Found problems: 85335
2021 USAMTS Problems, 1
A $5 \times 5$ Latin Square is a $5 \times 5$ grid of squares in which each square contains one
of the numbers $1$ through $5$ such that every number appears exactly once in each row and
column. A partially completed grid (with numbers in some of the squares) is puzzle-ready
if there is a unique way to fill in the remaining squares to complete a Latin Square.
Below is a partially completed grid with seven squares filled in and an additional three
squares shaded. Determine what numbers must be filled into the shaded squares to make
the grid (now with ten squares filled in) puzzle-ready, and then complete the Latin Square.
There is a unique solution, but you do not need to prove that your answer is the only
one possible. You merely need to find an answer that satisfies the constraints above. (Note:
In any other USAMTS problem, you need to provide a full proof. Only in this problem is
an answer without justification acceptable.)
[asy]
unitsize(1.5cm);
defaultpen(font("OT1","cmss","m","n"));
defaultpen(fontsize(48pt));
for (int i=0; i<6; ++i) {
draw((i,0)--(i,5));
draw((0,i)--(5,i));
}
label(scale(2)*"1",(0.5,4.5));
label(scale(2)*"1",(1.5,3.5));
label(scale(2)*"3",(2.5,3.5));
label(scale(2)*"2",(0.5,2.5));
label(scale(2)*"3",(1.5,2.5));
label(scale(2)*"5",(4.5,2.5));
label(scale(2)*"5",(3.5,1.5));
path p = (0,0)--(1,0)--(1,1)--(0,1)--cycle;
filldraw(shift(0,1)*p,gray,black);
filldraw(shift(4,1)*p,gray,black);
filldraw(shift(2,2)*p,gray,black);
[/asy]
2010 Contests, 4
Point $O$ is chosen in a triangle $ABC$ such that ${d_a},{d_b},{d_c}$ are distance from point $O$ to sides $BC,CA,AB$, respectively. Find position of point $O$ so that product ${d_a} \cdot {d_b} \cdot {d_c}$ becomes maximum.
2020 BMT Fall, Tie 1
An [i]exterior [/i] angle is the supplementary angle to an interior angle in a polygon. What is the sum of the exterior angles of a triangle and dodecagon ($12$-gon), in degrees?
2007 Moldova Team Selection Test, 3
Let $ABC$ be a triangle. A circle is tangent to sides $AB, AC$ and to the circumcircle of $ABC$ (internally) at points $P, Q, R$ respectively. Let $S$ be the point where $AR$ meets $PQ$. Show that \[\angle{SBA}\equiv \angle{SCA}\]
Russian TST 2022, P1
Non-zero polynomials $P(x)$, $Q(x)$, and $R(x)$ with real coefficients satisfy the identities
$$ P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0. $$
Prove that the degrees of the three polynomials are all even.
2011 Federal Competition For Advanced Students, Part 1, 4
Inside or on the faces of a tetrahedron with five edges of length $2$ and one edge of lenght $1$, there is a point $P$ having distances $a, b, c, d$ to the four faces of the tetrahedron. Determine the locus of all points $P$ such that $a+b+c+d$ is minimal and the locus of all points $P$ such that $a+b+c+d$ is maximal.
2003 Gheorghe Vranceanu, 1
Prove that any permutation group of an order equal to a power of $ 2 $ contains a commutative subgroup whose order is the square of the exponent of the order of the group.
2019 India IMO Training Camp, P2
Let $ABC$ be a triangle with $\angle A=\angle C=30^{\circ}.$ Points $D,E,F$ are chosen on the sides $AB,BC,CA$ respectively so that $\angle BFD=\angle BFE=60^{\circ}.$ Let $p$ and $p_1$ be the perimeters of the triangles $ABC$ and $DEF$, respectively. Prove that $p\le 2p_1.$
2025 Ukraine National Mathematical Olympiad, 10.4
It is known that a sequence of positive real numbers \(\left(x_n\right)\) satisfies the relation:
\[
x_{n+1} = x_n + \sqrt{x_n + \frac{1}{4}} + \sqrt{x_{n+1} + \frac{1}{4}}, \quad n \geq 1
\]
Prove that the following inequality holds:
\[
\frac{1}{x_2} + \frac{1}{x_3} + \cdots + \frac{1}{x_{2025}} < \frac{1}{\sqrt{x_1}}
\]
[i]Proposed by Oleksii Masalitin[/i]
2017 Saudi Arabia JBMO TST, 6
Find all pairs of prime numbers $(p, q)$ such that $p^2 + 5pq + 4q^2$ is a perfect square.
2006 MOP Homework, 5
Set $X$ has $56$ elements. Determine the least positive integer $n$ such that for any 15 subsets of $X$, if the union of any $7$ of
the subsets has at least $n$ elements, then 3 of the subsets have
nonempty intersection.
1983 AIME Problems, 2
Let $f(x) = |x - p| + |x - 15| + |x - p - 15|$, where $0 < p < 15$. Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \le x \le 15$.
2013 Sharygin Geometry Olympiad, 2
Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ meet at points $A$ and $B$. Points $C$ and $D$ on $\omega_1$ and $\omega_2$, respectively, lie on the opposite sides of the line $AB$ and are equidistant from this line. Prove that $C$ and $D$ are equidistant from the midpoint of $O_1O_2$.
1969 German National Olympiad, 5
Prove that for all real numbers $x$ holds:
$$\sin 5x = 16 \sin x \cdot \sin \left(x -\frac{\pi}{5} \right) \cdot \sin\left(x -\frac{2\pi}{5} \right) \sin \left(x +\frac{2\pi}{5} \right) $$
1999 Tournament Of Towns, 3
There are $n$ straight lines in the plane such that each intersects exactly $1999$ of the others . Find all posssible values of $n$.
(R Zhenodarov)
2021 Olimphíada, 5
Let $p$ be an odd prime. The numbers $1, 2, \ldots, d$ are written on a blackboard, where $d \geq p-1$ is a positive integer. A valid operation is to delete two numbers $x$ and $y$ and write $x + y - c \cdot xy$ in their place, where $c$ is a positive integer. One moment there is only one number $A$ left on the board. Show that if there is an order of operations such that $p$ divides $A$, then $p | d$ or $p | d + 1$.
Ukrainian From Tasks to Tasks - geometry, 2014.9
On a circle with diameter $AB$ we marked an arbitrary point $C$, which does not coincide with $A$ and $B$. The tangent to the circle at point $A$ intersects the line $BC$ at point $D$. Prove that the tangent to the circle at point $C$ bisects the segment $AD$.
2020 Azerbaijan Senior NMO, 3
Let $ABC$ be a scalene triangle, and let $I$ be its incenter. A point $D$ is chosen on line $BC$, such that the circumcircle of triangle $BID$ intersects $AB$ at $E\neq B$, and the circumcircle of triangle $CID$ intersects $AC$ at $F\neq C$. Circumcircle of triangle $EDF$ intersects $AB$ and $AC$ at $M$ and $N$, respectively. Lines $FD$ and $IC$ intersect at $Q$, and lines $ED$ and $BI$ intersect at $P$. Prove that $EN\parallel MF\parallel PQ$.
2005 Junior Balkan Team Selection Tests - Moldova, 8
The families of second degree functions $f_m, g_m: R\to R, $ are considered , $f_m (x) = (m^2 + 1) x^2 + 3mx + m^2 - 1$, $g_m (x) = m^2x^2 + mx - 1$, where $m$ is a real nonzero parameter.
Show that, for any function $h$ of the second degree with the property that $g_m (x) \le h (x) \le f_m (x)$ for any real $x$, there exists $\lambda \in [0, 1]$ which verifies the condition $h (x) = \lambda f_m (x) + (1- \lambda) g_m (x)$, whatever real $x$ is.
2019 ELMO Shortlist, C3
In the game of [i]Ring Mafia[/i], there are $2019$ counters arranged in a circle. $673$ of these counters are mafia, and the remaining $1346$ counters are town. Two players, Tony and Madeline, take turns with Tony going first. Tony does not know which counters are mafia but Madeline does.
On Tony’s turn, he selects any subset of the counters (possibly the empty set) and removes all counters in that set. On Madeline’s turn, she selects a town counter which is adjacent to a mafia counter and removes it. Whenever counters are removed, the remaining counters are brought closer together without changing their order so that they still form a circle. The game ends when either all mafia counters have been removed, or all town counters have been removed.
Is there a strategy for Tony that guarantees, no matter where the mafia counters are placed and what Madeline does, that at least one town counter remains at the end of the game?
[i]Proposed by Andrew Gu[/i]
1999 AMC 12/AHSME, 29
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $ P$ is selected at random inside the circumscribed sphere. The probability that $ P$ lies inside one of the five small spheres is closest to
$ \textbf{(A)}\ 0\qquad
\textbf{(B)}\ 0.1\qquad
\textbf{(C)}\ 0.2\qquad
\textbf{(D)}\ 0.3\qquad
\textbf{(E)}\ 0.4$
2013 NIMO Problems, 3
Find the integer $n \ge 48$ for which the number of trailing zeros in the decimal representation of $n!$ is exactly $n-48$.
[i]Proposed by Kevin Sun[/i]
2003 Estonia National Olympiad, 2
Prove that for all positive real numbers $a, b$, and $c$ , $\sqrt[3]{abc}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 2\sqrt3$.
When does the equality occur?
Oliforum Contest I 2008, 1
Consider the sequence of integer such that:
$ a_1 = 2$
$ a_2 = 5$
$ a_{n + 1} = (2 - n^2)a_n + (2 + n^2)a_{n - 1}, \forall n\ge 2$
Find all triplies $ (x,y,z) \in \mathbb{N}^3$ such that $ a_xa_y = a_z$.
2015 Purple Comet Problems, 19
Problem 19 The diagram below shows a 24×24 square ABCD. Points E and F lie on sides AD and CD, respectively, so that DE = DF = 8. Set X consists of the shaded triangle ABC with its interior, while set Y consists of
the shaded triangle DEF with its interior. Set Z consists of all the points that are midpoints of segments
connecting a point in set X with a point in set Y . That is, Z = {z | z is the midpoint of xy for x ∈ X and y ∈ Y}. Find the area of the set Z.
For diagram to http://www.purplecomet.org/welcome/practice