Found problems: 85335
2015 Online Math Open Problems, 29
Let $ABC$ be an acute scalene triangle with incenter $I$, and let $M$ be the circumcenter of triangle $BIC$. Points $D$, $B'$, and $C'$ lie on side $BC$ so that $ \angle BIB' = \angle CIC' = \angle IDB = \angle IDC = 90^{\circ} $. Define $P = \overline{AB} \cap \overline{MC'}$, $Q = \overline{AC} \cap \overline{MB'}$, $S = \overline{MD} \cap \overline{PQ}$, and $K = \overline{SI} \cap \overline{DF}$, where segment $EF$ is a diameter of the incircle selected so that $S$ lies in the interior of segment $AE$. It is known that $KI=15x$, $SI=20x+15$, $BC=20x^{5/2}$, and $DI=20x^{3/2}$, where $x = \tfrac ab(n+\sqrt p)$ for some positive integers $a$, $b$, $n$, $p$, with $p$ prime and $\gcd(a,b)=1$. Compute $a+b+n+p$.
[i]Proposed by Evan Chen[/i]
2011 QEDMO 8th, 3
Show that every rational number $r$ can be written as the sum of numbers in the form $\frac{a}{p^k}$ where $p$ is prime, $a$ is an integer and $k$ is natural.
2007 Chile National Olympiad, 4
$31$ guests at a party sit in equally spaced chairs around a round table , but they have not noticed that there are cards with the names of the guests on the stalls.
(a) Assuming they have been so unlucky that no one is in the room which corresponds to him, show that it is possible to get at least two people to stay in their correct position, without anyone getting up from their seat, turning the table.
(b) Show a configuration where exactly one guest is in his assigned place and where in no way that the table is turned it is possible to achieve that at least two remain right.
1998 Federal Competition For Advanced Students, Part 2, 2
Let $P(x) = x^3 - px^2 + qx - r$ be a cubic polynomial with integer roots $a, b, c$.
[b](a)[/b] Show that the greatest common divisor of $p, q, r$ is equal to $1$ if the greatest common divisor of $a, b, c$ is equal to $1$.
[b](b)[/b] What are the roots of polynomial $Q(x) = x^3-98x^2+98sx-98t$ with $s, t$ positive integers.
2009 Sharygin Geometry Olympiad, 2
Given nonisosceles triangle $ ABC$. Consider three segments passing through different vertices of this triangle and bisecting its perimeter. Are the lengths of these segments certainly different?
2025 Romania National Olympiad, 3
Prove that, for a function $f \colon \mathbb{R} \to \mathbb{R}$, the following $2$ statements are equivalent:
a) $f$ is differentiable, with continuous first derivative.
b) For any $a\in\mathbb{R}$ and for any two sequences $(x_n)_{n\geq 1},(y_n)_{n\geq 1}$, convergent to $a$, such that $x_n \neq y_n$ for any positive integer $n$, the sequence $\left(\frac{f(x_n)-f(y_n)}{x_n-y_n}\right)_{n\geq 1}$ is convergent.
2008 Harvard-MIT Mathematics Tournament, 4
In a triangle $ ABC$, take point $ D$ on $ BC$ such that $ DB \equal{} 14, DA \equal{} 13, DC \equal{} 4$, and the circumcircle of $ ADB$ is congruent to the circumcircle of $ ADC$. What is the area of triangle $ ABC$?
1996 AMC 12/AHSME, 8
If $3 = k \cdot 2^r$ and $15 = k \cdot 4^r$, then $r =$
$\text{(A)}\ - \log_2 5 \qquad \text{(B)}\ \log_5 2 \qquad \text{(C)}\ \log_{10} 5 \qquad \text{(D)}\ \log_2 5 \qquad \text{(E)}\ \displaystyle \frac{5}{2}$
2017 Princeton University Math Competition, A6/B8
Triangle $ABC$ has $\angle{A}=90^{\circ}$, $AB=2$, and $AC=4$. Circle $\omega_1$ has center $C$ and radius $CA$, while circle $\omega_2$ has center $B$ and radius $BA$. The two circles intersect at $E$, different from point $A$. Point $M$ is on $\omega_2$ and in the interior of $ABC$, such that $BM$ is parallel to $EC$. Suppose $EM$ intersects $\omega_1$ at point $K$ and $AM$ intersects $\omega_1$ at point $Z$. What is the area of quadrilateral $ZEBK$?
1975 Bundeswettbewerb Mathematik, 2
Prove that in each polyhedron there exist two faces with the same number of edges.
2025 239 Open Mathematical Olympiad, 4
The numbers from $1$ to $2025$ are arranged in some order in the cells of the $1 \times 2025$ strip. Let's call a [i]flip[/i] an operation that takes two arbitrary cells of a strip and swaps the numbers written in them, but only if the larger of these numbers is located to the left of the smaller one. A [i]flop[/i] is a set of several flips that do not contain common cells that are executed simultaneously. (For example, a simultaneous flip between the 2nd and 8th cells and a flip between the 5th and 101st cells.) Prove that there exists a sequence of $66$ flops such that for any initial arrangement, applying this sequence of flops to it will result in the numbers being ordered from left to right in ascending order.
1994 Baltic Way, 6
Prove that any irreducible fraction $p/q$, where $p$ and $q$ are positive integers and $q$ is odd, is equal to a fraction $\frac{n}{2^k-1}$ for some positive integers $n$ and $k$.
2021 IMO Shortlist, G7
Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\angle DAB = \angle CAD.$ The point $E$ on the segment $AC$ satisfies $\angle ADE =\angle BCD,$ the point $F$ on the segment $AB$ satisfies $\angle FDA =\angle DBC,$ and the point $X$ on the line $AC$ satisfies $CX = BX.$ Let $O_1$ and $O_2$ be the circumcenters of the triangles $ADC$ and $EXD,$ respectively. Prove that the lines $BC, EF,$ and $O_1O_2$ are concurrent.
May Olympiad L1 - geometry, 1995.5
A tortoise walks $60$ meters per hour and a lizard walks at $240$ meters per hour. There is a rectangle $ABCD$ where $AB =60$ and $AD =120$. Both start from the vertex $A$ and in the same direction ($A \to B \to D \to A$), crossing the edge of the rectangle. The lizard has the habit of advancing two consecutive sides of the rectangle, turning to go back one, turning to go forward two, turning to go back one and so on. How many times and in what places do the tortoise and the lizard meet when the tortoise completes its third turn?
2018 BMT Spring, Tie 2
Points $A, B, C$ are chosen on the boundary of a circle with center $O$ so that $\angle BAC$ encloses an arc of $120$ degrees. Let $D$ be chosen on $\overline{BA}$ so that $\angle AOD$ is a right angle. Extend $\overline{CD}$ so that it intersects with $O$ again at point $P$. What is the measure of the arc, in degrees, that is enclosed by $\angle ACP$? Please use the $tan^{-1}$ function to express your answer.
2017 Iran MO (3rd round), 2
For prime number $q$ the polynomial $P(x)$ with integer coefficients is said to be factorable if there exist non-constant polynomials $f_q,g_q$ with integer coefficients such that all of the coefficients of the polynomial $Q(x)=P(x)-f_q(x)g_q(x)$ are dividable by $q$ ; and we write:
$$P(x)\equiv f_q(x)g_q(x)\pmod{q}$$
For example the polynomials $2x^3+2,x^2+1,x^3+1$ can be factored modulo $2,3,p$ in the following way:
$$\left\{\begin{array}{lll}
X^2+1\equiv (x+1)(-x+1)\pmod{2}\\
2x^3+2\equiv (2x-1)^3\pmod{3}\\
X^3+1\equiv (x+1)(x^2-x+1)
\end{array}\right.$$
Also the polynomial $x^2-2$ is not factorable modulo $p=8k\pm 3$.
a) Find all prime numbers $p$ such that the polynomial $P(x)$ is factorable modulo $p$:
$$P(x)=x^4-2x^3+3x^2-2x-5$$
b) Does there exist irreducible polynomial $P(x)$ in $\mathbb{Z}[x]$ with integer coefficients such that for each prime number $p$ , it is factorable modulo $p$?
2011 Middle European Mathematical Olympiad, 7
Let $A$ and $B$ be disjoint nonempty sets with $A \cup B = \{1, 2,3, \ldots, 10\}$. Show that there exist elements $a \in A$ and $b \in B$ such that the number $a^3 + ab^2 + b^3$ is divisible by $11$.
2018 ASDAN Math Tournament, 5
In the expansion of $(x + b)^{2018}$, the coefficients of $x^2$ and $x^3$ are equal. Compute $b$.
1992 China Team Selection Test, 1
16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems.
2002 Iran MO (2nd round), 5
Let $\delta$ be a symbol such that $\delta \neq 0$ and $\delta^2 = 0$. Define $\mathbb R[\delta] = \{a + b \delta | a, b \in \mathbb R\}$, where $a+ b \delta = c+ d \delta$ if and only if $a = c$ and $b = d$, and define
\[(a + b \delta) + (c + d \delta) = (a + c) + (b + d) \delta,\]\[(a + b \delta) \cdot (c + d \delta) = ac + (ad + bc) \delta.\]
Let $P(x)$ be a polynomial with real coefficients. Show that $P(x)$ has a multiple real root if and only if $P(x)$ has a non-real root in $\mathbb R[\delta].$
2024 India IMOTC, 11
There are $n\ge 3$ particles on a circle situated at the vertices of a regular $n$-gon. All these particles move on the circle with the same constant speed. One of the particles moves in the clockwise direction while all others move in the anti-clockwise direction. When particles collide, that is, they are all at the same point, they all reverse the direction of their motion and continue with the same speed as before.
Let $s$ be the smallest number of collisions after which all particles return to their original positions. Find $s$.
[i]Proposed by N.V. Tejaswi[/i]
2015 Estonia Team Selection Test, 5
Find all functions $f$ from reals to reals which satisfy $f (f(x) + f(y)) = f(x^2) + 2x^2 f(y) + (f(y))^2$ for all real numbers $x$ and $y$.
2014 Iran MO (3rd Round), 1
Denote by $g_n$ the number of connected graphs of degree $n$ whose vertices are labeled with numbers $1,2,...,n$. Prove that $g_n \ge (\frac{1}{2}).2^{\frac{n(n-1)}{2}}$.
[b][u]Note[/u][/b]:If you prove that for $c < \frac{1}{2}$, $g_n \ge c.2^{\frac{n(n-1)}{2}}$, you will earn some point!
[i]proposed by Seyed Reza Hosseini and Mohammad Amin Ghiasi[/i]
2003 Iran MO (3rd Round), 13
here is the most difficult and the most beautiful problem occurs in 21th iranian (2003) olympiad
assume that P is n-gon ,lying on the plane ,we name its edge 1,2,..,n.
if S=s1,s2,s3,.... be a finite or infinite sequence such that for each i, si is in {1,2,...,n},
we move P on the plane according to the S in this form: at first we reflect P through the s1
( s1 means the edge which iys number is s1)then through s2 and so on like the figure below.
a)show that there exist the infinite sequence S sucth that if we move P according to S we cover all the plane
b)prove that the sequence in a) isn't periodic.
c)assume that P is regular pentagon ,which the radius of its circumcircle is 1,and D is circle ,with radius 1.00001 ,arbitrarily in the plane .does exist a sequence S such that we move P according to S then P reside in D completely?
1981 Putnam, A3
Find
$$ \lim_{t\to \infty} e^{-t} \int_{0}^{t} \int_{0}^{t} \frac{e^x -e^y }{x-y} \,dx\,dy,$$
or show that the limit does not exist.