Found problems: 85335
2018 Miklós Schweitzer, 6
Prove that if $a$ is an integer and $d$ is a positive divisor of the number $a^4+a^3+2a^2-4a+3$, then $d$ is a fourth power modulo $13$.
1989 Swedish Mathematical Competition, 6
On a circle $4n$ points are chosen ($n \ge 1$). The points are alternately colored yellow and blue. The yellow points are divided into $n$ pairs and the points in each pair are connected with a yellow line segment. In the same manner the blue points are divided into $n$ pairs and the points in each pair are connected with a blue segment. Assume that no three of the segments pass through a single point. Show that there are at least $n$ intersection points of blue and yellow segments.
2009 Denmark MO - Mohr Contest, 5
Imagine a square scheme consisting of $n\times n$ fields with edge length $1$, where $n$ is an arbitrary positive integer. What is the maximum possible length of a route you can follow along the edges of the fields from point $A$ in the lower left corner to point $B$ in the upper right corner if you must never return to one point where you have been before? (The figure shows for $n = 5$ an example of a permitted route and an example of a not permitted route).
[img]https://cdn.artofproblemsolving.com/attachments/6/e/92931d87f11b9fb3120b8dccc2c37c35a04456.png[/img]
2021 Durer Math Competition (First Round), 4
Determine all triples of positive integers $a, b, c$ that satisfy
a) $[a, b] + [a, c] + [b, c] = [a, b, c]$.
b) $[a, b] + [a, c] + [b, c] = [a, b, c] + (a, b, c)$.
Remark: Here $[x, y$] denotes the least common multiple of positive integers $x$ and $y$, and $(x, y)$ denotes their greatest common divisor.
2021 Nigerian MO Round 3, Problem 4
In the multiplication magic square below, $l, m, n, p, q, r, s, t, u$ are positive integers. The product of any three numbers in any row, column or diagonal is equal to a constant $k$, where $k$ is a number between $11, 000$ and $12, 500$. Find the value of $k$.
\begin{tabular}{|l|l|l|}
\hline
$l$ & $m$ & $n$ \\
\hline
$p$ & $q$ & $r$ \\
\hline
$s$ & $t$ & $u$ \\
\hline
\end{tabular}
IV Soros Olympiad 1997 - 98 (Russia), 11.7
On straight line $\ell$ there are points $A$, $B$, $C$ and $D$, following in the indicated order: $AB = a$, $BC = b$, $CD = c$. Segments $AD$ and $BC$ serve as chords of two circles, and the sum of the angular values of the arcs of these circles located on one side of $\ell$ is equal to $360^o$. A third circle passes through $A$ and $B$, intersecting the first two at points $K$ and $M$. The straight line $KM$ intersects $\ell$ at point $E$. Find $AE$.
2002 Singapore Team Selection Test, 1
Let $A, B, C, D, E$ be five distinct points on a circle $\Gamma$ in the clockwise order and let the extensions of $CD$ and $AE$ meet at a point $Y$ outside $\Gamma$. Suppose $X$ is a point on the extension of $AC$ such that $XB$ is tangent to $\Gamma$ at $B$. Prove that $XY = XB$ if and only if $XY$ is parallel $DE$.
2015 Peru IMO TST, 8
Let $I$ be the incenter of the $ABC$ triangle. The circumference that passes through $I$ and has center
in $A$ intersects the circumscribed circumference of the $ABC$ triangle at points $M$ and
$N$. Prove that the line $MN$ is tangent to the inscribed circle of the $ABC$ triangle.
1996 Estonia National Olympiad, 3
The vertices of the quadrilateral $ABCD$ lie on a single circle. The diagonals of this rectangle divide the angles of the rectangle at vertices $A$ and $B$ and divides the angles at vertices $C$ and $D$ in a $1: 2$ ratio. Find angles of the quadrilateral $ABCD$.
2014 Harvard-MIT Mathematics Tournament, 29
Natalie has a copy of the unit interval $[0,1]$ that is colored white. She also has a black marker, and she colors the interval in the following manner: at each step, she selects a value $x\in [0,1]$ uniformly at random, and
(a) If $x\leq\tfrac12$ she colors the interval $[x,x+\tfrac12]$ with her marker.
(b) If $x>\tfrac12$ she colors the intervals $[x,1]$ and $[0,x-\tfrac12]$ with her marker.
What is the expected value of the number of steps Natalie will need to color the entire interval black?
1972 Bulgaria National Olympiad, Problem 6
It is given a tetrahedron $ABCD$ for which two points of opposite edges are mutually perpendicular. Prove that:
(a) the four altitudes of $ABCD$ intersects at a common point $H$;
(b) $AH+BH+CH+DH<p+2R$, where $p$ is the sum of the lengths of all edges of $ABCD$ and $R$ is the radii of the sphere circumscribed around $ABCD$.
[i]H. Lesov[/i]
2013 Albania Team Selection Test, 4
It is given a triangle $ABC$ whose circumcenter is $O$ and orthocenter $H$.
If $AO=AH$ find the angle $\hat{BAC}$ of that triangle.
2011 NIMO Summer Contest, 15
Let
\[
N = \sum_{a_1 = 0}^2 \sum_{a_2 = 0}^{a_1} \sum_{a_3 = 0}^{a_2} \dots \sum_{a_{2011} = 0}^{a_{2010}} \left [ \prod_{n=1}^{2011} a_n \right ].
\]
Find the remainder when $N$ is divided by 1000.
[i]Proposed by Lewis Chen
[/i]
2021 China Second Round, 4
Find the minimum value of $c$ such that for any positive integer $n\ge 4$ and any set $A\subseteq \{1,2,\cdots,n\}$, if $|A| >cn$, there exists a function $f:A\to\{1,-1\}$ satisfying
$$\left| \sum_{a\in A}a\cdot f(a)\right| \le 1.$$
2024 German National Olympiad, 2
Six quadratic mirrors are put together to form a cube $ABCDEFGH$ with a mirrored interior. At each of the eight vertices, there is a tiny hole through which a laser beam can enter and leave the cube. A laser beam enters the cube at vertex $A$ in a direction not parallel to any of the cube's sides. If the beam hits a side, it is reflected; if it hits an edge, the light is absorbed, and if it hits a vertex, it leaves the cube.
For each positive integer $n$, determine the set of vertices where the laser beam can leave the cube after exactly $n$ reflections.
2018 CCA Math Bonanza, L5.3
Choose an integer $n$ from $1$ to $10$ inclusive as your answer to this problem. Let $m$ be the number of distinct values in $\left\{1,2,\ldots,10\right\}$ chosen by all teams at the Math Bonanza for this problem which are greater than or equal to $n$. Your score on this problem will be $\frac{mn}{15}$. For example, if $5$ teams choose $1$, $2$ teams choose $2$, and $6$ teams choose $3$ with these being the only values chosen, and you choose $2$, you will receive $\frac{4}{15}$ points.
[i]2018 CCA Math Bonanza Lightning Round #5.3[/i]
2024 India IMOTC, 2
Let $x_1, x_2 \dots, x_{2024}$ be non-negative real numbers such that $x_1 \le x_2\cdots \le x_{2024}$, and $x_1^3 + x_2^3 + \dots + x_{2024}^3 = 2024$. Prove that
\[\sum_{1 \le i < j \le 2024} (-1)^{i+j} x_i^2 x_j \ge -1012.\]
[i]Proposed by Shantanu Nene[/i]
2017 IMAR Test, 3
We consider $S$ a set of odd positive interger numbers with $n\geq 3$ elements such that no element divides another element. We say that a set $S$ is $beautiful$ if for any 3 elements from $S$, there is one the divides the sum of the other 2. We call a beautiful set $S$ $maximal$ if we can't add another number to the set such that $S$ will still be beautiful. Find the values of $n$ for which there exists a $maximal$ set.
2018 District Olympiad, 3
Let $(a_n)_{n\ge 1}$ be a sequence such that $a_n > 1$ and $a_{n+1}^2 \ge a_n a_{n + 2}$, for any $n\ge 1$. Show that the sequence $(x_n)_{n\ge 1}$ given by $x_n = \log_{a_n} a_{n + 1}$ for $n\ge 1$ is convergent and compute its limit.
2018 ISI Entrance Examination, 5
Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that its derivative $f'$ is a continuous function. Moreover, assume that for all $x\in\mathbb{R}$, $$0\leqslant \vert f'(x)\vert\leqslant \frac{1}{2}$$ Define a sequence of real numbers $\{a_n\}_{n\in\mathbb{N}}$ by :$$a_1=1~~\text{and}~~a_{n+1}=f(a_n)~\text{for all}~n\in\mathbb{N}$$ Prove that there exists a positive real number $M$ such that for all $n\in\mathbb{N}$, $$\vert a_n\vert \leqslant M$$
1959 AMC 12/AHSME, 47
Assume that the following three statements are true:
$I$. All freshmen are human. $II$. All students are human. $III$. Some students think.
Given the following four statements:
$ \textbf{(1)}\ \text{All freshmen are students.}\qquad$
$\textbf{(2)}\ \text{Some humans think.}\qquad$
$\textbf{(3)}\ \text{No freshmen think.}\qquad$
$\textbf{(4)}\ \text{Some humans who think are not students.}$
Those which are logical consequences of I,II, and III are:
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 2,3\qquad\textbf{(D)}\ 2,4\qquad\textbf{(E)}\ 1,2 $
2013 Estonia Team Selection Test, 1
Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\
a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$
1975 AMC 12/AHSME, 12
If $ a \neq b$, $ a^3 \minus{} b^3 \equal{} 19x^3$, and $ a\minus{}b \equal{} x$, which of the following conclusions is correct?
$ \textbf{(A)}\ a\equal{}3x \qquad
\textbf{(B)}\ a\equal{}3x \text{ or } a \equal{} \minus{}2x \qquad$
$ \textbf{(C)}\ a\equal{}\minus{}3x \text{ or } a \equal{} 2x \qquad
\textbf{(D)}\ a\equal{}3x \text{ or } a\equal{}2x \qquad
\textbf{(E)}\ a\equal{}2x$
1993 All-Russian Olympiad, 4
Thirty people sit at a round table. Each of them is either smart or dumb. Each of them is asked: "Is your neighbor to the right smart or dumb?" A smart person always answers correctly, while a dumb person can answer both correctly and incorrectly. It is known that the number of dumb people does not exceed $F$. What is the largest possible value of $F$ such that knowing what the answers of the people are, you can point at at least one person, knowing he is smart?
2004 Purple Comet Problems, 18
Find the number of addition problems in which a two digit number is added to a second two digit number to give a two digit answer, such as in the three examples: \[\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}23\\42\\\hline 65\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}36\\36\\\hline 72\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}42\\23\\\hline 65\end{tabular}\,.\]