Found problems: 85335
1999 Switzerland Team Selection Test, 3
Find all functions $f : R -\{0\} \to R$ that satisfy $\frac{1}{x}f(-x)+ f\left(\frac{1}{x}\right)= x$ for all $x \ne 0$.
2023-IMOC, G2
$P$ is a point inside $\triangle ABC$. $AP, BP, CP$ intersects $BC, CA, AB$ at $D, E, F$, respectively. $AD$ meets $(ABC)$ again at $D_1$. $S$ is a point on $(ABC)$. Lines $AS$, $EF$ intersect at $T$, lines $TP, BC$ intersect at $K$, and $KD_1$ meets $(ABC)$ again at $X$. Prove that $S, D, X$ are colinear.
LMT Guts Rounds, 2020 F24
In the Oxtingnle math team, there are $5$ students, numbered $1$ to $5$, all of which either always tell the truth or always lie. When Marpeh asks the team about how they did in a $10$ question competition, each student $i$ makes $5$ separate statements (so either they are all false or all true): "I got problems $i+1$ to $2i$, inclusive, wrong", and then "Student $j$ got both problems $i$ and $2i$ correct" for all $j \neq i$. What is the most problems the team could have gotten correctly?
[i]Proposed by Jeff Lin[/i]
1996 Canadian Open Math Challenge, 8
Determine all pairs of integers $(x,y)$ which satisfy the equation
\[ 6x^2-3xy-13x+5y = -11 \]
2019 BMT Spring, 2
A set of points in the plane is called [i]full[/i] if every triple of points in the set are the vertices of a non-obtuse triangle. What is the largest size of a full set?
2009 China National Olympiad, 2
Find all the pairs of prime numbers $ (p,q)$ such that $ pq|5^p\plus{}5^q.$
2000 Moldova National Olympiad, Problem 3
For every nonempty subset $X$ of $M=\{1,2,\ldots,2000\}$, $a_X$ denotes the sum of the minimum and maximum element of $X$. Compute the arithmetic mean of the numbers $a_X$ when $X$ goes over all nonempty subsets $X$ of $M$.
2011 Tournament of Towns, 3
(a) Does there exist an in nite triangular beam such that two of its cross-sections are similar but not congruent triangles?
(b) Does there exist an in nite triangular beam such that two of its cross-sections are equilateral triangles of sides $1$ and $2$ respectively?
2015 ASDAN Math Tournament, 1
A rectangle $ABCD$ is split into four smaller non-overlapping rectangles by two perpendicular line segments, whose endpoints are on the sides of $ABCD$. If the smallest three rectangles have areas of $48$, $18$, and $12$, what is the area of $ABCD$?
2021 Azerbaijan IZhO TST, 1
Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that:
$$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.
2019 Junior Balkan Team Selection Tests - Romania, 1
Let $n$ be a given positive integer. Determine all positive divisors $d$ of $3n^2$ such that $n^2 + d$ is the square of an integer.
2023 Baltic Way, 3
Denote a set of equations in the real numbers with variables $x_1, x_2, x_3 \in \mathbb{R}$ Flensburgian if there exists an $i \in \{1, 2, 3\}$ such that every solution of the set of equations where all the variables are pairwise different, satisfies $x_i>x_j$ for all $j \neq i$.
Find all positive integers $n \geq 2$, such that the following set of two equations $a^n+b=a$ and $c^{n+1}+b^2=ab$ in three real variables $a,b,c$ is Flensburgian.
2020 Greece JBMO TST, 4
Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, . . . , 8 \}$ with $A \cup B = X$ and $A \cap B = \emptyset$. Let $P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$. Find the minimum possible value of sum $P_A +P_B$.
PS. It is a variation of [url=https://artofproblemsolving.com/community/c6h2267998p17621980]JBMO Shortlist 2019 A3 [/url]
2018 Iran MO (1st Round), 23
Nadia bought a compass and after opening its package realized that the length of the needle leg is $10$ centimeters whereas the length of the pencil leg is $16$ centimeters! Assume that in order to draw a circle with this compass, the angle between the pencil leg and the paper must be at least $30$ degrees but the needle leg could be positioned at any angle with respect to the paper. Let $n$ be the difference between the radii of the largest and the smallest circles that Nadia can draw with this compass in centimeters. Which of the following options is closest to $n$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 20$
1957 Moscow Mathematical Olympiad, 372
Given $n$ integers $a_1 = 1, a_2,..., a_n$ such that $a_i \le a_{i+1} \le 2a_i$ ($i = 1, 2, 3,..., n - 1$) and whose sum is even. Find whether it is possible to divide them into two groups so that the sum of numbers in one group is equal to the sum of numbers in the other group.
1976 Putnam, 2
Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A^{n_1}B^{n_2}A^{n_3}\dots B^{n_k},$ where $n_1,\dots n_k$ are any integers, and $A^0=B^0=1$ as usual. Also suppose that $A^4=B^7=ABA^{-1}B=1, A^2\neq 1,$ and $B\neq 1.$
(a) How many elements of $G$ are of the form $C^2$ with $C$ in $G$?
(b) Write each such square as a word in $A$ and $B.$
2010 Contests, 4
The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.
2021 HMNT, 10
Real numbers $x, y, z$ satisfy $$x + xy + xyz = 1, y + yz + xyz = 2, z + xz + xyz = 4.$$
The largest possible value of $xyz$ is $\frac{a+b\sqrt{c}}{d}$, where $a$, $b$, $c$, $d$ are integers, $d$ is positive, $c$ is square-free, and gcd$(a,b, d) = 1$. Find $1000a + 100b + 10c + d$.
1958 Miklós Schweitzer, 4
[b]4.[/b] Let $P_1 P_2 P_3 P_4 P_5 P_6$ be a convex hexagon. Denote by $T$ its area and by $t$ the area of the triangle $Q_1 Q_2 Q_3$, where $Q_1,Q_2$ and $Q_3$ are the midpoints of $P_1P_4,P_2P_5,P_3P_6$ respectively. Prove that $t<\frac{1}{4}T$. [b](G. 3)[/b]
2022 Dutch IMO TST, 1
Consider an acute triangle $ABC$ with $|AB| > |CA| > |BC|$. The vertices $D, E$, and $F$ are the base points of the altitudes from $A, B$, and $C$, respectively. The line through F parallel to $DE$ intersects $BC$ in $M$. The angular bisector of $\angle MF E$ intersects $DE$ in $N$. Prove that $F$ is the circumcentre of $\vartriangle DMN$ if and only if $B$ is the circumcentre of $\vartriangle FMN$.
1993 Hungary-Israel Binational, 5
In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group.
Let $H \leq G, |H | = 3.$ What can be said about $|N_{G}(H ) : C_{G}(H )|$?
1997 IMC, 1
Let $f\in C^3(\mathbb{R})$ nonnegative function with $f(0)=f'(0)=0, f''(0)>0$. Define $g(x)$ as follows:
\[ \{ \begin{array}{ccc}g(x)= (\frac{\sqrt{f(x)}}{f'(x)})' &\text{for}& x\not=0 \\ g(x)=0 &\text{for}& x=0\end{array} \]
(a) Show that $g$ is bounded in some neighbourhood of $0$.
(b) Is the above true for $f\in C^2(\mathbb{R})$?
2002 AIME Problems, 8
Find the smallest integer $k$ for which the conditions
$(1)$ $a_1, a_2, a_3, \ldots$ is a nondecreasing sequence of positive integers
$(2)$ $a_n=a_{n-1}+a_{n-2}$ for all $n>2$
$(3)$ $a_9=k$
are satisfied by more than one sequence.
2019 Turkey Junior National Olympiad, 4
There are $27$ cardboard and $27$ plastic boxes. There are balls of certain colors inside the boxes. It is known that any two boxes of the same kind do not have a ball with the same color. Boxes of different kind have at least one ball of the same color. At each step we select two boxes that have a ball of same color and switch this common color into any other color we wish. Find the smallest number $n$ of moves required.
2009 Hungary-Israel Binational, 1
For a given prime $ p > 2$ and positive integer $ k$ let \[ S_k \equal{} 1^k \plus{} 2^k \plus{} \ldots \plus{} (p \minus{} 1)^k\] Find those values of $ k$ for which $ p \, |\, S_k$.