Found problems: 85335
2018 Vietnam National Olympiad, 5
For two positive integers $n$ and $d$, let $S_n(d)$ be the set of all ordered $d$-tuples $(x_1,x_2,\dots ,x_d)$ that satisfy all of the following conditions:
i. $x_i\in \{1,2,\dots ,n\}$ for every $i\in\{1,2,\dots ,d\}$;
ii. $x_i\ne x_{i+1}$ for every $i\in\{1,2,\dots ,d-1\}$;
iii. There does not exist $i,j,k,l\in\{1,2,\dots ,d\}$ such that $i<j<k<l$ and $x_i=x_k,\, x_j=x_l$;
a. Compute $|S_3(5)|$
b. Prove that $|S_n(d)|>0$ if and only if $d\leq 2n-1$.
2017 Princeton University Math Competition, A7
The sum
\[ \sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^{k}}+1}\]
can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2019 Simon Marais Mathematical Competition, A1
Consider the sequence of positive integers defined by $s_1,s_2,s_3, \dotsc $ of positive integers defined by
[list]
[*]$s_1=2$, and[/*]
[*]for each positive integer $n$, $s_{n+1}$ is equal to $s_n$ plus the product of prime factors of $s_n$.[/*]
[/list]
The first terms of the sequence are $2,4,6,12,18,24$.
Prove that the product of the $2019$ smallest primes is a term of the sequence.
2000 AIME Problems, 8
In trapezoid $ABCD,$ leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD},$ and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001},$ find $BC^2.$
2007 Danube Mathematical Competition, 1
Let $ n\ge2$ be a positive integer and denote by $ S_n$ the set of all permutations of the set $ \{1,2,\ldots,n\}$. For $ \sigma\in S_n$ define $ l(\sigma)$ to be $ \displaystyle\min_{1\le i\le n\minus{}1}\left|\sigma(i\plus{}1)\minus{}\sigma(i)\right|$. Determine $ \displaystyle\max_{\sigma\in S_n}l(\sigma)$.
2023 Malaysian Squad Selection Test, 5
Find the maximal value of $c>0$ such that for any $n\ge 1$, and for any $n$ real numbers $x_1, \cdots, x_n$ there exists real numbers $a ,b$ such that $$\{x_i-a\}+\{x_{i+1}-b\}\le \frac{1}{2024}$$ for at least $cn$ indices $i$. Here, $x_{n+1}=x_1$ and $\{x\}$ denotes the fractional part of $x$.
[i]Proposed by Wong Jer Ren[/i]
2022 Junior Balkan Mathematical Olympiad, 4
We call an even positive integer $n$ [i]nice[/i] if the set $\{1, 2, \dots, n\}$ can be partitioned into $\frac{n}{2}$ two-element subsets, such that the sum of the elements in each subset is a power of $3$. For example, $6$ is nice, because the set $\{1, 2, 3, 4, 5, 6\}$ can be partitioned into subsets $\{1, 2\}$, $\{3, 6\}$, $\{4, 5\}$. Find the number of nice positive integers which are smaller than $3^{2022}$.
I Soros Olympiad 1994-95 (Rus + Ukr), 10.1
The equation $x^2 + bx + c = 0$ has two different roots $x_1$ and $x_2$. It is also known that the numbers $b$, $x_1$, $c$, $x_2$ in the indicated order form an arithmetic progression. Find the difference of this progression.
2022 Girls in Math at Yale, 1
Charlotte is playing the hit new web number game, Primle. In this game, the objective is to guess a two-digit positive prime integer between $10$ and $99$, called the [i]Primle[/i]. For each guess, a digit is highlighted blue if it is in the [i]Primle[/i], but not in the correct place. A digit is highlighted orange if it is in the [i]Primle[/i] and is in the correct place. Finally, a digit is left unhighlighted if it is not in the [i]Primle[/i]. If Charlotte guesses $13$ and $47$ and is left with the following game board, what is the [i]Primle[/i]?
$$\begin{array}{c}
\boxed{1} \,\, \boxed{3} \\[\smallskipamount]
\boxed{4}\,\, \fcolorbox{black}{blue}{\color{white}7}
\end{array}$$
[i]Proposed by Andrew Wu and Jason Wang[/i]
1986 Federal Competition For Advanced Students, P2, 4
Find the largest $ n$ for which there is a natural number $ N$ with $ n$ decimal digits which are all different such that $ n!$ divides $ N$. Furthermore, for this largest $ n$ find all possible numbers $ N$.
2023 Tuymaada Olympiad, 6
An $\textit{Euclidean step}$ transforms a pair $(a, b)$ of positive integers, $a > b$, to the pair $(b, r)$, where $r$ is the remainder when a is divided by $b$. Let us call the $\textit{complexity}$ of a pair $(a, b)$ the number of Euclidean steps needed to transform it to a pair of the form $(s, 0)$. Prove that if $ad - bc = 1$, then the complexities of $(a, b)$ and $(c, d)$ differ at most by $2$.
1998 National Olympiad First Round, 35
What is the maximum number of subsets, having property that none of them is a subset of another, can a set with 10 elements have?
$\textbf{(A)}\ 126 \qquad\textbf{(B)}\ 210 \qquad\textbf{(C)}\ 252 \qquad\textbf{(D)}\ 420 \qquad\textbf{(E)}\ 1024$
2017 NIMO Problems, 8
Let $ABC$ be a triangle with $BC=49$ and circumradius $25$. Suppose that the circle centered on $BC$ that is tangent to $AB$ and $AC$ is also tangent to the circumcircle of $ABC$. Then \[\dfrac{AB \cdot AC}{-BC+AB+AC} = \frac{m}{n}\] where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
[i]Proposed by Michael Ren[/i]
2006 ISI B.Math Entrance Exam, 5
A domino is a $2$ by $1$ rectangle . For what integers $m$ and $n$ can we cover an $m*n$ rectangle with non-overlapping dominoes???
2013 USA Team Selection Test, 1
Two incongruent triangles $ABC$ and $XYZ$ are called a pair of [i]pals[/i] if they satisfy the following conditions:
(a) the two triangles have the same area;
(b) let $M$ and $W$ be the respective midpoints of sides $BC$ and $YZ$. The two sets of lengths $\{AB, AM, AC\}$ and $\{XY, XW, XZ\}$ are identical $3$-element sets of pairwise relatively prime integers.
Determine if there are infinitely many pairs of triangles that are pals of each other.
2007 Romania Team Selection Test, 3
Three travel companies provide transportation between $n$ cities, such that each connection between a pair of cities is covered by one company only. Prove that, for $n \geq 11$, there must exist a round-trip through some four cities, using the services of a same company, while for $n < 11$ this is not anymore necessarily true.
[i]Dan Schwarz[/i]
2025 China Team Selection Test, 16
In convex quadrilateral $ABCD, AB \perp AD, AD = DC$. Let $ E$ be a point on side $BC$, and $F$ be a point on the extension of $DE$ such that $\angle ABF = \angle DEC>90^{\circ}$. Let $O$ be the circumcenter of $\triangle CDE$, and $P$ be a point on the side extension of $FO$ satisfying $FB =FP$. Line BP intersects AC at point Q. Prove that $\angle AQB =\angle DPF.$
2015 Romania Team Selection Tests, 1
Let $ABC$ be a triangle. Let $P_1$ and $P_2$ be points on the side $AB$ such that $P_2$ lies on the segment $BP_1$ and $AP_1 = BP_2$; similarly, let $Q_1$ and $Q_2$ be points on the side $BC$ such that $Q_2$ lies on the segment $BQ_1$ and $BQ_1 = CQ_2$. The segments $P_1Q_2$ and $P_2Q_1$ meet at $R$, and the circles $P_1P_2R$ and $Q_1Q_2R$ meet again at $S$, situated inside triangle $P_1Q_1R$. Finally, let $M$ be the midpoint of the side $AC$. Prove that the angles $P_1RS$ and $Q_1RM$ are equal.
1989 AMC 8, 13
$\frac{9}{7\times 53} =$
$\text{(A)}\ \frac{.9}{.7\times 53} \qquad \text{(B)}\ \frac{.9}{.7\times .53} \qquad \text{(C)}\ \frac{.9}{.7\times 5.3} \qquad \text{(D)}\ \frac{.9}{7\times .53} \qquad \text{(E)}\ \frac{.09}{.07\times .53}$
2005 iTest, 20
If $A$ is the $3\times 3$ square matrix $\begin{bmatrix}
5 & 3 & 8\\
2 & 2 & 5\\
3 & 5 & 1
\end{bmatrix}$ and $B$ is the $4\times 4$ square matrix $\begin{bmatrix}
32 & 2 & 4 & 3 \\
3 & 4 & 8 & 3 \\
11 & 3 & 6 & 1 \\
5 & 5 & 10 & 1
\end{bmatrix} $ find the sum of the determinants of $A$ and $B$.
2020 BMT Fall, 21
Let $\vartriangle ABC$ be a right triangle with legs $AB = 6$ and $AC = 8$. Let $I$ be the incenter of $\vartriangle ABC$ and $X$ be the other intersection of $AI$ with the circumcircle of $\vartriangle ABC$. Find $\overline{AI} \cdot \overline{IX}$.
2023 AIME, 13
Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\sqrt{21}$ and $\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. A parallelepiped is a solid with six parallelogram faces such as the one shown below.
[asy]
unitsize(2cm);
pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70);
draw(o--u--(u+v));
draw(o--v--(u+v), dotted);
draw(shift(w)*(o--u--(u+v)--v--cycle));
draw(o--w);
draw(u--(u+w));
draw(v--(v+w), dotted);
draw((u+v)--(u+v+w));
[/asy]
2013 Finnish National High School Mathematics Competition, 1
The coefficients $a,b,c$ of a polynomial $f:\mathbb{R}\to\mathbb{R}, f(x)=x^3+ax^2+bx+c$ are mutually distinct integers and different from zero. Furthermore, $f(a)=a^3$ and $f(b)=b^3.$ Determine $a,b$ and $c$.
2017 BMT Spring, 5
Find the value of $y$ such that the following equation has exactly three solutions.
$$||x -1|-4|= y.$$
2019 Brazil Team Selection Test, 5
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?