt{a-\sqrt{a+...}}}}= 2017$$
(b) Show that
$$\sqrt{2018-\sqrt{2018+\sqrt{2018-\sqrt{2018+...}}}}=\sqrt{2017-\sqrt{2017-\sqrt{2017-\sqrt{2017-...}}}}$$
[b]p5.[/b] (a) Suppose that $m, n$ are positive integers such that $7n^2 - m^2 > 0$. Prove that, in fact, $7n^2 - m^2 \ge 3$.
(b) Suppose that $m, n$ are positive integers such that $\frac{m}{n} <\sqrt7$. Prove that, in fact, $\frac{m}{n}+\frac{1}{mn}
<\sqrt7$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2016 Thailand TSTST, 3
Determine whether there exists a positive integer $a$ such that $$2015a,2016a,\dots,2558a$$ are all perfect power.
2021 CMIMC, 1.5
Suppose $f$ is a degree 42 polynomial such that for all integers $0\le i\le 42$,
$$f(i)+f(43+i)+f(2\cdot43+i)+\cdots+f(46\cdot43+i)=(-2)^i$$
Find $f(2021)-f(0)$.
[i]Proposed by Adam Bertelli[/i]
2007 AIME Problems, 1
How many positive perfect squares less than $10^{6}$ are multiples of $24$?
1987 AMC 12/AHSME, 9
The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is
$ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ \frac{2}{3} \qquad\textbf{(E)}\ 2 $
1971 All Soviet Union Mathematical Olympiad, 148
The volumes of the water containing in each of three big enough containers are integers. You are allowed only to relocate some times from one container to another the same volume of the water, that the destination already contains. Prove that you are able to discharge one of the containers.
2006 May Olympiad, 4
With $150$ white cubes of $1 \times 1 \times 1$ a prism of $6 \times 5 \times 5$ is assembled, its six faces are painted blue and then the prism is disassembled. Lucrecia must build a new prism, without holes, exclusively using cubes that have at least one blue face and so that the faces of Lucrecia's prism are all completely blue.
Give the dimensions of the prism with the largest volume that Lucrecia can assemble.
2022 Purple Comet Problems, 17
There are real numbers $x, y,$ and $z$ such that the value of $$x+y+z-\left(\frac{x^2}{5}+\frac{y^2}{6}+\frac{z^2}{7}\right)$$ reaches its maximum of $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n + x + y + z.$
2011 Purple Comet Problems, 14
The five-digit number $12110$ is divisible by the sum of its digits $1 + 2 + 1 + 1 + 0 = 5.$ Find the greatest five-digit number which is divisible by the sum of its digits
1967 Miklós Schweitzer, 7
Let $ U$ be an $ n \times n$ orthogonal matrix. Prove that for any $ n \times n$ matrix $ A$, the matrices \[ A_m=\frac{1}{m+1} \sum_{j=0}^m U^{-j}AU^j\] converge entrywise as $ m \rightarrow \infty.$
[i]L. Kovacs[/i]
2019 IFYM, Sozopol, 8
We are given a $\Delta ABC$. Point $D$ on the circumscribed circle k is such that $CD$ is a symmedian in $\Delta ABC$. Let $X$ and $Y$ be on the rays $\overrightarrow{CB}$ and $\overrightarrow{CA}$, so that $CX=2CA$ and $CY=2CB$. Prove that the circle, tangent externally to $k$ and to the lines $CA$ and $CB$, is tangent to the circumscribed circle of $\Delta XDY$.
2018 Iran MO (3rd Round), 2
Prove that for every prime number $p$ there exist infinity many natural numbers $n$ so that they satisfy:
$2^{2^{2^{ \dots ^{2^n}}}} \equiv n^{2^{2^{\dots ^{2}}}} (mod p)$
Where in both sides $2$ appeared $1397$ times
2016 Latvia Baltic Way TST, 14
Let $ABC$ be a scalene triangle. Let $D$ and $E$ be the points where the incircle touches sides $BC$ and $CA$, respectively. Let $K$ be the common point of line $BC$ and the bisector of the angle $\angle BAC$. Let $AD$ intersect $EK$ in $P$. Prove that $PC$ is perpendicular to $AK$.
2000 France Team Selection Test, 2
$A,B,C,D$ are points on a circle in that order. Prove that $|AB-CD|+|AD-BC| \ge 2|AC-BD|$.
2007 Today's Calculation Of Integral, 235
Show that a function $ f(x)\equal{}\int_{\minus{}1}^1 (1\minus{}|\ t\ |)\cos (xt)\ dt$ is continuous at $ x\equal{}0$.
2009 USAMTS Problems, 2
Alice has three daughters, each of whom has two daughters, each of Alice's six grand-daughters has one daughter. How many sets of women from the family of $16$ can be chosen such that no woman and her daughter are both in the set? (Include the empty set as a possible set.)
2007 Peru IMO TST, 2
Let $ABC$ be a triangle such that $CA \neq CB$,
the points $A_{1}$ and $B_{1}$ are tangency points for the ex-circles relative to sides $CB$ and $CA$,
respectively, and $I$ the incircle.
The line $CI$ intersects the cincumcircle of the triangle $ABC$ in the point $P$.
The line that trough $P$ that is perpendicular to $CP$, intersects the line $AB$ in $Q$.
Prove that the lines $QI$ and $A_{1}B_{1}$ are parallels.
Geometry Mathley 2011-12, 16.2
Let $ABCD$ be a quadrilateral and $P$ a point in the plane of the quadrilateral. Let $M,N$ be on the sides $AC,BD$ respectively such that $PM \parallel BC, PN \parallel AD$. $AC$ meets $BD$ at $E$. Prove that the orthocenter of triangles $EBC, EAD, EMN$ are collinear if and only if $P$ is on the line $AB$.
Đỗ Thanh Sơn
PS. Instead of the word [b]collinear[/b], it was written [b]concurrent[/b], probably a typo.
2013 Danube Mathematical Competition, 4
Let $ABCD$ be a rectangle with $AB \ne BC$ and the center the point $O$. Perpendicular from $O$ on $BD$ intersects lines $AB$ and $BC$ in points $E$ and $F$ respectively. Points $M$ and $N$ are midpoints of segments $[CD]$ and $[AD]$ respectively. Prove that $FM \perp EN$ .