Found problems: 85335
2015 Czech-Polish-Slovak Junior Match, 1
In the right triangle $ABC$ with shorter side $AC$ the hypotenuse $AB$ has length $12$. Denote $T$ its centroid and $D$ the feet of altitude from the vertex $C$. Determine the size of its inner angle at the vertex $B$ for which the triangle $DTC$ has the greatest possible area.
2016 CMIMC, 2
Identical spherical tennis balls of radius 1 are placed inside a cylindrical container of radius 2 and height 19. Compute the maximum number of tennis balls that can fit entirely inside this container.
2004 China Team Selection Test, 1
Using $ AB$ and $ AC$ as diameters, two semi-circles are constructed respectively outside the acute triangle $ ABC$. $ AH \perp BC$ at $ H$, $ D$ is any point on side $ BC$ ($ D$ is not coinside with $ B$ or $ C$ ), through $ D$, construct $ DE \parallel AC$ and $ DF \parallel AB$ with $ E$ and $ F$ on the two semi-circles respectively. Show that $ D$, $ E$, $ F$ and $ H$ are concyclic.
2024 LMT Fall, 12
Call a number [i]orz[/i] if it is a positive integer less than $2024$. Call a number [i]admitting[/i] if it can be expressed as $a^2-1$ where $a$ is a positive integer. Finally call a number [i]muztaba[/i] if it has exactly $4$ positive integer factors. Find the number of [i]muztaba admitting orz[/i] numbers.
2025 Caucasus Mathematical Olympiad, 6
A point $P$ is chosen inside a convex quadrilateral $ABCD$. Could it happen that$$PA = AB, \quad PB = BC, \quad PC = CD \quad \text{and} \quad PD = DA?$$
2014 Cuba MO, 5
Determine all real solutions to the system of equations:
$$x^2 - y = z^2$$
$$y^2 - z = x^2$$
$$z^2 - x = y^2$$
2002 Mongolian Mathematical Olympiad, Problem 5
Let $A$ be the ratio of the product of sides to the product of diagonals in a circumscribed pentagon. Find the maximum possible value of $A$.
2021 Nordic, 3
Let $n$ be a positive integer. Alice and Bob play the following game. First, Alice picks $n + 1$ subsets $A_1,...,A_{n+1}$ of $\{1,... ,2^n\}$ each of size $2^{n-1}$. Second, Bob picks $n + 1$ arbitrary integers $a_1,...,a_{n+1}$. Finally, Alice picks an integer $t$. Bob wins if there exists an integer $1 \le i \le n + 1$ and $s \in A_i$ such that $s + a_i \equiv t$ (mod $2^n$). Otherwise, Alice wins.
Find all values of $n$ where Alice has a winning strategy.
2024 Bulgarian Spring Mathematical Competition, 12.4
Let $d \geq 3$ be a positive integer. The binary strings of length $d$ are splitted into $2^{d-1}$ pairs, such that the strings in each pair differ in exactly one position. Show that there exists an $\textit{alternating cycle}$ of length at most $2d-2$, i.e. at most $2d-2$ binary strings that can be arranged on a circle so that any pair of adjacent strings differ in exactly one position and exactly half of the pairs of adjacent strings are pairs in the split.
2020 CMIMC Team, 3
Let $ABC$ be a triangle with centroid $G$ and $BC = 3$. If $ABC$ is similar to $GAB$, compute the area of $ABC$.
1986 IMO Longlists, 6
In an urn there are one ball marked $1$, two balls marked $2$, and so on, up to $n$ balls marked $n$. Two balls are randomly drawn without replacement. Find the probability that the two balls are assigned the same number.
1970 IMO Longlists, 9
For even $n$, prove that $\sum_{i=1}^{n}{\left((-1)^{i+1}\cdot\frac{1}{i}\right)}=2\sum_{i=1}^{n/2}{\frac{1}{n+2i}}$.
2013 National Chemistry Olympiad, 46
What is the characteristic color of the flame test for potassium?
${ \textbf{(A)}\ \text{yellow}\qquad\textbf{(B)}\ \text{red}\qquad\textbf{(C)}\ \text{green}\qquad\textbf{(D)}}\ \text{violet}\qquad $
1983 IMO Shortlist, 20
Find all solutions of the following system of $n$ equations in $n$ variables:
\[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\]
where $a$ is a given number.
1972 IMO Longlists, 20
Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1 \cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$
1991 Iran MO (2nd round), 2
Triangle $ABC$ is inscribed in circle $C.$ The bisectors of the angles $A,B$ and $C$ meet the circle $C$ again at the points $A', B', C'$. Let $I$ be the incenter of $ABC,$ prove that
\[\frac{IA'}{IA} + \frac{IB'}{IB}+\frac{IC'}{IC} \geq 3\]\[, IA'+IB'+IC' \geq IA+IB+IC\]
2021 Azerbaijan Senior NMO, 1
At least how many numbers must be deleted from the product $1 \times 2 \times \dots \times 46 \times 47$ in order to make it a perfect square?
1966 IMO Shortlist, 51
Consider $n$ students with numbers $1, 2, \ldots, n$ standing in the order $1, 2, \ldots, n.$ Upon a command, any of the students either remains on his place or switches his place with another student. (Actually, if student $A$ switches his place with student $B,$ then $B$ cannot switch his place with any other student $C$ any more until the next command comes.)
Is it possible to arrange the students in the order $n,1, 2, \ldots, n-1$ after two commands ?
1995 Czech And Slovak Olympiad IIIA, 1
Suppose that tetrahedron $ABCD$ satisfies $\angle BAC+\angle CAD+\angle DAB = \angle ABC+\angle CBD+\angle DBA = 180^o$. Prove that $CD \ge AB$.
2021 HMNT, 5
Let $n$ be the answer to this problem. The polynomial $x^n+ax^2+bx+c$ has real coefficients and exactly $k$ real roots. Find the sum of the possible values of $k$.
1987 IMO Longlists, 66
At a party attended by $n$ married couples, each person talks to everyone else at the party except his or her spouse. The conversations involve sets of persons or cliques $C_1, C_2, \cdots, C_k$ with the following property: no couple are members of the same clique, but for every other pair of persons there is exactly one clique to which both members belong. Prove that if $n \geq 4$, then $k \geq 2n$.
[i]Proposed by USA.[/i]
2009 Harvard-MIT Mathematics Tournament, 8
Compute \[\int_1^{\sqrt{3}} x^{2x^2+1}+\ln\left(x^{2x^{2x^2+1}}\right)dx.\]
2023 China Girls Math Olympiad, 4
Let $ABCD$ be an inscribed quadrilateral of some circle $\omega$ with $AC\ \bot \ BD$. Define $E$ to be the intersection of segments $AC$ and $BD$. Let $F$ be some point on segment $AD$ and define $P$ to be the intersection point of half-line $FE$ and $\omega$. Let $Q$ be a point on segment $PE$ such that $PQ\cdot PF = PE^2$. Let $R$ be a point on $BC$ such that $QR\ \bot \ AD$. Prove that $PR=QR$.
2008 Puerto Rico Team Selection Test, 6
Let $n$ be a natural composite number. Prove that there are integers $a_1, a_2,. . . , a_k$ all greater than $ 1$, such that $$a_1 + a_2 +... + a_k = n \left(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}\right)$$
2021 Czech-Polish-Slovak Junior Match, 4
Find the smallest positive integer $n$ with the property that in the set $\{70, 71, 72,... 70 + n\}$ you can choose two different numbers whose product is the square of an integer.