Found problems: 85335
1989 Flanders Math Olympiad, 3
Show that:\[\alpha = \pm \frac{\pi}{12} + k\cdot \frac{\pi}2
(k\in \mathbb{Z}) \Longleftrightarrow\ |{\tan \alpha}| + |{\cot
\alpha}| = 4\]
2021-IMOC, G2
Let the midline of $\triangle ABC$ parallel to $BC$ intersect the circumcircle $\Gamma$ of $\triangle ABC$ at $P$, $Q$, and the tangent of $\Gamma$ at $A$ intersects $BC$ at $T$. Show that $\measuredangle BTQ = \measuredangle PTA$.
2000 Iran MO (3rd Round), 2
Find all f:N $\longrightarrow$ N that:
[list][b]a)[/b] $f(m)=1 \Longleftrightarrow m=1 $
[b]b)[/b] $d=gcd(m,n) f(m\cdot n)= \frac{f(m)\cdot f(n)}{f(d)} $
[b]c)[/b] $ f^{2000}(m)=f(m) $[/list]
2012 ELMO Shortlist, 9
For a set $A$ of integers, define $f(A)=\{x^2+xy+y^2: x,y\in A\}$. Is there a constant $c$ such that for all positive integers $n$, there exists a set $A$ of size $n$ such that $|f(A)|\le cn$?
[i]David Yang.[/i]
2024 Bulgarian Autumn Math Competition, 10.1
Find all real solutions to the system of equations: $$\begin{cases} (x^2+xy+y^2)\sqrt{x^2+y^2} = 88 \\ (x^2-xy+y^2)\sqrt{x^2+y^2} = 40 \end{cases}$$
Indonesia Regional MO OSP SMA - geometry, 2019.5
Given triangle $ABC$, with $AC> BC$, and the it's circumcircle centered at $O$. Let $M$ be the point on the circumcircle of triangle $ABC$ so that $CM$ is the bisector of $\angle ACB$. Let $\Gamma$ be a circle with diameter $CM$. The bisector of $BOC$ and bisector of $AOC$ intersect $\Gamma$ at $P$ and $Q$, respectively. If $K$ is the midpoint of $CM$, prove that $P, Q, O, K$ lie at one point of the circle.
2003 Poland - Second Round, 4
Prove that for any prime number $p > 3$ exist integers $x, y, k$ that meet conditions: $0 < 2k < p$ and $kp + 3 = x^2 + y^2$.
2010 China Western Mathematical Olympiad, 8
Determine all possible values of integer $k$ for which there exist positive integers $a$ and $b$ such that $\dfrac{b+1}{a} + \dfrac{a+1}{b} = k$.
2018 Junior Balkan Team Selection Tests - Romania, 1
Determine the prime numbers $p$ for which the number $a = 7^p - p - 16$ is a perfect square.
Lucian Petrescu
1995 AMC 8, 6
Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. The perimeter of $III$ is
[asy]
draw((0,0)--(15,0)--(15,6)--(12,6)--(12,9)--(0,9)--cycle);
draw((9,0)--(9,9));
draw((9,6)--(12,6));
label("$III$",(4.5,4),N);
label("$II$",(12,2.5),N);
label("$I$",(10.5,6.75),N);
[/asy]
$\text{(A)}\ 9 \qquad \text{(B)}\ 18 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 72 \qquad \text{(D)}\ 81$
1982 Spain Mathematical Olympiad, 1
On the puzzle page of a newspaper this problem is proposed:
“Two children, Antonio and José, have $160$ comics. Antonio counts his by $7$ by $7$ and there are $4$ left over. José counts his $ 8$ by $8$ and he also has $4$ left over. How many comics does he have each?" In the next issue of the newspaper this solution is given: “Antonio has $60$ comics and José has $100$.”
Analyze this solution and indicate what a mathematician would do with this problem.
2013 Harvard-MIT Mathematics Tournament, 3
Let $S$ be the set of integers of the form $2^x+2^y+2^z$, where $x,y,z$ are pairwise distinct non-negative integers. Determine the $100$th smallest element of $S$.
2022 MOAA, 15
Let $I_B, I_C$ be the $B, C$-excenters of triangle $ABC$, respectively. Let $O$ be the circumcenter of $ABC$. If $BI_B$ is perpendicular to $AO$, $AI_C = 3$ and $AC = 4\sqrt2$, then $AB^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Note: In triangle $\vartriangle ABC$, the $A$-excenter is the intersection of the exterior angle bisectors of $\angle ABC$ and $\angle ACB$. The $B$-excenter and $C$-excenter are defined similarly.
1983 IMO Shortlist, 13
Let $E$ be the set of $1983^3$ points of the space $\mathbb R^3$ all three of whose coordinates are integers between $0$ and $1982$ (including $0$ and $1982$). A coloring of $E$ is a map from $E$ to the set {red, blue}. How many colorings of $E$ are there satisfying the following property: The number of red vertices among the $8$ vertices of any right-angled parallelepiped is a multiple of $4$ ?
2023 Moldova EGMO TST, 12
Let there be an integer $n\geq2$. In a chess tournament $n$ players play between each other one game. No game ended in a draw. Show that after the end of the tournament the players can be arranged in a list: $P_1, P_2, P_3,\ldots,P_n$ such that for every $i (1\leq i\leq n-1)$ the player $P_i$ won against player $P_{i+1}$.
2013 CIIM, Problem 5
Let $A,B$ be $n\times n$ matrices with complex entries. Show that there exists a matrix $T$ and an invertible matrix $S$ such that \[ B=S(A+T)S^{-1}\ -T \iff \operatorname{tr}(A) = \operatorname{tr}(B) \]
2022 CMIMC, 2.7
For polynomials $P(x) = a_nx^n + \cdots + a_0$, let $f(P) = a_n\cdots a_0$ be the product of the coefficients of $P$. The polynomials $P_1,P_2,P_3,Q$ satisfy $P_1(x) = (x-a)(x-b)$, $P_2(x) = (x-a)(x-c)$, $P_3(x) = (x-b)(x-c)$, $Q(x) = (x-a)(x-b)(x-c)$ for some complex numbers $a,b,c$. Given $f(Q) = 8$, $f(P_1) + f(P_2) + f(P_3) = 10$, and $abc>0$, find the value of $f(P_1)f(P_2)f(P_3)$.
[i]Proposed by Justin Hsieh[/i]
2023 UMD Math Competition Part I, #14
Let $m \neq -1$ be a real number. Consider the quadratic equation
$$
(m + 1)x^2 + 4mx + m - 3 =0.
$$
Which of the following must be true?
$\quad\rm(I)$ Both roots of this equation must be real.
$\quad\rm(II)$ If both roots are real, then one of the roots must be less than $-1.$
$\quad\rm(III)$ If both roots are real, then one of the roots must be larger than $1.$
$$
\mathrm a. ~ \text{Only} ~(\mathrm I)\rm \qquad \mathrm b. ~(I)~and~(II)\qquad \mathrm c. ~Only~(III) \qquad \mathrm d. ~Both~(I)~and~(III) \qquad \mathrm e. ~(I), (II),~and~(III)
$$
2006 Iran MO (3rd Round), 1
Prove that in triangle $ABC$, radical center of its excircles lies on line $GI$, which $G$ is Centroid of triangle $ABC$, and $I$ is the incenter.
2017 India IMO Training Camp, 3
Prove that for any positive integers $a$ and $b$ we have $$a+(-1)^b \sum^a_{m=0} (-1)^{\lfloor{\frac{bm}{a}\rfloor}} \equiv b+(-1)^a \sum^b_{n=0} (-1)^{\lfloor{\frac{an}{b}\rfloor}} \pmod{4}.$$
2015 NIMO Summer Contest, 14
We say that an integer $a$ is a quadratic, cubic, or quintic residue modulo $n$ if there exists an integer $x$ such that $x^2\equiv a \pmod n$, $x^3 \equiv a \pmod n$, or $x^5 \equiv a \pmod n$, respectively. Further, an integer $a$ is a primitive residue modulo $n$ if it is exactly one of these three types of residues modulo $n$.
How many integers $1 \le a \le 2015$ are primitive residues modulo $2015$?
[i] Proposed by Michael Ren [/i]
2002 Iran Team Selection Test, 5
A school has $n$ students and $k$ classes. Every two students in the same class are friends. For each two different classes, there are two people from these classes that are not friends. Prove that we can divide students into $n-k+1$ parts taht students in each part are not friends.
2012 AMC 12/AHSME, 3
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid $3$ acorns in each of the holes it dug. The squirrel hid $4$ acorns in each of the holes it dug. They each hid the same
number of acorns, although the squirrel needed $4$ fewer holes. How many acorns did the chipmunk hide?
${{ \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48}\qquad\textbf{(E)}\ 54} $
1966 IMO Longlists, 29
A given natural number $N$ is being decomposed in a sum of some consecutive integers.
[b]a.)[/b] Find all such decompositions for $N=500.$
[b]b.)[/b] How many such decompositions does the number $N=2^{\alpha }3^{\beta }5^{\gamma }$ (where $\alpha ,$ $\beta $ and $\gamma $ are natural numbers) have? Which of these decompositions contain natural summands only?
[b]c.)[/b] Determine the number of such decompositions (= decompositions in a sum of consecutive integers; these integers are not necessarily natural) for an arbitrary natural $N.$
[b]Note by Darij:[/b] The $0$ is not considered as a natural number.
2000 National Olympiad First Round, 14
What is the last two digits of the decimal representation of $9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}$?
$ \textbf{(A)}\ 81
\qquad\textbf{(B)}\ 61
\qquad\textbf{(C)}\ 41
\qquad\textbf{(D)}\ 21
\qquad\textbf{(E)}\ 01
$