Found problems: 85335
2013 Online Math Open Problems, 21
Let $ABC$ be a triangle with $AB = 5$, $AC = 8$, and $BC = 7$. Let $D$ be on side $AC$ such that $AD = 5$ and $CD = 3$. Let $I$ be the incenter of triangle $ABC$ and $E$ be the intersection of the perpendicular bisectors of $\overline{ID}$ and $\overline{BC}$. Suppose $DE = \frac{a\sqrt{b}}{c}$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.
[i]Proposed by Ray Li[/i]
2007 Sharygin Geometry Olympiad, 8
Three circles pass through a point $P$, and the second points of their intersection $A, B, C$ lie on a straight line. Let $A_1 B_1, C_1$ be the second meets of lines $AP, BP, CP$ with the corresponding circles. Let $C_2$ be the intersections of lines $AB_1$ and $BA_1$. Let $A_2, B_2$ be defined similarly. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal,
2015 India IMO Training Camp, 2
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.
[i]Proposed by Belgium[/i]
2006 ISI B.Stat Entrance Exam, 2
Suppose that $a$ is an irrational number.
(a) If there is a real number $b$ such that both $(a+b)$ and $ab$ are rational numbers, show that $a$ is a quadratic surd. ($a$ is a quadratic surd if it is of the form $r+\sqrt{s}$ or $r-\sqrt{s}$ for some rationals $r$ and $s$, where $s$ is not the square of a rational number).
(b) Show that there are two real numbers $b_1$ and $b_2$ such that
i) $a+b_1$ is rational but $ab_1$ is irrational.
ii) $a+b_2$ is irrational but $ab_2$ is rational.
(Hint: Consider the two cases, where $a$ is a quadratic surd and $a$ is not a quadratic surd, separately).
2000 Moldova National Olympiad, Problem 1
Find all positive integers $a$ for which $a^{2000}-1$ is divisible by $10$.
2024 Kazakhstan National Olympiad, 4
Prove that for any positive integers $a$, $b$, $c$, at least one of the numbers $a^3b+1$, $b^3c+1$, $c^3a+1$ is not divisible by $a^2+b^2+c^2$.
2008 Balkan MO Shortlist, A3
Let $(a_m)$ be a sequence satisfying $a_n \geq 0$, $n=0,1,2,\ldots$ Suppose there exists $A >0$, $a_m - a_{m+1}$ $\geq A a_m ^2$ for all $m \geq 0$. Prove that there exists $B>0$ such that
\begin{align*} a_n \le \frac{B}{n} \qquad \qquad \text{for }1 \le n \end{align*}
2018 Moldova EGMO TST, 3
Let $\triangle ABC $ be an acute triangle.$O$ denote its circumcenter.Points $D$,$E$,$F$ are the midpoints of the sides $BC$,$CA$,and $AB$.Let $M$ be a point on the side $BC$ . $ AM \cap EF = \big\{ N \big\} $ . $ON \cap \big( ODM \big) = \big\{ P \big\} $ Prove that $M'$ lie on $\big(DEF\big)$ where $M'$ is the symmetrical point of $M$ thought the midpoint of $DP$.
2015 Moldova Team Selection Test, 3
Consider an acute triangle $ABC$, points $E,F$ are the feet of the perpendiculars from $B$ and $C$ in $\triangle ABC$. Points $I$ and $J$ are the projections of points $F,E$ on the line $BC$, points $K,L$ are on sides $AB,AC$ respectively such that $IK \parallel AC$ and $JL \parallel AB$. Prove that the lines $IE$,$JF$,$KL$ are concurrent.
1976 Czech and Slovak Olympiad III A, 6
Consider two non-parallel half-planes $\pi,\pi'$ with the common boundary line $p.$ Four different points $A,B,C,D$ are given in the half-plane $\pi.$ Similarly, four points $A',B',C',D'\in\pi'$ are given such that $AA'\parallel BB'\parallel CC'\parallel DD'$. Moreover, none of these points lie on $p$ and the points $A,B,C,D'$ form a tetrahedron. Show that the points $A',B',C',D$ also form a tetrahedron with the same volume as $ABCD'.$
2021 Malaysia IMONST 1, 10
Determine the number of integer solutions $(x, y, z)$, with $0 \le x, y, z \le 100$, for the equation $$(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.$$
1999 USAMTS Problems, 4
We say a triangle in the coordinate plane is [i]integral[/i] if its three vertices have integer coordinates and if its three sides have integer lengths.
(a) Find an integral triangle with perimeter of $42$.
(b) Is there an integral triangle with perimeter of $43$?
2017 NIMO Problems, 3
Let $ABCD$ be a cyclic quadrilateral with circumradius $100\sqrt{3}$ and $AC=300$. If $\angle DBC = 15^{\circ}$, then find $AD^2$.
[i]Proposed by Anand Iyer[/i]
2000 Abels Math Contest (Norwegian MO), 3
a) Each point, on the perimeter of a square, is colored either red, or blue. Show that, there is a right-angled triangle where all the corners are on the square of the square and so that all the corners are on points of the same color.
b) Show that, it is possible to color each point on the perimeter of one square, red, white, or blue so that, there is not a right-angled triangle where all the three corners are at points of same color.
2007 Grigore Moisil Intercounty, 3
Let be two functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ such that $ g $ has infinite limit at $ \infty . $
[b]a)[/b] Prove that if $ g $ continuous then $ \lim_{x\to\infty } f(x) =\lim_{x\to\infty } f(g(x)) $
[b]b)[/b] Provide an example of what $ f,g $ could be if $ f $ has no limit at $ \infty $ and $ \lim_{x\to\infty } f(g(x)) =0. $
2011 Lusophon Mathematical Olympiad, 3
Consider a sequence of equilateral triangles $T_{n}$ as represented below:
[asy]
defaultpen(linewidth(0.8));size(350);
real r=sqrt(3);
path p=origin--(2,0)--(1,sqrt(3))--cycle;
int i,j,k;
for(i=1; i<5; i=i+1) {
for(j=0; j<i; j=j+1) {
for(k=0; k<j; k=k+1) {
draw(shift(5*i-5+(i-2)*(i-1)*1,0)*shift(2(j-k)+k, k*r)*p);
}}}[/asy]
The length of the side of the smallest triangles is $1$. A triangle is called a delta if its vertex is at the top; for example, there are $10$ deltas in $T_{3}$. A delta is said to be perfect if the length of its side is even. How many perfect deltas are there in $T_{20}$?
2023 Math Prize for Girls Olympiad, 1
Let $n \ge 2023$ be an integer. Prove that there exists a permutation $(p_1, p_2, \dots, p_n)$ of $(1, 2, \dots, n)$ such that
\[
p_1 + 2p_2 + 3p_3 + \dots + np_n
\]
is divisible by $n$.
2013 Nordic, 1
Let ${(a_n)_{n\ge1}} $ be a sequence with ${a_1 = 1} $ and ${a_{n+1} = \lfloor a_n +\sqrt{a_n}+\frac{1}{2}\rfloor }$ for all ${n \ge 1}$, where ${\lfloor x \rfloor}$ denotes the greatest integer less than or equal to ${x}$. Find all ${n \le 2013}$ such that ${a_n}$ is a perfect square
2013 Tournament of Towns, 2
A math teacher chose $10$ consequtive numbers and submitted them to Pete and Basil. Each boy should split these numbers in pairs and calculate the sum of products of numbers in pairs. Prove that the boys can pair the numbers differently so that the resulting sums are equal.
2023 Bulgarian Spring Mathematical Competition, 9.2
Given is triangle $ABC$ with angle bisector $CL$ and the $C-$median meets the circumcircle $\Gamma$ at $D$. If $K$ is the midpoint of arc $ACB$ and $P$ is the symmetric point of $L$ with respect to the tangent at $K$ to $\Gamma$, then prove that $DLCP$ is cyclic.
1984 All Soviet Union Mathematical Olympiad, 388
The $A,B,C$ and $D$ points (from left to right) belong to the straight line. Prove that every point $E$, that doesn't belong to the line satisfy: $$|AE| + |ED| + | |AB| - |CD| | > |BE| + |CE|$$
1996 Singapore Team Selection Test, 1
Let $P$ be a point on the side $AB$ of a square $ABCD$ and $Q$ a point on the side $BC$. Let $H$ be the foot of the perpendicular from $B$ to $PC$. Suppose that $BP = BQ$. Prove that $QH$ is perpendicular to $HD$.
2020 CHKMO, 2
Let $S={1,2,\ldots,100}$. Consider a partition of $S$ into $S_1,S_2,\ldots,S_n$ for some $n$, i.e. $S_i$ are nonempty, pairwise disjoint and $\displaystyle S=\bigcup_{i=1}^n S_i$. Let $a_i$ be the average of elements of the set $S_i$. Define the score of this partition by
\[\dfrac{a_1+a_2+\ldots+a_n}{n}.\]
Among all $n$ and partitions of $S$, determine the minimum possible score.
2014 India PRMO, 20
What is the number of ordered pairs $(A,B)$ where $A$ and $B$ are subsets of $\{1,2,..., 5\}$ such that neither $A \subseteq B$ nor $B \subseteq A$?
1994 Brazil National Olympiad, 3
We are given n objects of identical appearance, but different mass, and a balance which can be used to compare any two objects (but only one object can be placed in each pan at a time). How many times must we use the balance to find the heaviest object and the lightest object?